diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzziasb" "b/data_all_eng_slimpj/shuffled/split2/finalzziasb" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzziasb" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n Molecular machine learning (ML) has been important in solving various molecular problems; these include prediction of aqueous solubility\\cite{intro1}, binding of drugs to specific target proteins\\cite{intro2} and blood brain-barrier permeability\\cite{intro3}. Specifically, molecular ML tackles these problems by linking structural information (these could be descriptors\\cite{intro4} or molecular fingerprints\\cite{intro5}) to some label (e.g., aqueous solubility measurements, binding or no binding to a protein, or permeability). As molecules are naturally represented as graphs $G = (V,E)$, where $V$ is a set of vertices (or nodes) corresponding to the atoms, and $E$ a set of edges corresponding to the bonds, it is strongly desired to encode molecules as graphs. By then subjecting these molecular graphs to graph neural networks\\cite{intro6, intro7} (GNNs), highly expressive and meaningful numerical representations of molecules for downstream classification or regression tasks can be obtained. As the application of GNNs for molecular ML has already proven useful for both predictive modeling\\cite{intro6, intro7, intro8, intro9, intro10} and interpretability\\cite{intro9, intro11}, it was desired to develop a Python package, MolGraph (available at \\url{https:\/\/github.com\/akensert\/molgraph}), for building GNNs specifically for those purposes (see Figure \\ref{fig:fig1}). Importantly, as MolGraph is built to be highly compatible with the TensorFlow\\cite{intro12} (TF) and Keras\\cite{intro13} application programming interface (API), a GNN model can be built and utilized with ease (see Appendix \\ref{appendix:a}).\n \n \\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.9\\textwidth]{Figure_1.png}\n \\caption{Schematic visualization of the different modules of MolGraph, and how they are utilized for molecular ML.}\n \\label{fig:fig1}\n \\end{figure}\n \n \\subsection{Related works and motivation}\n \n Many ML libraries exist in Python, these include Scikit-learn (for implementing classical ML models such as random forest, gradient boosting, multi-layer perceptrons, support vector machines, etc.), TF, Keras and Pytorch\\cite{intro14} (for implementing deep neural networks). Built on top of these libraries, people have developed Python packages that specialize in certain applications. For instance, for GNN applications, these include Spektral\\cite{intro15} (based on Keras\/TF), Deep Graph Library\\cite{intro16} (based on PyTorch, TF and Apache MXNet), DeepChem\\cite{intro17} (based on Keras\/TF and PyTorch), PyTorch Geometric\\cite{intro18}, and Graph Nets\\cite{intro19} (based on TF and Sonnet). In contrast to the aforementioned Python packages, MolGraph focuses exclusively on GNN applications for small molecular graphs, and therefore specializes in the generation of these graphs. Additionally, MolGraph is developed with the aim to be integrated seamlessly with the TF and Keras API. This includes compatibility with the Keras \\code{Sequential} model API and TF's \\code{saved\\_model} and \\code{Dataset} API. \n \n Both DeepChem (a prominent Python package for molecular ML) and Spektral (a prominent Python package for GNN applications) have a broader focus than MolGraph and thus a larger set of tools and utilities. While DeepChem implements its own dataset and model API, MolGraph relies directly on the dataset and model API of TF and Keras. Although the latter requires knowledge of TF and Keras, it allows for greater flexibility, where MolGraph's modules (mainly its \\code{layers} module) can be used with TF's and Keras' existing modules to tailor GNN implementations to the specific need of the user. Spektral, like MolGraph, relies (at least in part) on the model API of TF and Keras. However, while Spektral implements four different Data modes, depending on the problem at hand, MolGraph only implements one: the \\code{GraphTensor} (from hereon referred to as \\emph{graph tensor}). The graph tensor is indexable, conveniently updatable, batchable, and easily convertible between batches of (sub)graphs and a single disjoint graph --- both of which can be passed to e.g. a Keras model, comprising GNNs (see Appendix \\ref{appendix:b}). Ultimately, MolGraph is implemented to reduce cognitive load.\n\n\n\n\\begin{table}[!hb]\n \\caption{List of available atom and bond features of MolGraph}\n \\label{tab:table1}\n \\centering\n \\begin{threeparttable}[t]\n \\begin{tabular}{llll}\n \\toprule\n \\multicolumn{2}{c}{Atom features} & \\multicolumn{2}{c}{Bond features} \\\\\n \\midrule\n Description & Encoding & Description & Encoding \\\\\n \\midrule \n Chiral center & Binary & Bond type & One-hot \\\\\n Chirality (R, S or None) & One-hot & Conjugated & Binary \\\\\n Crippen Log P contribution & Float & Rotatable & Binary \\\\\n Crippen Molar refractivity contribution & Float & Stereo & One-hot \\\\\n Degree & One-hot & Part of ring & Binary \\\\\n Formal charge & One-hot & Part of ring of size \\emph{n} & One-hot \\\\\n Gasteiger charge & Float & & \\\\\n Hybridization & One-hot & & \\\\\n Aromatic & Binary & & \\\\\n Hydrogen donor & Binary & & \\\\\n Hydrogen acceptor & Binary & & \\\\\n Hetero & Binary & & \\\\\n Part of ring & Binary & & \\\\\n Part of ring of size \\emph{n} & One-hot & & \\\\\n Labute accessible surface area contribution & Float & & \\\\\n Number of hydrogens & One-hot & & \\\\\n Number of radical electrons & One-hot & & \\\\\n Number of valence electrons & One-hot & & \\\\\n Symbol (atom type) & One-hot & & \\\\\n Topological polar surface area contribution & Float & & \\\\\n \\bottomrule\n \\end{tabular}\n \\end{threeparttable}\n\\end{table}\n\n\\section{API}\n\n \\subsection{Graph tensor}\n To make MolGraph compatible with TF, it was important to define an appropriate Tensor class (namely, the graph tensor) that can hold graph data (specifically molecular graphs). The graph tensor has two \"configurations\": a \"ragged\" and a \"non-ragged\" configuration (see explanation later). The former makes the graph tensor batchable, while the latter makes the graph tensor efficient for the GNN. Specifically, the graph tensor contains nested tensors, either ragged or non-ragged, which together encode a molecular graph. These nested tensors include:\n\n \\begin{itemize}\n \\item {\\B[required]} a sparse adjacency matrix, represented as two vectors (tensors) indicating the indices of the destination nodes and source nodes. Row $j$ corresponds to row $j$ of edge features; and entry (index) $i$ corresponds to row $i$ of node features;\n \\item {\\B[required]} node features (or node states);\n \\item {\\B[optional]} edge features (or edge states); \n \\item {\\B[optional]} graph indicator (wherein each entry indicates which graph the node belongs to, relevant for the non-ragged configuration). Row $i$ corresponds to row $i$ of node features;\n \\item {\\B[optional]} a positional encoding. Row $i$ corresponds to row $i$ of node features.\n \\end{itemize}\n \n The graph tensor implements methods (specifically, \\code{merge} and \\code{separate}) to go between the two configurations; and an \\code{update} (and \\code{remove}) method that allows to update (and remove) the nested tensors within the graph tensor (see Appendix \\ref{appendix:b}). For the ragged configuration, the graph tensor separates each subgraph (molecule), causing the second dimension of the nested tensors to be ragged (as the molecules have different numbers of nodes\/edges); for the non-ragged configuration, all subgraphs are merged into a single disjoint graph, separable by the graph indicator.\n\n \\subsection{Molecular graph}\n The initial state of the graph tensor is computed via the \\code{chemistry} module of MolGraph, wherein a \\code{MolecularGraphEncoder} transforms string representations of molecules (e.g., SMILES or InChI) to a molecular graph encoded as a graph tensor (see Appendix \\ref{appendix:a}). Internally, the node (atom) and edge (bond) features are computed from atomic encoders based on a set of features (Table \\ref{tab:table1}). As atoms of a molecule may be assigned 3D coordinates, MolGraph also supplies a molecular encoder to compute, in addition to the required sparse adjacency matrix and node features, the distance geometry of a molecule. Specialized GNNs, such as DTNN\\cite{other20}, can then exploit such information to generate a potentially improved representation of the molecule (for downstream regression or classification tasks).\n\n \\subsection{GNN layers}\n Similar to how a feed-forward neural network (FFNN) is usually a stack of fully-connected layers, a GNN is usually a stack of GNN layers. Although the GNN layers of MolGraph take many shapes and forms, they are interchangeable, making it easy to replace one GNN layer with another. These GNN layers simply take as input a graph tensor (containing [at least] a sparse adjacency matrix and a set of node features), and outputs a new graph tensor with an updated set of node features. In essence, the GNN layers operate in two steps: (1) transformation of the node features and (2) aggregation of the node features (based on the sparse adjacency matrix). In the simplest case, the transformation is a single linear transformation via a learnable weight matrix on the node features, directly followed by an aggregation, by averaging, for each destination node, all source nodes' features. For some of the GNN layers however, these two steps are more complicated, wherein the updated node features might depend on the relevant edge features and\/or an attention mechanism.\n The GNN layers of MolGraph can be separated into three different categories: convolutional (e.g., GCN\\cite{other21, other22} and GIN\\cite{other22, other23}), attentional (e.g., GAT\\cite{other22, other24} and GMM\\cite{other22, other25}) and message passing (e.g., MPNN\\cite{intro8}) (see Figure \\ref{fig:fig2}). \n \n Convolutional layers perform an arbitrary transformation on the node states, and subsequently aggregate neighboring node states $j \\in N(i)$ to node $i$ via normalization coefficients $c_{ij}$, as follows:\n \n \\begin{equation}\n h_{i}^{(l+1)} = \\sigma\\left(\\sum_{j \\in N(i)}{c_{ij}\\psi(h_{j}^{(l)})}\\right),\n \\end{equation}\n \n were $\\sigma$ is a non-linear activation function, e.g., \\code{ReLU}. The attentional layer adds an attention mechanism, as a replacement for $c_{ij}$:\n \n \\begin{equation}\n h_{i}^{(l+1)} = \\sigma\\left(\\sum_{j \\in N(i)}{\\alpha_{ij}^{(l)}\\psi(h_{j}^{(l)})}\\right),\n \\end{equation}\n \n where $\\alpha$ denotes the attention coefficients, computed based on $h_{i}^{(l)}$ and $h_{j}^{(l)}$:\n \n \\begin{equation}\n \\alpha_{ij}^{(l)} = a(h_{i}^{(l)}, h_{j}^{(l)}),\n \\end{equation}\n \n and $a$ is a function which computes the attention coefficients $\\alpha_{ij}^{(l)}$. The message-passing layer uses a so-called message-function which computes the \"message\" to be propagated to the destination (target) nodes. Specifically, in comparison with the attentional (and convolutional) layer, the message-passing layer does not pass $h_{j}^{(l)}$ directly, but indirectly, through the message-function; which computes the message based on $h_{i}^{(l)}$ and $h_{j}^{(l)}$:\n \n \\begin{equation}\n h_{i}^{(l+1)} = \\sigma\\left(\\sum_{j \\in N(i)}{\\psi(h_{i}^{(l)}, h_{j}^{(l)})}\\right)\n \\end{equation}\n \n As can be seen, the transformation function depends on both $h_{i}^{(l)}$ and $h_{j}^{(l)}$, and not only $h_{j}^{(l)}$, which makes this approach most expressive.\n \n In addition to the above, a given GNN layer may implement edge states $e_{ij}$, in addition to the node states; where $i$ and $j$ indicate that it is the edge state of the edge between node $i$ and $j$. For example, the graph attention network (GAT) of MolGraph optionally incorporates edge states, wherein the attention coefficients $\\alpha_{ij}^{(l)}$ are computed based on both the node states and the associated edge state:\n \n \\begin{equation}\n \\alpha_{ij}^{(l)} = a(h_{i}^{(l)}, h_{j}^{(l)}, e_{ij}^{(l)})\n \\end{equation}\n \n \\clearpage\n \n \\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.95\\textwidth]{Figure_2.png}\n \\caption{Schematic illustration of the three main types of graph neural networks.}\n \\label{fig:fig2}\n \\end{figure}\n \n \\subsection{Readout}\n \n After $L$ steps of transformation and aggregation, $h_{i}^{(L)}$ is obtained for all nodes $i$ in the [molecular] graph. $h_{i}^{(L)}$ can be used to perform e.g., node predictions, edges predictions or graph predictions. For graph (molecule) predictions, node representations $h^{(L)} \\in \\mathbb{R}^{m x k}$ (where $m$ is the number of nodes and $k$ the number of features) need to be reduced to a one-dimensional vector representation $z \\in \\mathbb{R}^{k}$ of the graph (molecule). MolGraph implements several different layers that perform this reduction (known as \\emph{readout}). The most basic readout layer simply sums over the nodes (dimension $m$) as follows:\n \n \\begin{equation}\n z_{k} = \\sum_{i \\in m}{h_{ik}^{(L)}}\n \\end{equation}\n \n Subsequently, $z$ can be passed to an FFNN (denoted $f_{\\theta}$) to get a prediction (denoted $\\hat{y}$):\n \n \\begin{equation}\n \\hat{y} = f_{\\theta}(z)\n \\end{equation}\n\n \\subsection{GNN models}\n \n \\paragraph{Classification and regression}\n As the GNN layers can be implemented with either Keras' \\code{Functional} API, \\code{Sequential} API, or via Keras' subclassed models, GNN models for classification and regression can both easily and flexibly be created. The \\code{fit}, \\code{evaluate} and \\code{predict} methods of these models then accept either a graph tensor or a TF \\code{Dataset} constructed from a graph tensor (see Appendix \\ref{appendix:b}). Importantly, to allow for batching, the graph tensor needs to be in its ragged configuration. \n \n \\paragraph{Saliency and gradient activation maps}\n There is an increasing interest in better understanding and interpreting deep learning models (including GNNs). Among many techniques that can be used for this purpose, two prominent and well researched techniques are saliency and gradient activation mapping\\cite{intro11}. To allow better understanding and interpretability of the GNNs, both saliency and gradient activation maps are implemented in MolGraph (see Appendix \\ref{appendix:a}). These two techniques can be used to better understand what molecular substructures are important for the prediction; in other words, what parts of the molecules the GNN is \"looking\". \n\n \\subsection{TF records}\n To support larger datasets (exceeding 50 or 100 thousand molecules), which may not fit into memory (RAM), a TF Records module is implemented in MolGraph (see Appendix \\ref{appendix:c}). In brief, TF records allow the user to save data to disk, which can then efficiently be read (from disk) for modeling. Hence, the user may write data, namely molecular graphs (graph tensors) and associated labels, as TF records, which can then be read in a computationally and memory efficient way later on. Depending on the size of the molecular graph, as well as the computational resources, it may take anything from approximately 30 to 300 seconds to write 100 thousand small molecular graphs to TF record files; reading the TF records is significantly faster. \n\n\\section{Experiments and results}\n\n \\subsection{Predictive performance}\n To verify and evaluate the different GNNs, they were implemented with the Keras \\code{Sequential} model API, and subsequently fitted to, and evaluated on, a selection of benchmarks (datasets). The benchmarks included 15 datasets from MoleculeNet\\cite{other26}, one dataset from Domingo-Almenara et al.\\cite{other27} (in this study, referred to as \"SMRT\"; an abbreviation for \\emph{small molecular retention times}), and two datasets from Bonini et al.\\cite{other28} (in this study, referred to as \"RPLC\" and \"HILIC\"; reflecting which chromatographic separation mode was used, namely, \\emph{reversed-phase liquid chromatography} and \\emph{hydrophilic interaction liquid chromatography}). To get a better understanding of how well these GNNs performed, prediction errors were compared between the models of this study and the original publications of the benchmarks (referenced above). Importantly, the exact train, validation and test splits could not be extracted from the original publications (except for RPLC and HILIC), and thus exact conclusions cannot be drawn about which models performed better or worse. Instead, the comparisons give an indication of whether the models of this study (MolGraph) performed as expected. Furthermore, no hyper-parameter search was performed for any of the models of MolGraph, due to limited computational resources. Hence, the default hyperparameters were used for the GNNs. For optimization, Adam\\cite{other29} was used, with a starting learning rate of 0.0001 and ending learning rate of 0.000001, which decayed, on every plateau (defined as no improvement for 10 epochs), by a factor of 0.1. The training stopped when no improvement was made for 20 epochs, wherein the weights of the best epoch were restored for evaluation (on the test sets). The loss functions used were, for classification, binary cross entropy (BCE); and for regression, depending on the evaluation metric, mean absolute error (MAE), root mean squared error (RMSE) or Huber.\n\n As can be seen in Table \\ref{tab:table2} and Table \\ref{tab:table3}, the GNNs implemented in MolGraph performed as expected as they showed comparable performance to the models of MoleculeNet (\\emph{GC}, \\emph{Weave}, \\emph{MPNN} and \\emph{DTNN}). Furthermore, comparing the GNNs of MolGraph, the attentional and message-passing type models seem, overall, to perform marginally better than the convolutional type models. This suggests that increasing the complexity of the models could improve the performance --- though at the cost of significantly increasing the computational cost. Although the models (of MolGraph) were run only once against each dataset (one replica), consequently disregarding the variation in performance over replicate runs, it seems that the relative performance of each model heavily depended on the task it tried to solve (which dataset was used). For instance, the GT (graph transformer) models showed impressive performance on the quantum-chemistry datasets (\\emph{qm7}, \\emph{qm8} and \\emph{qm9}), but less so for the physiology datasets (\\emph{muv}, \\emph{hiv}, \\emph{pcba} and \\emph{bace}). An explanation for this could be that the GT models, which use a relatively complex attention mechanism, work well when the molecular graphs are smaller in size (in the case of the quantum-chemistry datasets).\n\n Similar to the observations on MoleculeNet, the performance of the GNNs of MolGraph performed as expected on the chromatographic retention time (RT) datasets (Table \\ref{tab:table2}). In contrast to the MoleculeNet models, the two models from literature (namely, for SMRT: \\emph{DLM}; and for RPLC and HILIC: \\emph{Keras}), were not GNNs, but FFNNs. Notably, the FFNNs were based on molecular descriptors --- vectors of precomputed features of molecules. It was of interest to compare these descriptor-based models (which are commonly used in molecular ML) with the GNNs. Although a direct comparison cannot be made to the DLM (as the exact splitting of the dataset could not be derived), the results of this study indicate superior performance of the GNNs. However, a study by Jiang et al.\\cite{intro10} showed somewhat different results, where descriptor-based models performed comparable to the GNNs on the MoleculeNet datasets.\n \n \\begin{table}[t]\n \\caption{Predictive performance on the three quantum-chemistry datasets, three physical-chemistry datasets and five chromatographic retention time datasets. Dashed line visually separate the models (and results) of this study (MolGraph) and models (and results) of other studies (see footnote of table). \"(E)\" indicates that edge features were used. Each value associated with the quantum-chemistry datasets indicates the mean absolute error (MAE); each value associated with the physical-chemistry datasets indicates the root mean squared error (RMSE); and each value associated with the chromatographic RT datasets indicate the mean relative error (MRE). Convolutional type models: GCN, RGCN(E), GIN, and GraphSage; attentional type models: GAT, GAT(E), GatedGCN, GatedGCN(E), GMM, GT, and GT(E); and message-passing type models: MPNN(E).}\n \\label{tab:table2}\n \\centering\n \\begin{threeparttable}[t]\n \\begin{tabular}{lccccccccccc}\n \\toprule\n & \\multicolumn{3}{c}{quantum-chemistry} & \\multicolumn{3}{c}{physical-chemistry} & \\multicolumn{5}{c}{chromatography} \\\\\n \\midrule\n & qm7 & qm8 & qm9 & esol & lipoph. & freesolv & SMRT & \\multicolumn{2}{c}{RPLC\\tnote{4}} & \\multicolumn{2}{c}{HILIC\\tnote{4}}\\\\\n & & & & & & & & test & ext. & test & ext. \\\\\n \\midrule\n GCN & 18.9011 & 0.0086 & 2.4795 & 0.5682 & 0.5412 & \\B 0.8453 \n & 0.038 & 0.09 & \\B 0.13 & 0.16 & 0.35 \\\\\n RGCN(E) & 13.6039 & 0.0086 & 2.0556 & 0.5992 & 0.5639 & 1.3143 \n & 0.037 & 0.11 & 0.19 & 0.18 & 0.33 \\\\\n GIN & 19.2505 & \\B 0.0084 & 2.0315 & 0.5865 & 0.5381 & 0.9066 \n & 0.037 & 0.10 & 0.17 & 0.17 & 0.30 \\\\\n GraphSage & 20.4176 & 0.0092 & 2.4739 & 0.5919 & 0.5449 & 1.6146 \n & 0.039 & 0.10 & 0.15 & 0.17 & 0.32 \\\\\n GAT & 22.0590 & 0.0088 & 2.6678 & 0.6471 & 0.5380 & 1.0530 \n & 0.038 & 0.09 & 0.15 & 0.18 & \\B 0.28 \\\\\n GAT(E) & 17.9556 & 0.0087 & 2.1902 & 0.5961 & 0.5402 & 1.1959 \n & 0.038 & 0.11 & 0.16 & 0.16 & 0.29 \\\\\n GatedGCN & 12.2210 & 0.0088 & 2.3740 & 0.5914 & \\B 0.5253 & 1.1264 \n & 0.037 & 0.10 & 0.15 & 0.16 & 0.31 \\\\ \n GatedGCN(E) & 10.2520 & 0.008 & 1.9858 & 0.5755 & 0.5422 & 1.4468 \n & \\B 0.037 & 0.10 & 0.14 & 0.16 & 0.32 \\\\\n GMM & 18.5327 & 0.0091 & 2.4415 & \\B 0.5453 & 0.5417 & 0.8613 \n & 0.038 & \\B 0.09 & 0.15 & 0.16 & 0.28 \\\\\n GT & 11.1366 & 0.0101 & 2.2205 & 0.6713 & 0.6195 & 1.0952 \n & & & & & \\\\\n GT(E) & \\B 7.5603 & 0.0095 & 1.7375 & 0.6992 & 0.5703 & 1.1807 \n & & & & & \\\\\n MPNN(E) & 15.0094 & 0.0089 & 2.5013 & 0.5505 & 0.5365 & 0.9538 \n & 0.038 & 0.09 & 0.16 & \\B 0.15 & 0.32 \\\\\n DTNN & 10.8403 & 0.0114 & \\B 1.3970 & & & & & & & & \\\\\n \\hdashline\n GC\\tnote{1} & 77.90 & 0.0148 & 4.7 & 0.97 & 0.655 & 1.40 & & & & & \\\\\n MPNN\\tnote{1} & & 0.0143 & 3.2 & 0.58 & 0.719 & 1.15 & & & & & \\\\\n DTNN\\tnote{1} & 8.8 & 0.0169 & 2.4 & & & & & & & & \\\\\n DLM\\tnote{2} & & & & & & & 0.068 & & & & \\\\\n Keras\\tnote{3} & & & & & & & & 0.13 & 0.18 & 0.2 & 0.46 \\\\ \n \\bottomrule\n \\end{tabular}\n \\begin{tablenotes}\n \\small\n \\item[1] Values are directly extracted from the MoleculeNet paper\\cite{other26}. The exact dataset splits producing these results may differ from the dataset splits of this study.\n \\item[2] Values are directly extracted from Domingo-Almenara et al.\\cite{other27} The exact dataset splits producing these results differ from the dataset splits of this study. In their study, a training and test set were used, with 75\\% and 25\\% of the data respectively. In this study, a training, validation and test set were used, with 70\\%, 5\\% and 25\\% of the data respectively.\n \\item[3] Values are extracted from Bonini et al.\\cite{other28}. \"Keras\" refers to the name of their model; a deep fully-connected neural network, implemented in Keras. \n \\item[4] These datasets include two test sets: a \"test set\" and an \"external set\". See Bonini et al.\\cite{other28} for more information.\n \\end{tablenotes}\n \\end{threeparttable}\n \\end{table}\n \n \\clearpage\n \n \\begin{table}[t]\n \\caption{Predictive performance on the four biophysics datasets and five physiology datasets. Dashed line visually separate the models (and results) of this study (MolGraph) and models (and results) of other studies (see footnote of table). \"(E)\" indicates that edge features were used. Each value associated with \\emph{muv} and \\emph{pcba} indicates the area under the precision-recall curve (PRC-AUC). Each remaining value indicates the area under the receiver operating characteristic curve (ROC-AUC). Convolutional type models: GCN, RGCN(E), GIN, and GraphSage; attentional type models: GAT, GAT(E), GatedGCN, GatedGCN(E), GMM, GT, and GT(E); and message-passing type models: MPNN(E).}\n \\label{tab:table3}\n \\centering\n \\begin{threeparttable}[t]\n \\begin{tabular}{lccccccccc}\n \\toprule\n & \\multicolumn{4}{c}{biophysics} & \\multicolumn{5}{c}{physiology} \\\\\n \\midrule\n & muv & hiv & pcba & bace & clintox & sider & toxcast & tox21 & bbbp \\\\\n \\midrule\n GCN & 0.0373 & 0.7964 & 0.1387 & 0.8111 & 0.7800 & 0.6230 & 0.7472 & 0.8233 & 0.6872 \\\\\n RGCN(E) & 0.0377 & 0.7945 & 0.1478 & 0.8213 & 0.7268 & \\B 0.6583 & 0.7501 & 0.8209 & 0.6948 \\\\\n GIN & 0.0033 & 0.7677 & \\B 0.1524 & 0.8231 & 0.8188 & 0.6202 & 0.7447 & \\B 0.8420 & 0.6763 \\\\ \n GraphSage & 0.0331 & 0.7763 & 0.1345 & 0.8129 & 0.8320 & 0.6376 & 0.7357 & 0.8264 & 0.6969 \\\\ \n GAT & 0.0308 & \\B 0.8009 & 0.1450 & 0.8294 & 0.7515 & 0.6091 & 0.7413 & 0.8336 & \\B 0.7196 \\\\ \n GAT(E) & 0.0279 & 0.7756 & 0.1500 & 0.7806 & 0.7483 & 0.6204 & 0.7342 & 0.8151 & 0.6938 \\\\\n GatedGCN & 0.0693 & 0.7669 & 0.1475 & 0.8063 & 0.7754 & 0.5995 & 0.7397 & 0.8348 & 0.6807 \\\\\n GatedGCN(E) & 0.0160 & 0.7689 & 0.1482 & 0.7974 & \\B 0.8746 & 0.6302 & 0.7384 & 0.8287 & 0.6967 \\\\\n GMM & 0.0380 & 0.7643 & 0.1466 & 0.7732 & 0.8128 & 0.6280 & 0.7473 & 0.8283 & 0.6903 \\\\\n GT & 0.0034 & 0.7493 & 0.1356 & 0.7325 & 0.8322 & 0.5704 & 0.7309 & 0.8057 & 0.6857 \\\\\n GT(E) & 0.0145 & 0.7543 & 0.1404 & 0.7958 & 0.8430 & 0.5922 & 0.7264 & 0.7982 & 0.6726 \\\\ \n MPNN(E) & 0.0400 & 0.7603 & 0.1427 & \\B 0.8320 & 0.8329 & 0.6400 & \\B 0.7524 & 0.8345 & 0.6942 \\\\ \n \\hdashline\n GC\\tnote{1} & 0.046 & 0.763 & 0.136 & 0.783 & 0.807 & 0.638 & 0.716 & 0.829 & 0.690 \\\\\n Weave\\tnote{1} & \\B 0.109 & 0.703 & & 0.806 & 0.832 & 0.581 & 0.742 & 0.820 & 0.671 \\\\\n \\bottomrule\n \\end{tabular}\n \\begin{tablenotes}\n \\small\n \\item[1] Values are directly extracted from the MoleculeNet paper\\cite{other26}. The exact dataset splits producing these results may differ from the dataset splits of this study.\n \\end{tablenotes}\n \\end{threeparttable}\n \\end{table}\n \n \\clearpage\n\n\n \\subsection{Molecular identification}\n In addition to benchmarking the predictive performance of the GNNs of MolGraph on the MoleculeNet datasets, it was also of interest to verify the capabilities of a graph convolutional network (GCN) to assist in molecular identification. For that purpose, a plant metabolite database (PLaSMA, available at \\url{http:\/\/plasma.riken.jp\/}) consisting of liquid chromatography-tandem mass spectrometry (LC-MS\/MS) data was used. In brief, the identity (structure) of an unknown molecule (analyte) in a given sample can be predicted based on LC-MS\/MS data; where the MS\/MS data is used to predict the structure of the analyte (using a software such as MSFinder\\cite{other30, other31}), and the LC RTs are used as an additional confirmation of the proposed structure. Thus, the GCN was trained to correlate the structure of known molecules in the PLaSMA dataset to their associated RTs, and subsequently used to obtain more confidence about proposed structures of analytes. Similar to the benchmarking, the PLaSMA dataset was divided into training, validation and test sets; in which the training set was used to train the model; the validation set was used to supervise the training and to determine thresholds for a RT filter; and the test set was used to evaluate how well the GCN model could filter out structures (candidates) suggested by the MSFinder software. \n \n Figure \\ref{fig:fig3} aims to visualize the process of eliminating candidates proposed by MSFinder. The upper plots of Figure \\ref{fig:fig3} illustrate how the validation set was used to define the RT filter (two \\emph{thresholds}, or \\emph{bounds}) that would decide whether a candidate would be discarded (or not) as a possible identity of the analyte. Specifically, if the distance between the predicted RT of the candidate and the RT of the analyte was within the bounds, it was kept, otherwise, discarded. The bounds were defined by the mean and standard deviation of the errors, specifically, $\\mu \\pm 2.58 \\sigma$ ($-1.313$ and $1.337$). The bottom subplot of Figure \\ref{fig:fig3} illustrates to what extent the GCN could filter out candidates with the RT filter. As can be seen, a significant portion (about 40\\%) of the candidates were filtered out (although in specific cases, none, or very few, got filtered out). In one case, a false negative occurred \u2014 i.e., the candidate matching the identity of the analyte was discarded. Overall, the results show evidence that the utilization of the GCN for RT predictions may facilitate the process of identifying analytes.\n\n \\begin{figure}[hb]\n \\centering\n \\includegraphics[width=0.95\\textwidth]{Figure_3.png}\n \\caption{Retention time (RT) filter. Thresholds ($-1.313$ and $1.337$) were computed from the validation set and used to filter out candidates suggested by MSFinder. The red bar indicates a false negative.}\n \\label{fig:fig3}\n \\end{figure}\n \n To inspect the filtering procedure in more detail, the MSFinder candidate scores as well as predicted candidate RTs (of the GCN) corresponding to analyte 10 (of Figure \\ref{fig:fig3}) are shown in Table \\ref{tab:table4}. The candidate in bold is the most likely identity based on the MSFinder score and RT filter. In this specific case, the predicted RT proved to be important for correctly identifying the analyte. Comparing MSFinder's three highest scoring candidates and fourth highest scoring candidate (which ended up being the most likely candidate after applying the RT filter), it can be observed that the three highest scoring candidates are significantly more polar --- hence low RTs in an RPLC setting. As the analyte had a significantly higher RT (because it is significantly less polar), the RT filter sufficed to discard them. Although this is just a single example, it aims to illustrate how combining information of predicted structures based on MS\/MS data and predicted RTs can increase the chances of correctly identifying the analyte. \n \n \\begin{table}[ht]\n \\caption{In bold: the most likely identity of the analyte, based on both the MSFinder score and the difference between predicted RT and analyte RT.}\n \\label{tab:table4}\n \\centering\n \\begin{threeparttable}[t]\n \\begin{tabular}{rcccc}\n \\toprule\n \\multicolumn{1}{l}{Target Identity:} & \\multicolumn{4}{l}{CS(=O)CCCCCCCN=C=S} \\\\\n \\multicolumn{1}{l}{Target RT:} & \\multicolumn{4}{l}{5.41} \\\\ \n \\midrule\n \\multicolumn{1}{c}{Candidates} & Score (MSFinder) & Predicted candidate RT & Difference & Filtered out \\\\\n \\midrule\n NC(CCOC(=O)CCC(=O)O)C(=O)O & 7.32 & 1.39 & 4.02 & Yes \\\\ \n CC(=O)NC1C(=O)OC(CO)C(O)C1O & 7.12 & 1.71 & 3.70 & Yes \\\\ \n CC(O)C(=O)NC(CCC(=O)O)C(=O)O & 7.06 & 2.03 & 3.38 & Yes \\\\\n \\B CS(=O)CCCCCCCN=C=S & \\B 7.06 & \\B 6.07 & \\B -0.66 & \\B No \\\\\n CC(OC(=O)CCC(N)C(=O)O)C(=O)O & 6.87 & 1.63 & 3.78 & Yes \\\\\n Cc1ccc(C=NNc2cn[nH]c(=O)n2)o1 & 6.73 & 2.60 & 2.81 & Yes \\\\\n Cc1nonc1NC(=O)Nc1cccnc1 & 5.95 & 1.94 & 3.47 & Yes \\\\\n CSC=CC(=O)NC=Cc1ccccc1 & 5.78 & 4.63 & 0.78 & No \\\\\n CSC=CNC(=O)C=Cc1ccccc1 & 5.72 & 4.95 & 0.46 & No \\\\\n CCOC(=O)NP(=O)(N1CC1)N1CC1 & 5.34 & 2.12 & 3.29 & Yes \\\\\n \\bottomrule\n \\end{tabular}\n \\end{threeparttable}\n \\end{table}\n\n \\subsection{Gradient activation maps}\n One of the potential strengths of the GNN models is their ability to compute saliency and\/or gradient activation maps to better interpret e.g. retention data; i.e., to better understand what substructures contribute to the (RT) prediction. Figure \\ref{fig:fig4} visualizes four different molecules from the RPLC and HILIC datasets, with gradient activation maps superimposed on 2D structures of these molecules. More contour lines and greener color indicate more importance; i.e., structures that are more important for RT prediction, relative to substructures with fewer (or no) contour lines. Interestingly, though expectedly, non-polar substructures are clearly most important in the RPLC setting, while polar substructures are clearly most important in the HILIC setting. These results are expected as the non-polar stationary phase of RPLC is expected to interact with the non-polar substructures of the molecules (causing a difference in retention); and similarly, the polar stationary phase of HILIC is expected to interact with the polar substructures of the molecules. These observations agree with the observations of Kensert et al.\\cite{intro9}, where saliency maps were used (instead of gradient activation maps). However, the advantage of using gradient activation maps over saliency maps, is that the former makes use of resulting node states of every GNN layer --- consequently generating more robust maps.\n \n \\begin{figure}[!t]\n \\centering\n \\includegraphics[width=0.75\\textwidth]{Figure_4.png}\n \\caption{Comparison of gradient activation maps between the RPLC dataset and the HILIC dataset. Gradient activation maps (in green) indicate what substructures were important for the RT prediction.}\n \\label{fig:fig4}\n \\end{figure}\n\n\n\\section{Conclusions}\n One of the main motivations behind MolGraph was to develop a GNN package highly compatible with TF and Keras. Consequently, MolGraph's GNN layers can be built and utilized with ease \u2014 using the TF \\code{Dataset} API for efficient data pipelines, and the Keras \\code{Sequential} (or \\code{Functional}) API for concise implementation of models. MolGraph has an intended limited scope, which on the one hand accommodated an improved chemistry module, but on the other hand limited the capabilities of MolGraph. In prospect, it would be interesting to implement modules for more advanced GNN models, including generative and self-supervised models\\cite{other32, other33}; and furthermore, to focus on larger graphs, such as proteins and oligonucleotides, which are of great interest in pharmaceutical science\\cite{other34, other35}.\n\n\n\\subsection*{Availability}\nSource code and license can be found at \\url{https:\/\/github.com\/akensert\/molgraph}. Documentation and tutorials can be found at \\url{https:\/\/molgraph.readthedocs.io\/en\/latest\/}. \n\\subsection*{Authors' contributions}\nAK wrote the source code and documentation for the MolGraph package. AK performed the experiments and wrote the original draft of the manuscript; GD and DC reviewed and edited the manuscript. GD and DC supervised, and acquired funding for, the project. AK, GD and DC conceptualized the project. \n\\subsection*{Acknowledgement}\nAlexander Kensert is funded by a joint-initiative of the Research Foundation Flanders (FWO) and the Walloon Fund for Scientific Research (FNRS) (EOS \u2013 research project \"Chimic\" (EOS ID: 30897864)).\n\\subsection*{Conflict of interests}\nThe authors declare that there is no conflict of interest.\n\n\n\\bibliographystyle{unsrt} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection{Introduction}\nCharacterizing the properties of the Higgs boson is arguably the most concrete particle physics goal \nof our time. This is further motivated by the dearth of any signs of physics beyond the Standard Model \n(BSM) in LHC data so far. One well-motivated course of action in this situation is to probe heavy \nnew physics outside the reach of direct searches via precise indirect measurements. A historic example \nof constraining high energy physics even beyond the energy coverage of a collider is the LEP experiment, \nwhich was able to probe scales up to the few TeV via indirect precision measurements although it ran at \na much smaller collision energy. \n\nAs the Higgs boson could not be produced before the LHC experiment under controlled conditions, one \nmight naively think that any measurement of interactions involving the Higgs boson is complementary \nto past measurements. However, an Effective Field Theory (EFT) perspective allows us to correlate \nmeasurements at different energy scales only on the basis of SM symmetry and matter content. In fact, \nthere are fewer $SU(2)_L\\times U(1)_Y$ invariant operators at the lowest order in the LHC-relevant \nEFT expansion at dimension 6~\\cite{Grzadkowski:2010es} than the number of (pseudo-)observables they \ncontribute to. As a result, correlations between LHC and LEP measurements can be exploited. For \ninstance the LEP measurements of $Z$-boson pole observables and anomalous Triple Gauge Couplings (TGCs) inform the measurement of Higgs observables at the LHC as they can be related \nto a common gauge-invariant set of SMEFT operators. The analysis in Ref.~\\cite{Gupta:2014rxa} reveals, \nin fact, that apart from eight Higgs observables, the so called ``Higgs primaries'', all other Higgs \ninteractions present in the dimension-6 Lagrangian can be constrained already by $Z$-pole and diboson \nmeasurements at LEP. \n\nFor the set of already constrained Higgs coupling deformations, the LHC has to compete with LEP's \nprecision to add new information in order to gain a more complete picture. This might seem challenging \ngiven that the LHC is intrinsically less accurate compared to LEP. The key advantage of the LHC (and other \nfuture colliders), however, is a much larger energy reach compared to LEP, thus allowing us to constrain \nnew physics from a plethora of available kinematical information~(see \nalso~\\cite{Banerjee:2013apa,Amar:2014fpa,Buschmann:2014sia,Craig:2014una, Banerjee:2015bla,Cohen:2016bsd,Ge:2016zro,Denizli:2017pyu,Barklow:2017suo,Barklow:2017awn,Khanpour:2017cfq}). As we will see, the high energy sensitivity of the LHC will allow us to strongly constrain EFT-induced anomalous couplings involved in processes that grow with energy:\n\\begin{equation}\n\\label{eq:growth}\n \\frac{\\delta\\sigma (\\hat{s})}{\\sigma_{SM} (\\hat{s})}\\sim \\delta g_i \\frac{\\hat{s}}{m_Z^2}\\,.\n\\end{equation}\nFrom Eq.~\\eqref{eq:growth}, we see that the anomalous coupling $g_i$ can be measured\/constrained at \nthe per-mille to percent level even if the underlying sensitivity to the fractional cross-section \ndeviation, $\\delta\\sigma\/\\sigma_{SM}$ is only ${\\cal O}(30\\%)$ at high energies. \n\nThe specific process we are interested in here is Higgs-strahlung, $pp \\to Z(\\ell^+ \\ell^-)h(b\\bar \nb)$. Studying the $h \\to b\\bar b$ mode instead of the $h \\to \\gamma \\gamma$ leads to a big enhancement \nin the rate but the Higgs-strahlung process still remains challenging with an ${\\cal O}(50)$ background-to-signal\nratio. Relating such a systematics limited result to the extraction of Higgs couplings can be at odds \nwith the implicit assumption of perturbativity of the EFT expansion. We technically rely on the latter\nto perform proof-of-principle analyses and eventually full searches at ATLAS and CMS. As $\\delta\\sigma\/\\sigma_{SM} \\gtrsim 1$ signals the \nbreakdown of EFT validity for weakly coupled UV completions~\\cite{Contino:2016jqw}, a sensitivity \nto smaller values of $\\delta\\sigma\/\\sigma_{SM}$ is essential. To gain such \nprecision, we need high luminosities (at least 300 fb$^{-1}$) and advanced boosted Higgs tagging \ntechniques which can reduce the ratio of the number of $Zb\\bar b$ to the SM $Zh(b\\bar b)$ events to \nan ${\\cal O}(1)$ number as shown earlier in Refs.~\\cite{Butterworth:2008iy, Soper:2010xk, Soper:2011cr}. \nThis work is, therefore, an example of a study at the ``high energy-luminosity'' frontier in the \nspirit of Ref.~\\cite{deBlas:2013qqa,Farina:2016rws,Franceschini:2017xkh}. \n\nWhile adding the channel $pp \\to Z(\\nu \\bar{\\nu})h(b\\bar \nb)$ can further improve the limits on the effective operators we study \\cite{Butterworth:2008iy}, this channel is subjected to backgrounds and employs observables with larger systematic uncertainties. We therefore leave an inclusion of this channel for future work.\n\n\nAs we will see, the leading high energy contribution to the $pp \\to Zh$ process comes from the four \ncontact interactions $hZ_\\mu \\bar{u}_{L,R} \\gamma^\\mu u_{L,R}$ and $hZ_\\mu \\bar{d}_{L,R} \\gamma^\\mu \nd_{L,R}$ that are present in the dimension-6 extended Lagrangian. Thus, although many more operators \ncontribute to the $pp \\to Zh$ process, the high energy limit isolates the four linear combinations of \noperators that generate the above contact terms. An interesting observation, first made in \nRef.~\\cite{Franceschini:2017xkh}, is that the same four EFT directions (that the authors call ``high \nenergy primaries'') also control $Wh$ and $WW\/WZ$ production. The reason is that at high energies these \nfour final processes correspond to the production of different components of the Higgs doublet due \nto the Goldstone Boson Equivalence Theorem~\\cite{Chanowitz:1985hj}. They are therefore related by \n$SU(2)_L$ symmetry for $\\hat{s}\\gg m^2_Z$. Hence, although these four diboson processes may be very \ndifferent from a collider physics point of view, they are intimately related by gauge symmetry, which \nstands at the heart of an EFT interpretation. This enables an elegant understanding of the connection of pseudo-observables in $WW$ production \n(such as TGCs) with those in $Zh$ production. It will also allow us to present our results in a \ncombined way with the projections for $WZ$ production in Ref.~\\cite{Franceschini:2017xkh}.\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{adjustbox}{width=\\columnwidth}\n\\small\n\\begin{tabular}{c|c\nSILH Basis&Warsaw Basis\\\\\\hline\n\\rule[-1.2em]{0pt}{3em}$\\displaystyle{\\cal O}_W=\\frac{ig}{2}\\left( H^\\dagger \\sigma^a \\lra {D^\\mu} H \\right )D^\\nu W_{\\mu \\nu}^a$&$\\displaystyle{\\cal O}^{(3)}_L= (\\bar{Q}_L \\sigma^a\\gamma^\\mu Q_L)(i H^\\dagger \\sigma^a\\lra D_\\mu H)$\\\\\n\\rule[-1.2em]{0pt}{3em}$\\displaystyle{\\cal O}_B=\\frac{ig'}{2}\\left( H^\\dagger \\lra {D^\\mu} H \\right )\\partial^\\nu B_{\\mu \\nu}$&$\\displaystyle{\\cal O}_L= (\\bar{Q}_L \\gamma^\\mu Q_L)(i H^\\dagger \\lra D_\\mu H)$\\\\\n\\rule[-.6em]{0pt}{1.5em}$\\displaystyle{\\cal O}_{HW}=i g(D^\\mu H)^\\dagger\\sigma^a(D^\\nu H)W^a_{\\mu\\nu}$&\n$\\displaystyle{\\cal O}^u_R= (\\bar{u}_R \\gamma^\\mu u_R)(i H^\\dagger \\lra D_\\mu H)$\\\\\n\\rule[-.6em]{0pt}{1.5em}$\\displaystyle{\\cal O}_{HB}=i g'(D^\\mu H)^\\dagger(D^\\nu H)B_{\\mu\\nu}$&$\\displaystyle{\\cal O}^d_R= (\\bar{d}_R \\gamma^\\mu d_R)(i H^\\dagger \\lra D_\\mu H)$\\\\\n\\rule[-1.2em]{0pt}{3em}$\\displaystyle{\\cal O}_{2W}=-\\frac{1}{2} ( D^\\mu W_{\\mu \\nu}^a)^2$&\\\\\n\\rule{0pt}{1.5em}$\\displaystyle{\\cal O}_{2B}=-\\frac{1}{2}( \\partial^\\mu B_{\\mu \\nu})^2$&\n \\end{tabular}\n \\end{adjustbox} \n \\caption{Dimension-six operators that give dominant contribution to $pp \\to Vh$ at high energies \n in the Warsaw~\\cite{Grzadkowski:2010es} and SILH~\\cite{Giudice:2007fh} bases. \\label{list}}\n\\end{center}\n\\end{table}\n\n\\begin{table*}[!t]\n\\begin{center}\n\\small\n\\begin{tabular}{|c|c|\n\\hline\n&EFT directions probed by high energy $ff \\to Vh$ production \\\\\n\\hline\n\\hline\n Warsaw Basis~\\cite{Grzadkowski:2010es} & $- \\frac{2 g}{c_{\\theta_W}}\\frac{v^2}{\\Lambda^2}(|T_3^f|c^{1}_L-T_3^f c^{3}_L+(1\/2-|T_3^f|)c_f)$\\\\\nBSM Primaries~\\cite{Gupta:2014rxa} &$ \\frac{2 g}{c_{\\theta_W}}Y_f t_{\\theta_W}^2 \\delta \\kappa_\\gamma+2 \\delta g^Z_{f}- \\frac{2 g}{c_{\\theta_W}}(T^f_3 c_{\\theta_W}^2 + Y_f s_{\\theta_W}^2)\\delta g_1^Z $\\\\\n SILH Lagrangian~\\cite{Giudice:2007fh} &$ \\frac{g}{c_{\\theta_W}}\\frac{m_W^2}{\\Lambda^2}(2 T_3^f\\hat{c}_W-2t_{\\theta_W}^2 Y_f \\hat{c}_B)$\\\\\n Universal observables &$ \\frac{2 g}{c_{\\theta_W}}Y_f t_{\\theta_W}^2 (\\delta \\kappa_\\gamma-\\hat{S}+Y)- \\frac{2 g}{c_{\\theta_W}}(T^f_3 c_{\\theta_W}^2 + Y_f s_{\\theta_W}^2)\\delta g_1^Z- \\frac{2 g}{c_{\\theta_W}}T^f_3 W$\\\\\n High Energy Primaries~\\cite{Franceschini:2017xkh} &$ -\\frac{2 m_W^2}{g c_{\\theta_W} }(|T_3^f| a_q^{(1)}-T_3^f a_q^{(3)}+(1\/2-|T_3^f|)a_f)$\\\\\n\\hline\n \\end{tabular}\n \\caption{ The linear combinations of Wilson coefficients contributing to the contact interaction couplings $g^h_{Zf}$ that control the $ff \\to Vh$ process at high energies. The four directions relevant for hadron colliders (corresponding to $f=u_L, d_L, u_R, d_R$) can be read off from this table by substituting the value of the $SU(2)_L$ and $U(1)_Y$ quantum numbers $T_3^f$ and $Y_f$ for the corresponding $f$. Here $\\hat{c}_W=c_W+c_{HW}-c_{2W}$ and $\\hat{c}_B= c_B+c_{HB}-c_{2B}$. For the nomenclature of the operators, their corresponding Wilson coefficients and observables see for eg. Ref.~\\cite{Franceschini:2017xkh}.}\n \\label{dirn}\n\\end{center}\n\\end{table*}\n\n\\subsection{The high energy $Vh$-amplitude in the SMEFT}\nLet us first study $Vh$ production at high energy in the SMEFT where $V=W,Z$. Although we focus on \n$pp \\to Zh$ production in the subsequent sections, here we keep the discussion more general \nconsidering also the $Wh$ final state. We will see that $Vh$ production at \nhadron colliders at high energies, isolates four independent directions in the full 59 dimensional \nspace of dimension 6 operators. To derive this fact, consider first the vertices in the dimension 6 Lagrangian that contribute \nto the $ff \\to Zh$ process in unitary gauge, \n\\begin{multline}\n\\Delta {\\cal L}_6\\supset \\sum_f {\\delta g^Z_{f}} Z_\\mu \\bar{f} \\gamma^\\mu f +\\delta g^W_{ud} (W^+_\\mu \\bar{u}_L \\gamma^\\mu d_L+h.c.)\\\\\n+ {g^h_{VV}}\\, h \\left[W^{+\\, \\mu} W^-_\\mu+\\frac{1}{2c_{\\theta_W}^2}Z^\\mu Z_\\mu\\right]\n+ \\delta g^h_{ZZ}\\, h \\frac{Z^\\mu Z_\\mu}{2c_{\\theta_W}^2}\\\\ \n+ \\sum_f g^h_{Zf}\\,\\frac{h}{v}Z_\\mu \\bar{f} \\gamma^\\mu f+g^h_{Wud}\\,\\frac{h}{v}(W^+_\\mu \\bar{u}_L \\gamma^\\mu d_L+h.c.)\\\\\n+{\\kappa_{Z \\gamma }}\\frac{h}{v} A^{\\mu\\nu} Z_{\\mu\\nu}+\\kappa_{WW}\\,\\frac{h}{v}\n W^{+\\, \\mu\\nu}W^-_{\\mu\\nu}\n+\\kappa_{ZZ}\\,\\frac{h}{2v} Z^{\\mu\\nu}Z_{\\mu\\nu}\\,.\n\\label{lagr}\n\\end{multline}\nWe are using the formalism presented in Ref.~\\cite{Gupta:2014rxa, Pomarol:2014dya} where $\\alpha_{em}$, $m_Z$ and $m_W$ have been used as input parameters and any corrections \nto the SM vector propagators, \\textit{i.e.} the terms $V_\\mu V^\\mu, V_{\\mu\\nu} V^{\\mu\\nu}$ and $V_{\\mu\\nu} F^{\\mu\\nu}$, have \nbeen traded in favor of the vertex corrections. Note that the above parameterisation is equivalent to the one in Ref.~\\cite{Gonzalez-Alonso:2014eva, Greljo:2015sla} (see Ref.~\\cite{Greljo:2017spw}). Keeping only the leading terms in $\\hat{s}\/m_Z^2$ in the BSM correction, we obtain for the amplitude $ \\mathcal{M}(ff\\to V_{T,L} h)$,\n\\begin{equation}\n \\label{amps}\n\\begin{split}\n Z_T h&: \\, {g^Z_{{f}}} \\frac{ \\epsilon^*\\cdot J_{f}}{v} \\frac{2 m_Z^2}{\\hat{s}}\\ \\, \\, \\Bigg[1+ \\left(\\frac{g^h_{Zff}}{g^Z_f}- \\kappa_{ZZ}\\right) \\frac{\\hat{s}}{2 m_Z^2} \\Bigg] \\, ,\\\\\nZ_L h&: \\, {g^Z_{{f}}} \\frac{ q\\cdot J_{f}}{v} \\frac{2 m_Z}{\\hat{s}}\\ \\, \\, \\Bigg[1+ \\frac{{g}^h_{Zff}}{g^Z_{f}} \\frac{\\hat{s}}{2 m_Z^2} \\Bigg] \\,,\\\\\n W_T h &: \\, {g^W_{{f}}} \\frac{ \\epsilon^*\\cdot J_{f}}{v} \\frac{2 m_W^2}{\\hat{s}}\\ \\, \\, \\Bigg[1+\\left(\\frac{g^h_{Wff'}}{g^W_f}- \\kappa_{WW}\\right) \\frac{\\hat{s}}{2 m_W^2} \\Bigg] \\, ,\\\\\n W_L h &: \\, g^W_{f} \\frac{ q\\cdot J_{f}}{v} \\frac{2 m_W}{\\hat{s}}\\ \\, \\, \\Bigg[1+\\frac{g^h_{Wff'}}{g^W_f} \\frac{\\hat{s}}{2 m_W^2} \\Bigg] \\,, \n\\end{split}\n\\end{equation}\n where $g^Z_{f}=g(T_3^f-Q_f s_{\\theta_W}^2)\/c_{\\theta_W}$, and $g^W_f={g}\/{\\sqrt{2}}$. $J^\\mu_f$ is the fermion current $\\bar{f} \\gamma^\\mu f$, the subscript $L$ ($T$) denotes the longitudinal (transverse) polarization of the gauge boson, $q$ denotes its four momentum and $\\epsilon$ the polarization vector. \n \nWe see that only the ${g}^h_{Vf}$ and $\\kappa_{VV}$ couplings lead to an amplitude \ngrowing with energy. In the case of the $\\kappa_{VV}$ couplings, the energy growth arises because of \nthe extra powers of momenta in the $hVV$ vertex, whereas for the contact interaction, ${g}^h_{Vf}$, \nthe energy growth is due to the fact that there is no propagator in the diagram involving this vertex.\nIn fact for the latter interaction, the only difference in the amplitude with respect to the SM is \nthe absence of the propagator. Thus, angular distributions \nare expected to be identical for BSM and SM production. Therefore, the only way to probe this interaction is through the \ndirect energy-dependence of differential cross-sections.\n\nOn the other hand, the $\\kappa_{VV}$ interactions contribute only to the transverse $V$ amplitude as \na consequence of their vertex structure. Hence, they cannot interfere with the dominant longitudinal \npiece in the SM amplitude. As a result, after summing over all $V$-polarizations, the leading \npiece in the high energy cross-section deviation, is controlled only by the couplings ${g}^h_{Vf}$ \nwhereas the $\\kappa_{VV}$ contribution is suppressed by an additional ${\\cal O} (m_V^2\/\\hat{s})$ \nfactor. \n\nNote that the couplings, $\\delta g^Z_{f}$ and $\\delta g^h_{ZZ}$ also lead to deviations from the SM amplitude but these corrections do not grow with energy and are also suppressed by an additional ${\\cal O} (m_V^2\/\\hat{s})$ \nfactor with respect to the ${g}^h_{Vf}$ contribution. We have checked explicitly that including these couplings have no noticeable impact on our analysis.\n\nAt hadron colliders, the $pp \\to Vh$ process at high energies and at leading order are therefore \ncontrolled by the five contact interactions: $g^h_{Zf}$, with $f=u_L, u_R,d_L$ and $d_R$ and \n$g^h_{Wud}$. These five couplings correspond to different linear combinations of Wilson coefficients \nin any given basis. In Tab.~\\ref{list} we show all operators in the ``Warsaw''~\\cite{Grzadkowski:2010es} \nand strongly-interacting light Higgs (SILH)~\\cite{Giudice:2007fh} bases that generate these contact \nterms. As there are only four independent operators contributing to these five interactions in the \nWarsaw basis, there exists a basis independent constraint at the dimension-6 level,\n\\begin{equation}\ng^h_{Wud}=c_{\\theta_W}\\frac{g^h_{Zu_L}-g^h_{Zd_L}}{\\sqrt{2}}\n\\end{equation}\nleaving only the four independent $g^h_{Zf}$ couplings. \n\nIn Table~\\ref{dirn}, we show the linear combinations of Wilson coefficients contributing to the four \n$g^h_{Zf}$ couplings in different EFT bases. The first row gives these directions in the Warsaw basis. \nThe second row provides the aforementioned directions in the BSM Primary basis of Ref.~\\cite{Gupta:2014rxa}, \nwhere the Wilson coefficients can be written in terms of already constrained pseudo-observables. It is clear in this basis that the directions to be probed by high energy $Vh$ production can be \nwritten in terms of the LEP (pseudo)observables. The couplings $\\delta g^Z_{f}$ defined in \\eq{lagr} \nare strongly constrained by $Z$-pole measurements at LEP, whereas the anomalous TGCs, $\\delta \\kappa_\\gamma$ and \n$\\delta g_1^Z$ (in the notation of Ref.~\\cite{Hagiwara:1986vm}), were constrained by $WW$ \nproduction during LEP2. \n\nFor the physically motivated case where the leading effects of new physics \ncan be parametrized by universal (bosonic) operators, the SILH Lagrangian provides a convenient \nformulation and we show the above directions in this basis in the third row of Table~\\ref{dirn}. For this case, as\nshown in the fourth row of Table~\\ref{dirn}, one \ncan again write the directions in terms of only the ``oblique''\/universal pseudo-observables, \n\\textit{viz.}, the TGCs $\\delta \\kappa_\\gamma$ and $\\delta g_1^Z$ and the Peskin-Takeuchi \n$\\hat{S}$-parameter~\\cite{Peskin:1991sw} in the normalization of Ref.~\\cite{Barbieri:2004qk}. For a definition of these observables we refer to the Lagrangian presented in Ref.~\\cite{Elias-Miro:2013eta} (see also Ref.~\\cite{Wells:2015uba}). As we already mentioned, upon using the Goldstone \nEquivalence Principle, one finds that the same 4 dimensional subspace of operators also controls the \nlongitudinal $VV$ production at high energies. This space is defined in Ref.~\\cite{Franceschini:2017xkh} \nin terms of the four high energy primaries which are linear combinations of the four ${g}^h_{Vf}$ \ncouplings, as shown in the last row of Table~\\ref{dirn}.\n\nAs it is not possible to control the polarization of the initial state partons in a hadron collider, \nthe process can, in reality, only probe two of the above four directions. Taking only the interference \nterm, these directions are \n\\begin{equation}\n\\begin{split}\ng^Z_{\\textbf{u}}&= g^h_{Zu_L}+\\frac{g^Z_{u_R}}{g^Z_{u_L}} g^h_{Zu_R}\\,, \\\\\ng^Z_{\\textbf{d}}&= g^h_{Zd_L}+\\frac{g^Z_{d_R}}{g^Z_{d_L}} g^h_{Zd_R}\\,,\n\\end{split}\n\\end{equation}\n where $g^Z_f$ is defined below \\eq{amps}. Also, at a given energy, the interference term for the \n $pp \\to Zh$ process is sensitive only to a linear combination of the up-type and down-type coupling \n deviations, \\textit{i.e.}, to the direction, \n\\begin{equation}\ng^Z_{\\textbf{p}}= g^Z_{\\textbf{u}}+ \\frac{{\\cal L}_d(\\hat{s})}{{\\cal L}_u(\\hat{s})}g^Z_{\\textbf{d}}\n\\end{equation} \nwhere ${\\cal L}_{u,d}$ is the $u \\bar{u}$, $d \\bar{d}$ luminosity at a given partonic centre of mass \nenergy. We find that the luminosity ratio changes very little with energy (between 0.65 and 0.59 if \n$\\sqrt{\\hat{s}}$ is varied between 1 and 2 TeV). Thus, to a good approximation, $pp \\to Zh$ probes \nthe single direction in EFT space given by \n\\begin{equation}\ng^Z_{\\textbf{p}}=g^h_{Zu_L} -0.76~g^h_{Zd_L} - 0.45~g^h_{Zu_R} + 0.14~g^h_{Zd_R} \\,,\n\\label{compdir}\n\\end{equation} \nwhere we have substituted the values for $g^Z_f$ and evaluated the luminosities at the energy \n${\\hat{s}}=(1.5~\\text{TeV})^2$. This can now be written in terms of the LEP-constrained pseudo-observables in the second and fourth row of Tab.~\\ref{dirn},\n\\begin{eqnarray}\ng^h_{Z\\textbf{p}}&=&2~\\delta g^h_{Zu_L} -1.52~\\delta g^h_{Zd_L} - 0.90~\\delta g^h_{Zu_R} + 0.28~\\delta g^h_{Zd_R}\\nonumber\\\\\n&&-0.14~\\delta \\kappa_\\gamma -0.89~\\delta g^Z_1 \\nonumber\\\\\ng^h_{Z\\textbf{p}}&=&-0.14~(\\delta \\kappa_\\gamma-\\hat{S}+Y) -0.89~\\delta g^Z_1 -1.3~W\n\\label{diru}\n\\end{eqnarray}\nwhere the first line applies to the general case and the second line to the universal case.\n\nNote that in the discussion so far we have not considered the $gg \\to Zh$ production channel at \nhadron colliders~\\cite{Dicus:1988yh,Kniehl:1990iva,Matsuura:1990ba,Kniehl:2011aa,Harlander:2013mla,Englert:2013vua,Altenkamp:2012sx,Hespel:2015zea,Harlander:2018yns}. \nWhile formally a higher order correction, after all the cuts are applied, this subprocess contributes \nan appreciable 15\\% of the total SM $pp \\to Zh$ cross-section in our analysis due to the top-threshold \ninducing boosted final states~\\cite{Englert:2013vua}. We find, however, that introduction of the EFT \noperators does not lead to a energy growing amplitude with this initial state, and thus this channel \nhas a subdominant contribution to the EFT signal. While we fully include this contribution in our \ncollider analysis, the introduction of this channel does not alter the discussion so far in an \nimportant way.\n\n \n\\medskip\n \nWe now turn to the crucial issue of estimating the scale of new physics (and thus the cut-off for \nour EFT treatment) for a given size of the couplings, ${g}^h_{Vf}$. This will also give us an idea of \nthe new physics scenarios that our analysis can probe. As is clear from the operators in Tab.~\\ref{list}, \nthe ${g}^h_{Vf}$ couplings arise from current-current operators that can be generated, for instance, \nby integrating out at tree-level a heavy $SU(2)_L$ triplet (singlet) vector $W'^a$ ($Z'$) that couples \nto SM fermion currents, $\\bar{f} \\sigma^a \\gamma_\\mu f$ ($\\bar{f} \\gamma_\\mu f)$ with a coupling \n$g_f$ and to the Higgs current $i{H}^\\dagger \\sigma^a \\lra{D}_\\mu H$ ($i{H}^\\dagger \\lra{D}_\\mu H$) \nwith a coupling $g_H$, \n\\begin{equation}\n{g}^h_{Zf}\\sim \\frac{g_H g g_f v^2}{\\Lambda^2}\\,,\n\\end{equation}\nwhere $\\Lambda$ is the mass of the vector and therefore the matching scale or cut-off of the low \nenergy EFT. The coupling to the SM fermions can be universal if the heavy vector couples to them only \nvia kinetic mixing with the SM gauge bosons. This results in a coupling of the heavy vector to the \n$SU(2)_L$ and hypercharge currents given by $g_W=g\/2$ and $g_B=g' Y_f$, $Y_f$ being the SM hypercharge. \nAs we want our results to be applicable to the universal case, we assume the coupling $g_f$ to be \ngiven by a combination of $g_B$ and $g_W$ to obtain,\n\\begin{equation}\n \\label{cutoff}\n \\begin{split}\n{g}^h_{Zu_L,d_L} &\\sim \\frac{g_H g^2 v^2}{ 2\\Lambda^2}\\,,\\\\\n{g}^h_{Zu_R,d_R} &\\sim \\frac{g_H g g' Y_{u_R,d_R} v^2}{\\Lambda^2} \\,,\n\\end{split}\n\\end{equation}\n and then further assume a weakly coupled scenario with $g_H=1$ (note that this is a bit larger than the corresponding value $g_H=g\/(2 c_{\\theta_W})$ for the SM $hZZ$ coupling). In the above equation, we have \n ignored the smaller contributions from $g_B$ to the left-handed couplings. For any set of couplings \n $\\{{g}^h_{Zu_L} ,{g}^h_{Zd_L} ,{g}^h_{Zu_R} ,{g}^h_{Zd_R} \\}$, we evaluate the cut-off using \n \\eq{cutoff} with $g_H=1$ and take the smallest of the four values. It is clear that for strongly \n coupled scenarios with larger values of $g_H$, the cut-off assumed in our analysis is smaller than \n necessary and thus our projected bounds will be conservative.\n \n \n \n\\begin{table*}[ht]\n\\begin{center}\n\\small\n\\begin{tabular}{||c|c|c||\n\\hline\nCuts & $Zb\\bar{b}$ & $Zh$ (SM) \\\\\n\\hline \\hline\nAt least 1 fat jet with 2 $B$-mesons with $p_T > 15$ GeV & 0.23 & 0.41 \\\\\n2 OSSF isolated leptons & 0.41 & 0.50 \\\\\n$80 \\; \\text{GeV} < M_{\\ell\\ell} < 100$ GeV, $p_{T,\\ell\\ell} > 160$ GeV, $\\Delta R_{\\ell\\ell} > 0.2$ & 0.83 & 0.89 \\\\\nAt least 1 fat jet with 2 $B$-meson tracks with $p_T > 110$ GeV & 0.96 & 0.98 \\\\\n2 Mass drop subjets and $\\ge 2$ filtered subjets & 0.88 & 0.92 \\\\\n2 $b$-tagged subjets & 0.38 & 0.41 \\\\\n$115 \\; \\text{GeV} < m_h < 135$ GeV & 0.15 & 0.51 \\\\\n$\\Delta R(b_i, \\ell_j) > 0.4$, $\\slashed{E}_T < 30$ GeV, $|y_h| < 2.5$, $p_{T,h\/Z} > 200$ GeV & 0.47 & 0.69 \\\\ \n\\hline \n \\end{tabular}\n\\caption{Cut-flow table showing the effect of each cut on $Zb\\bar{b}$ and SM $Zh$.}\n\\label{tab:cut-flow}\n\\end{center}\n\\end{table*}\n\n\\subsection{Analysis}\nIn order to probe the reach of the high luminosity runs of the LHC in constraining the EFT directions in Tab.~\\ref{dirn}, we \noptimize a hadron-level analysis to obtain maximum sensitivity to the BSM signal, which is well-pronounced in the \nhigh energy bins. To achieve this, we consider the $Z(\\ell^+\\ell^-)h$ production from a pair of quarks as well as from a \npair of gluons. As far as the decay of the Higgs boson is concerned, we find that at an integrated luminosity of 300 fb$^{-1}$, the diphoton mode yields less than 5 events at high energies ($p_{T,Z}>150$ GeV) and is thus not sensitive to the effects we want to probe. We thus scrutinize the $h(b\\bar{b})Z(\\ell^+\\ell^-)$ final state where the dominant backgrounds are composed of\n$Zb\\bar{b}$ and the irreducible SM production of $Zh$. For the $Zb\\bar{b}$ process, \nwe consider the tree-level production as well as the $gg \\to ZZ$ production at one-loop.\nReducible contributions arise from $Z+$ jets production ($c$-quarks included but not explicitly\ntagged), where the light jets can be misidentified as $b$-jets, and the fully leptonic decay for \n$t\\bar{t}$. Instead of performing a standard resolved analysis, where one would demand two separate\n$b$-tagged jets, we demand a fat jet with a cone radius of $R=1.2$. We employ the so-called BDRS \napproach~\\cite{Butterworth:2008iy} with minor modifications to maximize sensitivity. In a nutshell, \nthis technique helps in discriminating boosted electroweak-scale resonances from large QCD backgrounds. \n\nWe will see that using this approach will allow us to reduce the ratio of $Zbb$ to SM $Zh$ events from about 40 to an ${\\cal O}(1)$ number with about 40 SM events still surviving at 300 fb$^{-1}$. This shows that the kind of analysis performed here would not be possible at integrated luminosities smaller than 300 fb$^{-1}$.\n\nThe BDRS approach recombines jets using the Cambridge-Aachen (CA) algorithm~\\cite{Dokshitzer:1997in,Wobisch:1998wt} with a significantly large\ncone radius to contain all the decay products of the resonance. One then works backwards\nthrough the jet clustering and stops when a significant {\\emph{mass drop}}, \n$m_{j_1} < \\mu m_j$ with $\\mu=0.66$, ($m_j$ being the mass of the fatjet) occurs for a not too \nasymmetric splitting\n, \n$$\\frac{\\text{min}(p_{T,j_1}^2,p_{T,j_2}^2)}{m_j^2}\\Delta R_{j_1,j_2}^2>y_{\\text{cut}},$$ \nwith\n$y_{\\text{cut}} = 0.09$. If this condition is not met, the softer subjet, $j_2$ is removed and the subjets of $j_1$ are tested\nfor the aforementioned criteria to be satisfied in an iterative process. The algorithm stops as soon as one obtains two subjets,\n$j_1$ and $j_2$ abiding by the mass drop condition. \n\nTo improve the resonance reconstruction, the technique considers a further step called \\textit{filtering}. In this\nstep, the constituents of $j_1$ and $j_2$ are again recombined using the CA algorithm with a cone radius\n$R_{\\text{filt}} = \\text{min}(0.3, R_{b\\bar{b}}\/2)$. Only the hardest three filtered subjets are retained\nto reconstruct the resonance. In the original work of Ref.~\\cite{Butterworth:2008iy}, this resonance is the SM-like Higgs and thus the two\nhardest filtered subjets are $b$-tagged. In our work, we find that the filtered cone radius \n$R_{\\text{filt}} = \\text{max}(0.2, R_{b\\bar{b}}\/2)$ works better in removing the backgrounds.\\footnote{The criteria $R_{\\text{filt}} = \\text{max}(0.2, R_{b\\bar{b}}\/2)$ followed by $R_{\\text{filt}} = \n\\text{min}(0.3, R_{b\\bar{b}}\/2)$ hardly changes the results.} The filtering step greatly reduces the active area\nof the initial fatjet. \n\nWe use the \\texttt{FeynRules}~\\cite{Alloul:2013bka} and \\texttt{UFO}~\\cite{Degrande:2011ua} toolkits to implement the signal contributions (we will comment on the effect of including squared dimension 6 interactions as compared to interference-only terms below).\nBoth signal and background processes are generated including the full decay chain with \\texttt{MG5$\\_$aMC@NLO}~\\cite{Alwall:2014hca}, at leading order.\nFor the gluon initiated part of the SM and BSM $Zh$ production, we employ the \\texttt{FeynArts\/FormCalc\/LoopTools}~\\cite{Hahn:2000kx,Hahn:1998yk} \nframework and the decays are performed using \\texttt{MadSpin}~\\cite{Frixione:2007zp,Artoisenet:2012st}. We shower and hadronize the samples using \\texttt{Pythia 8}~\\cite{Sjostrand:2001yu,Sjostrand:2014zea} and \nperform a simplified detector analysis. \n\nBecause our ultimate goal is to look for new physics effects in \nhigh energy bins, we generate the $Zh$, $Zb\\bar{b}$ and $t\\bar{t}$ samples with the following cuts: \n$p_{T,(j,b)} > 15$ GeV, $p_{T,\\ell} > 5$ GeV, $|y_j| < 4$, $|y_{b\/\\ell}| < 3$, $\\Delta R_{bb\/bj\/bl} > 0.2$, \n$\\Delta R_{\\ell\\ell} > 0.15$, $70 \\; \\text{GeV} < m_{\\ell \\ell} < 110$ GeV, $75 \\; \\text{GeV} < m_{bb} < \n155$ GeV and $p_{T,\\ell\\ell} > 150$ GeV. The former two processes are generated upon merging with an \nadditional matrix element (ME) parton upon using the MLM scheme~\\cite{Mangano:2006rw}. \nFor the $Z+$jets process, we generate the samples without the invariant mass cuts on the jets; we \nfurther merge the sample up to three ME partons. \n\nTo account for higher order QCD corrections for the $q\\bar{q}$-initiated $Zh$ \nprocess, we apply a bin-by-bin (in $M_{Zh}$, the invariant mass of the filtered double $b$-tagged fat jet and the reconstructed $Z$-boson) $K$-factor reweighting to the NLO-accurate distribution both for the SM background and the EFT signal using Ref.~\\cite{Greljo:2017spw}. For \nthe $gg$ initiated $Zh$ process, we consider a conservative NLO $K$-factor of 2~\\cite{Altenkamp:2012sx}. \nFor the tree-level $Zb\\bar{b}$ and $Z+$jets processes, flat $K$-factors of 1.4 (computed within \\texttt{MG5$\\_$aMC@NLO}) and 0.91~\\cite{Campbell:2002tg} are applied.\nFor the $gg \\to ZZ$ production, a flat $K$-factor of $\\sim1.8$~\\cite{Alioli:2016xab} has been\nused.\n\nWe first test the power of a cut-based analysis. In doing so, we construct\nthe fatjets with a cone radius of $R=1.2$, having $p_T > 80$ GeV and rapidity, $|y| < 2.5$ using \n\\texttt{FastJet}~\\cite{Cacciari:2011ma}. We isolate the leptons ($e,\\mu$) by demanding that the total hadronic activity around a cone radius of $R=0.3$ must be less than\n10\\% of its $p_T$ and the leptons are required to have $p_T > 20$ GeV and $|y| < 2.5$. All non-isolated objects\nare considered while constructing the fatjets. In selecting our events, we consider only those with\nexactly two isolated leptons having opposite charge and same flavour (OSSF). Moreover, we demand the \ninvariant mass of the pair of leptons to lie in the range $[80~\\text{GeV},100~\\text{GeV}]$ in order to reconstruct the \n$Z$-peak. The reconstructed $Z$ is required to be boosted with $p_T > 160$ GeV and the separation between\nthe two isolated leptons is required to be $\\Delta R > 0.2$. In reconstructing the Higgs boson, we demand\nthat each event has at least one fatjet containing no less than two $B$-meson tracks with $p_T > 15$ GeV.\nThe minimum transverse momentum of the fatjet is required to be $p_T > 110$ GeV. After satisfying the \nmass drop and filtering criteria, we require exactly two subjets after the former step\nand at least two subjets after filtering. We proceed with $b$-tagging the two hardest \nsubjets. We choose a $b$-tagging efficiency of 70\\% and a misidentification rate for light \njets of 2\\%. After the filtering and $b$-tagging steps, we require events with exactly two $b$-tagged \nsubjets, which are well-separated from the isolated leptons: \n$\\Delta R(b_i,\\ell_j) > 0.4$ for both leptons $\\ell_{1,2}$ and $b$-tagged subjets $b_i$. \nWe reconstruct the Higgs by requiring its invariant mass to lie in the range [115 GeV, 135 GeV]. \n\nIn order to further reduce the backgrounds, we demand both the reconstructed $Z$ and the Higgs\nbosons to have $p_T > 200$ GeV. The $t\\bar{t}$ background can be removed almost entirely by requiring \n$\\slashed{E}_T < 30$ GeV. The cut-flow affecting the most dominant background $Zb\\bar{b}$ and the SM \n$Zh$ channel, is summarized in Table~\\ref{tab:cut-flow}. \n\nBefore focussing on the very high-energy effects by imposing\ncuts on $M_{Zh}$, we find that the ratio of cross-section between SM $Zh$ and $Zb\\bar{b}$ is $\\sim 0.26$.\nA multivariate implementation at this level strengthens this ratio further. In order\nto be quantitative, we impose looser cuts on the aforementioned variables $70 \\; \\text{GeV} < \nm_{\\ell\\ell} < 110$ GeV, $p_{T,\\ell\\ell} > 160$ GeV, $\\Delta R_{\\ell\\ell} > 0.2$, \n$p_{T,\\text{fatjet}} > 60$ GeV, $95 \\; \\text{GeV} < m_h < 155$ GeV, $\\Delta R_{b_i,\\ell_j} > 0.4$ and \n$\\slashed{E}_T < 30$ GeV. Because $Z+$jets and $t\\bar{t}$ are much less significant than $Zb\\bar{b}$, we\ntrain the boosted decision trees only with the SM $q\\bar{q}$-initiated $Zh$ and $Zb\\bar{b}$ samples \nusing the following variables: $p_T$ of the two isolated leptons, $\\Delta R$ between pairs of $b$-subjets \nand isolated leptons, between the two isolated leptons and between the hardest two $b$-subjets in the Higgs \nfatjet, the reconstructed $Z$-boson mass and its $p_T$, $\\Delta \\Phi$ separation\nbetween the fatjet and the reconstructed $Z$-boson, $\\slashed{E}_T$, mass of the reconstructed Higgs jet\nand its $p_T$, $p_T$ of the two $b$-tagged filtered subjets, the ratio of their $p_T$ and the rapidity of\nthe Higgs jet. We ensure that we do not have variables which are $\\sim100\\%$ correlated but we retain all \nother variables. Because our final distribution of interest is the invariant mass of the $Zh$-system, we\ndo not consider it as an input variable. We use the \\texttt{TMVA}~\\cite{2007physics3039H} framework to\ntrain our samples and always ensure that the Kolmogorov-Smirnov statistic is at least of the order $\\sim0.1$\nin order to avoid overtraining of the samples~\\cite{KS}. We find that the aforementioned ratio increases\nto $\\sim 0.50$ upon using the boosted decision tree algorithm showing that a further optimisation of the cut-based\nanalysis was necessary. Finally, we test all our samples with the training obtained from the SM $q\\bar{q}$\ninitiated $Zh$ and the $Zb\\bar{b}$ samples. \n\n\n\\begin{figure}[!t]\n\\includegraphics[scale=0.5]{bars}\n\\caption{The differential distribution of events at an integrated luminosity of 300 fb$^{-1}$ with respect to $M_{Zh}$ for the EFT signal as well as the different backgrounds. For the EFT signal we have taken the point $\\{{g}^h_{Zu_L} ,{g}^h_{Zd_L} ,{g}^h_{Zu_R} ,{g}^h_{Zd_R} \\}=\\{-0.005, 0.0001 , -0.010, 0.005 \\}$ which is allowed by the LEP bounds.}\n\\label{bars}\n\\end{figure}\n\nTo distinguish between the EFT signal and the irreducible SM $Zh(b\\bar b)$ background we utilise the \ngrowth of the EFT cross-section at high energies. The effects are readily seen in the $M_{Zh}$ distribution, \nour observable of interest. In Fig.~\\ref{bars} we show the differential distribution with respect to \nthis variable for the EFT signal as well as the different backgrounds for an integrated luminosity of \n300 fb$^{-1}$. For the EFT signal we take a point that can be excluded in our analysis but is well within the LEP allowed region. We see that the \nEFT cross-section keeps growing with energy, but much of this growth is unphysical at energies above \nthe cut-off, \\textit{i.e.}, $M_{Zh} >\\Lambda$, where $\\Lambda$ is the cut-off evaluated as described \nbelow \\eq{cutoff} and shown by a vertical line in Fig.~\\ref{bars}. For our analysis we dropped all events above this cut-off. For $M_{Zh} <\\Lambda$, the EFT \ndeviations are never larger than an ${\\cal O}(1)$ factor with respect to the SM background as \nexpected on general grounds. Note, however, that even for $M_{Zh} <\\Lambda$, even though \nthe underlying anomalous couplings, $g^h_{Zf}$, are per-mille to percent level, the \nfractional deviations are much larger because of the energy growth of the BSM rate. To \nmake full use of the shape deviation of the EFT signal with respect to the background, \nwe perform a binned log likelihood analysis assuming a 5$\\%$ systematic error. The \nlikelihood function is taken to be the product of Poisson distribution functions for \neach bin with the mean given by the number of events expected for a given BSM point. \nTo account for the 5$\\%$ systematic error we smear the mean with a Gaussian \ndistribution. To obtain the projection for the 95$\\%$ CL exclusion curve we assume that \nthe observed number of events agrees with the SM\n\\subsection{Discussion}\n\n \n\n\\begin{table}[t]\n\\begin{center}\n\n\\begin{tabular}{c|c|c\n&Our Projection &LEP Bound\\\\\\hline\n$\\delta g^Z_{u_L}$ & $\\pm0.002~(\\pm0.0007)$ & $-0.0026\\pm 0.0016$\\\\\n$\\delta g^Z_{d_L}$ & $\\pm0.003~(\\pm0.001)$ & $0.0023\\pm 0.001$\\\\\n$\\delta g^Z_{u_R}$ & $\\pm0.005~(\\pm0.001)$ & $-0.0036\\pm 0.0035$\\\\\n$\\delta g^Z_{d_R}$ & $\\pm0.016~(\\pm0.005)$ & $0.016\\pm 0.0052$\\\\\n$\\delta g^Z_1$ & $\\pm0.005~(\\pm0.001)$ & $0.009^{+0.043}_{-0.042}$\\\\\n$\\delta \\kappa_\\gamma$ & $\\pm0.032~(\\pm0.009)$ & $0.016^{+0.085}_{-0.096}$\\\\\n$\\hat{S}$ & $\\pm0.032~(\\pm0.009)$ & $0.0004 \\pm 0.0007$\\\\\n$W$ & $\\pm0.003~(\\pm0.001)$ & $0.0000 \\pm 0.0006$\\\\\n$Y$ & $\\pm0.032~(\\pm0.009)$ & $0.0003\\pm 0.0006$\n \\end{tabular}\n \\caption{Comparison of the bounds obtained in this work with existing LEP bounds. The numbers outside (inside) brackets, \n in the second column, denote our bounds with $\\mathcal{L} = 300 \\; (3000)$ fb$^{-1}$. To obtain our projection we turn on the LEP observables in \\eq{diru} one by one and use \\eq{obo}. The LEP bounds on the $Z$ coupling to quarks has been obtained from Ref.~\\cite{Falkowski:2014tna}, the bound on the TGCs from Ref.~\\cite{LEP2}, the bound on $\\hat{S}$ from Ref.~\\cite{Baak:2012kk} and finally the bounds on $W,Y$ have been obtained from Ref.~\\cite{Barbieri:2004qk}. Except for the case of the bounds on $\\delta g^Z_f$, all of the bounds in the last column were derived by turning on only the given parameter and putting all other parameters to zero. \\label{lepb}}\n\\end{center}\n\\end{table}\n\\begin{figure}[!b]\n\\includegraphics[scale=0.6]{300plot}\n\\caption{We show in light blue (dark blue) the projection for the allowed region with 300 fb$^{-1}$ (3 ab$^{-1}$) data from the \n$pp \\to Zh$ process for universal models in the $\\delta \\kappa_\\gamma-\\hat{S}$ vs $\\delta g^Z_1$ plane. The allowed region after LEP bounds (taking the TGC $\\lambda_\\gamma=0$, a conservative choice) are imposed is shown in grey. The pink (dark pink) region corresponds to the projection from the $WZ$ process with 300 fb$^{-1}$ (3 ab$^{-1}$) data derived in Ref.~\\cite{Franceschini:2017xkh} and the purple (green) region shows the region that survives after our projection from the $Zh$ process is combined with the above $WZ$ projections with 300 fb$^{-1}$ (3 ab$^{-1}$) data.}\\label{bounds}\n\\end{figure} \nConsidering only the SM-BSM interference term, we find the per-mille level bounds, \n\\begin{eqnarray}\ng^h_{Z\\textbf{p}} &\\in & \\left[-0.004,0.004\\right]~~~~(300{\\rm~fb}^{-1}) \\nonumber\\\\\ng^h_{Z\\textbf{p}} &\\in & \\left[-0.001,0.001\\right]~~~~(3000{\\rm~fb}^{-1}).\n\\label{obo}\n\\end{eqnarray}\nUsing \\eq{cutoff} the above bounds can be translated to a lower bound on the scale of new physics given by 2.4 TeV (4.4 TeV) at 300 fb$^{-1}$ (3000 fb$^{-1}$). One can now compare the above projections with existing LEP bounds by turning on the LEP observables contributing to $g^h_{Z\\textbf{p}}$ in \\eq{diru} one by one. This is equivalent to assuming that there are no large cancellations in \\eq{diru} so that each individual term is bounded by \\eq{obo}. The results are shown in Tab.~\\ref{lepb}. We see that our projections are much stronger than the LEP bounds for the TGCs $\\delta g^Z_1$ and $\\delta \\kappa_\\gamma$ and comparable in the case of the $Z$-pole observables $\\delta g^Z_f$, that parametrize the deviations of the $Z$ coupling to quarks. \n\nTo obtain \\eq{obo}, we have used the cut-off as defined in \\eq{cutoff} with $g_h=1$. One might expect a stronger bound by taking a larger $g_h$. We find, however, that while taking stronger couplings can increase the cut-off many times, this does not lead to an appreciably higher sensitivity because the high energy bins have very few or no SM\/BSM events being suppressed by the small PDFs at these energies.\n\nFor the universal case, the EFT directions presented in Table~\\ref{dirn} can be visualized in the \n$\\delta \\kappa_\\gamma-\\hat{S}$ vs. $\\delta g^Z_1$ plane as shown in Fig.~\\ref{bounds} for the interesting class of models where $W=Y=0$~\\cite{Franceschini:2017xkh}. The flat \ndirection related to the $pp \\to Zh$ interference term, \\textit{i.e.}, $g^h_{Z\\textbf{p}}=0$, \n\\eq{compdir}, is shown by the dashed blue line, where the direction $g^h_{Z\\textbf{p}}$ is now given \nby the second line of \\eq{diru}. The grey shaded area shows the allowed region after the LEP II bounds~\\cite{LEP2} from the $e^+e^- \\to W^+W^-$ \nprocess are imposed. The results of this work are shown in blue (light (dark) blue for results at\n300 (3000) fb$^{-1}$). To understand the shape of the blue bands, note that along the dashed line, \nthe SM-BSM interference term vanishes. If the interference was the only dominant effect, the \nprojected allowed region would be a band along this direction. The BSM squared term thus plays a \nrole in determining the shape of the blue region. To the left of the dashed blue line, the squared \nand the interference terms have the same sign while there is a partial cancellation between these \ntwo terms on the right hand side of the dashed line. This results in the curvature of the blue band \nwith stronger bounds to the left of the dashed line and weaker bounds to its right. \n\nWe see that, as we move further from the origin, the effect of the squared term becomes more pronounced. This is expected, as along the dashed line, the interference term is accidentally zero, even for energies below the cut-off, and thus, the parametrically sub-dominant squared term is larger. To \nachieve a partial cancellation between these two terms one needs to \ndeviate more and more from the dashed line. While EFT validity has been carefully imposed to derive \nour bounds, the fact that the interference term vanishes along the flat direction and the squared \nterm becomes important, does imply that for weakly coupled UV completions our bounds are susceptible \nto ${\\cal O}(1)$ dimension 8 deformations in this direction. In the orthogonal direction shown by the \ndotted line, on the other hand, our projections are more robust and not sensitive to such effects. Such an ambiguity also exists in the results in Tab.~\\ref{lepb}, for the pseudo-observables such as $\\delta g^Z_{d_R}$ and $\\delta \\kappa_\\gamma$, that are somewhat aligned to the above flat direction. This ambiguity can be resolved by performing a global fit upon combining analyses of all the $Vh, VV$ channels, that will avoid such flat directions.\n \nAs we have emphasized already, $VV$ production constrains the same set of operators as the $Vh$ \nproduction. In Fig.~\\ref{bounds}, we also show the projected bound from the $WZ$ process at 300 \nfb$^{-1}$ obtained in Ref.~\\cite{Franceschini:2017xkh}. When both these bounds are combined, only the \npurple region remains. At 3000 fb$^{-1}$, this region shrinks further to the green region shown in \nFig.~\\ref{bounds}. Thus, we see a drastic reduction in the allowed LEP region is possible \nby investigating $pp \\to Zh$ at high energies.\n\n\\subsection{Conclusions}\nAs hints for new physics beyond the SM remain elusive with the LHC entering a new energy territory, model-independent approaches based on the assumption of no additional light propagating degrees of freedom are gaining ground. The power of effective field theory is that theoretical correlations between independent measurements can be exploited to formulate tight constraints on the presence of new physics, solely based on the SM symmetries and matter content.\n\nThe high precision measurements performed during the LEP era are therefore the driving forces behind combined constraints early in the LHC program. To enter new territory, the LHC has to push beyond the LEP sensitivity for interactions that relate the phenomenology at both collider experiments. The Higgs boson, as arguably the most significant TeV scale degree of freedom, can be placed at the core of such a program, that will naturally involve LHC measurements at high luminosity. \n\nIn this paper, we focussed on the impact of associated Higgs production that provides complementary information to the diboson production modes observed at LEP2, which determine the precision of the associated coupling constraints. Using a dedicated investigation of expected signal and backgrounds, we find that the LHC will ultimately be able to improve the sensitivity expected from LEP measurements. Our results are summarised in \\eq{obo}, Tab.~\\ref{lepb} and Fig.~\\ref{bounds}. Higgs-strahlung is also complementary to diboson production at LHC investigated in Ref.~\\cite{Franceschini:2017xkh}. Combining Higgs-strahlung measurements with diboson results in the high energy limit will allow us to drastically improve the sensitivity to the underlying new physics parameters in an unparalleled way.\n\nBoth high energies and luminosities are crucial for a study like ours. Potentially even higher new physics scales can thus be probed at the High Energy LHC or other future colliders.\n \n\\subsection{Acknowledgements}\nWe thank Matthew~McCullough for helpful discussions and collaboration on this project in its early stage.\nWe also thank Biplob Bhattacherjee, Mikael Chala, Shilpi Jain, Giuliano Panico, Michael Peskin and Francesco Riva for \nhelpful discussions at various stages of this work.\n\nC.E. is supported by the IPPP Associateship scheme and by the UK Science and Technology Facilities Council (STFC) under grant ST\/P000746\/1.\nS.B. is supported by a Durham Junior Research Fellowship COFUNDed by Durham University and the \nEuropean Union, under grant agreement number 609412.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sect:intro}\n\nIn a previous work \\cite{CP98,PC00}, we performed \nhypernetted chain (HNC)\ncalculations and proposed analytic formulae\nfor the equation of state (EOS) of electron-ion\nplasmas (EIP). An alternative \nanalytic approximation for the EOS of EIP was proposed in\n\\cite{SB96,SB00}. \nA comparison (e.g., \\cite{CP98,SB00}) shows that \nthe formulae in \\cite{PC00} have a higher accuracy,\nin particular\nfor the thermodynamic\ncontributions of ion-ion and ion-electron correlations\nin the regime of moderate\nCoulomb coupling.\nRecently~\\cite{PCR09,PCCDR09}, we studied classical ion mixtures\nand proposed a\ncorrection to the linear mixing rule.\nIn this paper we \n review the analytic expressions\nfor all contributions to thermodynamic functions\nand\nintroduce some\npractical modifications to the previously published formulae.\nThe reviewed analytic\ndescription of the thermodynamic\nfunctions of Coulomb plasmas\nis realized in a publicly available computer code.\n\n\nLet $n_e$ and $n_\\mathrm{i}$ be the electron and ion number densities,\n$A$ and $Z$ the ion mass and charge numbers, respectively. The\nelectric neutrality implies $n_e = Z n_\\mathrm{i}$. In this paper we\nneglect positrons (they can be described using the same formulae\nas the electrons; see, e.g., Ref.~\\cite{Blin}) and\nfree neutrons (see, e.g., Ref.~\\cite{NSB1}), and consider mainly\nplasmas containing a single type of ions (for the extension to\nmulticomponent mixtures, see \\cite{PCR09,PCCDR09}).\n\nThe state of a free electron gas\nis determined by the electron number density $n_e$ and\ntemperature $T$.\nInstead of $n_e$ it is convenient to introduce\nthe dimensionless density parameter $r_s=a_e\/a_0$,\nwhere $a_0$ is the Bohr radius and\n$a_e=(\\frac43\\pi n_e)^{-1\/3}$.\nThe parameter $r_s$ can be easily evaluated from the relations\n$\n r_s = 1.1723 \\,n_{24}^{-1\/3},\n$\nwhere $n_{24} \\equiv n_e\/10^{24}{\\rm~cm}^{-3}$,\nor\n$\n r_s=(\\rho_0\/\\rho)^{1\/3},\n$\nwhere $\\rho_0=2.6752\\,(A\/ Z)$ g~cm$^{-3}$.\nThe analogous density parameter for the\nion one-component plasma (OCP) is \n$R_S=a_\\mathrm{i} m_\\mathrm{i} (Ze)^2\/\\hbar^2 = 1822.89\\, r_s AZ^{7\/3}$,\nwhere $m_\\mathrm{i}$ is the ion mass\nand $a_\\mathrm{i}\\equiv(\\frac43\\pi n_\\mathrm{i})^{-1\/3}= a_e Z^{1\/3}$ \nis the ion sphere radius.\n\nAt stellar densities it is convenient to use, instead of $r_s$,\nthe relativity parameter \\cite{Salpeter61}\n$\n x_\\mathrm{r} = p_\\mathrm{F} \/ m_e c = 1.00884 \\,\n \\left( \\rho_6 Z \/ A \n \\right)^{1\/3}\\!\\! = 0.014005\\,r_s^{-1},\n$\nwhere $ p_\\mathrm{F} = \\hbar \\,(3 \\pi^2 n_e)^{1\/3}\n$\nis the electron Fermi momentum and $\\rho_6\\equiv\\rho\/10^6$ g~cm$^{-3}$.\nThe Fermi kinetic\nenergy is\n$\\epsilon_\\mathrm{F} = c\\,\\sqrt{(m_e c)^2 + p_\\mathrm{F}^2}-m_e c^2 ,\n$\nand the Fermi temperature equals\n$T_\\mathrm{F} \\equiv \\epsilon_\\mathrm{F}\/k_B = T_\\mathrm{r} \\, ( \\gamma_\\mathrm{r} - 1) ,$\nwhere \n$T_\\mathrm{r} \\equiv {m_e c^2 \/ k_B } = 5.93\\times\n10^9~\\mathrm{K}$,\n$\\gamma_\\mathrm{r} \\equiv \\sqrt{1+ x_\\mathrm{r}^2}$,\nand $k_B$ is the Boltzmann constant.\nIf $x_\\mathrm{r}\\ll1$, then $T_\\mathrm{F}\\approx 1.163\\times10^6\\, r_s^{-2}$~K.\nThe effects of special relativity are controlled by $x_\\mathrm{r}$\nin degenerate plasmas ($T\\ll T_\\mathrm{F}$) and by $\\tau \\equiv {T \/T_\\mathrm{r}}$\nin nondegenerate plasmas ($T\\gg T_\\mathrm{F}$).\n\nThe ions are nonrelativistic in most\napplications. The strength of the Coulomb interaction of ions\nis characterized by the Coulomb coupling parameter\n$\n \\Gamma =\n {(Z e)^2}\/{a_\\mathrm{i} k_B T} = \\Gamma_e Z^{5\/3},\n$\nwhere\n$\n \\Gamma_e \\equiv { e^2 }\/{ a_e k_B T}\n \\approx {22.747}\\,{ T_6^{-1}}\n \\left(\\rho_6 { Z }\/{ A}\n \\right)^{1\/3}\n$\nand $T_6\\equiv T\/10^6$~K.\n\nThermal de Broglie wavelengths of the ions and electrons are\nusually defined as\n$\n \\lambda_\\mathrm{i} = \\left({2\\pi\\hbar^2}\/{ m_\\mathrm{i} k_B T}\\right)^{1\/2}\n$\nand\n$\n \\lambda_e = \\left({2\\pi\\hbar^2}\/{ m_e k_B T}\\right)^{1\/2}.\n$\nThe quantum effects on ion motion are\nimportant either at $\\lambda_\\mathrm{i}\\gtrsim a_\\mathrm{i}$ or at \n$T \\ll T_\\mathrm{p}$, where\n$\n T_\\mathrm{p} \\equiv (\\hbar \\omega_\\mathrm{p}\/ k_B)\n \\approx 7.832 \\times 10^6\\, (Z\/A)\\,\\sqrt{\\rho_6}\\,\\textrm{~K}\n$\nis the \\emph{plasma temperature} determined by\nthe ion plasma frequency\n$\n \\omega_\\mathrm{p} = \\left( {4 \\pi e^2 \\,n_\\mathrm{i}}\n Z^2 \/m_\\mathrm{i} \\right)^{1\/2}.\n$\nThe corresponding dimensionless parameter is\n$\\eta \\equiv T_\\mathrm{p}\/T$.\n\n\nAssuming commutativity of the kinetic\nand potential energy operators and separation of the\ntraces of the electronic and ionic parts of\nthe Hamiltonian,\nthe total Helmholtz free energy $F$\ncan be conveniently written as\n$\n F = \n F_\\mathrm{id}^{(i)} + F_\\mathrm{id}^{(e)} \n + F_{ee} + F_\\mathrm{ii} + F_{ie}, \n$\nwhere $F_\\mathrm{id}^{(i)}$ and $F_\\mathrm{id}^{(e)}$ denote the ideal free energy of ions\nand electrons, and the last three terms represent\nan excess free energy arising from the electron-electron,\nion-ion, and ion-electron interactions, respectively. \nThis decomposition induces analogous decompositions\nof pressure $P$, internal energy $U$, entropy $S$, the heat capacity\n$C_V$, and\nthe pressure derivatives\n$\n \\chi_T=(\\partial\\ln P\/\\partial\\ln T)_V\n$ \nand\n$\n \\chi_\\rho=-(\\partial\\ln P\/\\partial\\ln V)_T.\n$\nOther second-order functions can be expressed through these\nones by Maxwell relations.\n\n\\section{Ideal part of the free energy}\n\\label{sect:id}\n\nThe free energy of a gas of\n$N_\\mathrm{i}=n_\\mathrm{i} V$ nonrelativistic classical ions is\n$\n F_\\mathrm{id}^{(i)} =\n N_\\mathrm{i} k_B T \\left[\\ln(n_\\mathrm{i}\\lambda_\\mathrm{i}^3\/M)-1 \\right],\n$\nwhere $M$ is the spin multiplicity.\nIn the OCP, it can be written in terms of the\ndimensionless plasma parameters\n as\n$\n {F_\\mathrm{id}^{(i)}}={N_\\mathrm{i}k_B T}\n \\left[ 3 \\ln \\eta - 1.5 \\ln \\Gamma\n - 0.5 \\ln(6\/\\pi)-\\ln M -1 \\right].\n$\n\nThe free energy of the electron gas is given by\n$\n F_\\mathrm{id}^{(e)} =\n \\mu_e N_e - P_\\mathrm{id}^{(e)}\\,V,\n$\nwhere $\\mu_e$ is the electron chemical potential.\nThe pressure and the number density are functions of\n$\\mu_e$ and $T$:\n\\begin{equation}\n P_\\mathrm{id}^{(e)} =\n \\frac{8}{3\\sqrt\\pi}\\,\\frac{k_B T }{ \\lambda_e^3}\n \\left[ I_{3\/2}(\\chi_e,\\tau)\n + \\frac{\\tau}{ 2}I_{5\/2}(\\chi_e,\\tau) \\right],\n\\quad\n n_e =\n \\frac{4}{\\sqrt{\\pi}\\,\\lambda_e^3}\n \\left[ I_{1\/2}(\\chi_e,\\tau)\n + \\tau I_{3\/2}(\\chi_e,\\tau) \\right],\n\\label{n_e}\n\\end{equation}\nwhere \n$\\chi_e=\\mu_e\/k_B T$\n(here, we do not include\nthe rest energy $m_e c^2$ in $\\mu_e$) and\n\\begin{equation}\n I_\\nu(\\chi_e,\\tau) \\equiv \\int_0^\\infty\n \\frac{ x^\\nu\\,(1+\\tau x\/2)^{1\/2}\n }{ \\exp(x-\\chi_e)+1 }\\,{\\mathrm{d}}x\n\\label{I_nu}\n\\end{equation}\nis a Fermi-Dirac integral.\nAn analytic approximation for $\\mu_e(n_e)$\nhas been derived in \\cite{CP98}.\n\nIn Ref.~\\cite{CP98} we calculated $I_\\nu(\\chi_e,\\tau)$\nusing analytical approximations \\cite{Blin}. These approximations\nare piecewise: below, within, and above the interval\n$0.6\\leq\\chi_e<14$. Their typical fractional accuracy is a few\n$\\times10^{-4}$, the maximum error and discontinuities at the\nboundaries reach $\\sim0.2$\\%. For the first and second\nderivatives, the errors and discontinuities lie within 1.5\\%. At\n$\\chi_e\\geq14$ we use the Sommerfeld expansion\n(e.g., \\cite{Chandra})\n\\begin{equation}\n I_\\nu(\\chi_e,\\tau) \\approx\n \\mathcal{I}_\\nu^{(0)}(\\tilde\\mu)\n +\\frac{\\pi^2}{6}\\tau^2 \\mathcal{I}_\\nu^{(2)}(\\tilde\\mu)\n+ \\frac{7\\pi^4}{360}\\tau^4 \\mathcal{I}_\\nu^{(4)}(\\tilde\\mu)\n + \\ldots,\n\\label{Sommer}\n\\end{equation}\nwhere we have defined $\\tilde\\mu=\\chi_e\\tau=\\mu_e\/m_e c^2$,\n\\begin{equation}\n \\mathcal{I}_\\nu^{(0)}(\\epsilon)\n = \\int_0^\\epsilon \\mathcal{I}_\\nu^{(1)}(\\epsilon')\\,\\mathrm{d} \\epsilon' \n = \\int_0^{x_0} \\big(\\sqrt{1+x^2}-1\\big)^{\\nu-1\/2}\n \\frac{x^2\\,\\mathrm{d} x}{\\sqrt{1+x^2}},\n\\quad\n \\mathcal{I}_\\nu^{(n+1)}(\\tilde\\mu)\n = \\frac{\\mathrm{d} \\mathcal{I}_\\nu^{(n)}(\\tilde\\mu)}{\\mathrm{d}\\tilde\\mu} ,\n\\end{equation}\n$\\mathcal{I}_\\nu^{(1)}(\\epsilon)=\\epsilon^\\nu\n\\sqrt{2+\\epsilon}$\nand $x_0 \\equiv \\sqrt{\\tilde\\mu(2+\\tilde\\mu)}$.\nIn particular,\n$$\n\\mathcal{I}_{1\/2}^{(0)}(\\tilde\\mu)\n = [x_0\\gamma_0-\\ln(x_0+\\gamma_0) ]\/2,\n\\quad\n\\mathcal{I}_{3\/2}^{(0)}(\\tilde\\mu)\n = x_0^3\/3 - \\mathcal{I}_{1\/2}^{(0)}(\\tilde\\mu),\n\\quad\n\\mathcal{I}_{5\/2}^{(0)}(\\tilde\\mu)\n = x_0^3\\gamma_0\/4 - 2 x_0^3\/3\n + 1.25\\,\\mathcal{I}_{1\/2}^{(0)}(\\tilde\\mu),\n$$\nwhere\n$\\gamma_0 \\equiv \\sqrt{1+x_0^2} = 1+\\tilde\\mu\\,$\n(note that, if $\\tilde\\mu=\\tilde\\epsilon$, where\n$\\tilde\\epsilon\\equiv\\epsilon_\\mathrm{F}\/m_e c^2$,\nthen $x_0=x_\\mathrm{r}$ and $\\gamma_0=\\gamma_\\mathrm{r}$).\n\nAt small $\\tilde\\mu$, accuracy can be lost\nbecause of numerical cancellations\nof close terms of opposite signs \nin \\req{Sommer} and in the respective partial derivatives. \nIn this case we use the expansion\n\\begin{equation}\n I_\\nu(\\chi,\\tau) = \\! I^\\mathrm{nr}_\\nu(\\chi)\n + \\!\\sum_{m=0}^\\infty (-1)^m\n \\frac{(2m-1)!!\\,\\tau^{m+1}}{4^{m+1} m!}\n I^\\mathrm{nr}_{\\nu+m+1}(\\chi),\n\\quad\n I^\\mathrm{nr}_\\nu(\\chi) = \\int_0^\\infty \\frac{ x^\\nu\n \\,\\mathrm{d} x }{ e^{x-\\chi}+1 },\n\\label{expan1}\n\\end{equation}\nwhere $(2m-1)!!\\equiv\\prod_{k=1}^m (2k-1)$ should be replaced by\n1 for $m=0$. At large $\\chi$, the nonrelativistic Fermi integrals\n$I^\\mathrm{nr}_\\nu(\\chi)$ are calculated with the use of \nthe Sommerfeld expansion.\nSufficiently smooth and accurate overall approximations for\nfunctions $I_\\nu(\\chi_e,\\tau)$ with $\\nu=\\frac12$, $\\frac32$, and\n$\\frac 52$ are\nprovided by switching to Eq.~(\\ref{expan1})\nat $\\chi_e\\geq14$ and $\\tilde\\mu<0.1$, \nwhile retaining the terms\nup to $m=3$.\n\nThe chemical potential at fixed $n_e$ is obtained\nwith fractional accuracy $\\sim T^2\/T_\\mathrm{F}^2\\,$ by \nusing Eqs.~(\\ref{n_e}), (\\ref{Sommer}), setting\n$\n\\mathcal{I}_\\nu^{(n)}(\\tilde\\mu)\\approx\n\\mathcal{I}_\\nu^{(n)}(\\tilde\\epsilon)\n+\\mathcal{I}_\\nu^{(n+1)}(\\tilde\\epsilon)\\,(\\tilde\\mu-\\tilde\\epsilon) ,\n$\nand dropping the terms that contain the product\n$(\\tilde\\mu-\\tilde\\epsilon)\\,\\tau^2$.\nThen at $T\\llT_\\mathrm{F}$ \n\\begin{equation}\n\\Delta\\tilde\\epsilon \\equiv\n \\tilde\\epsilon-\\tilde\\mu \\approx\n \\frac{\\pi^2\\tau^2}{6\\,\\tilde\\epsilon}\\,\n \\frac{1+2x_\\mathrm{r}^2}{\\gamma_\\mathrm{r}(1+\\gamma_\\mathrm{r})} \\, ,\n\\quad\n \\Delta F\\equiv F_\\mathrm{id}^{(e)} - F_0^{(e)}\n \\approx -\\frac12\\,TS_\\mathrm{id}^{(e)}\\approx\n -P_\\mathrm{r} V\\frac{x_\\mathrm{r}\\gamma_\\mathrm{r}\\tau^2}{6},\n\\label{DeltaEF}\n\\end{equation}\nwhere \n$\n F_0^{(e)} =\n ({ P_\\mathrm{r} V}\/{ 8\\pi^2 })\n [ x_\\mathrm{r}\\,(1+2x_\\mathrm{r}^2)\\,\\gamma_\\mathrm{r}\n - \\ln(x_\\mathrm{r}+\\gamma_\\mathrm{r}) ]-N_e m_e c^2 \n$\nis the zero-temperature limit \\cite{Frenkel} (without\nthe rest energy $N_e m_e c^2$),\nand\n$P_\\mathrm{r} \\equiv m_e c^2\\,(m_e c\/\\hbar)^3\n = 1.4218\\times 10^{25}\\mathrm{~dyn~cm}^{-2}$\nis the relativistic unit of pressure.\n\nAccordingly,\n$\nP_\\mathrm{id}^{(e)} \\approx\nP_0^{(e)}+\\Delta P,\n$\nwhere\n$\n P_0^{(e)} =\n ({P_\\mathrm{r}}\/{8\\pi^2})\\,\n \\left[x_\\mathrm{r}\\left(\\frac23\\,x_\\mathrm{r}^2-1\\right) \\gamma_\\mathrm{r}\n + \\ln(x_\\mathrm{r}+\\gamma_\\mathrm{r})\\right],\n$\nand $\\Delta P = (P_\\mathrm{r}\/18)\\,\\tau^2 x_\\mathrm{r}\\,(2+x_\\mathrm{r}^2)\/\\gamma_\\mathrm{r}$. In this case,\n$\\chi_\\rho^{(e)} \\approx\nP_\\mathrm{r}x_\\mathrm{r}^5\/9\\pi^2\\gamma_\\mathrm{r} P_\\mathrm{id}^{(e)},\n$\n$\n\\chi_T^{(e)} \\approx\n2\\Delta P\/P_\\mathrm{id}^{(e)},\n$\n$\nC_V^{(e)} \\approx \\pi^2 k_B\nN_e\\,\\gamma_\\mathrm{r}\\tau\/x_\\mathrm{r}^2.\n$\nIn order to minimize numerical\njumps at the transition between the fit at \n$\\chi_e<14$ and the Sommerfeld expansion at $\\chi_e>14$,\n we multiply the expressions for $\\Delta F$\nby empirical correction factor\n$(1+\\Delta\\tilde\\epsilon\/\\tilde\\epsilon)^{-1}$,\nand those for $\\chi_T^{(e)}$ and $C_V^{(e)}$ by\n$[1+(4-2x_\\mathrm{r}\/\\gamma_\\mathrm{r})\\,\\Delta\\tilde\\epsilon\/\\tilde\\epsilon]^{-1}$.\n\nAt $x_\\mathrm{r}<10^{-5}$ we replace $F_0^{(e)}$ and\n$P_0^{(e)}$ by their nonrelativistic limits,\n$F_0^{(e)}\/V = P_\\mathrm{r} x_\\mathrm{r}^5\/10\\pi^2$ and \n$P_0^{(e)} = P_\\mathrm{r}x_\\mathrm{r}^5\/15\\pi^2\\propto n_e^{5\/3}$. In the\nopposite case of $x_\\mathrm{r}\\gg1$, one has $P_0^{(e)} =\nP_\\mathrm{r}x_\\mathrm{r}^4\/12\\pi^2\\propto n_e^{4\/3}$.\n\n\\section{Electron exchange and correlation}\n\\label{sect:ee}\n\nElectron exchange-correlation effects were studied by\nmany authors. For the reasons explained in \\cite{CP98}, we\nadopt the fit to $F_{ee}$ presented in Ref.~\\cite{IIT}. It is\nvalid at any densities and temperatures, provided that \n$x_\\mathrm{r}\\ll1$.\n\nIn Ref.~\\cite{PC00} we implemented an interpolation between the\nnonrelativistic fit \\cite{IIT} and approximation \\cite{SB96,SB00}\nvalid for strongly degenerate\nrelativistic electrons. However, later on we found that such\nan interpolation may cause an unphysical behavior of the heat\ncapacity in a certain density-temperature domain. Therefore we\nhave reverted to the original formula \\cite{IIT},\ntaking into account that in applications, as long as the electrons\nare relativistic, their exchange and correlation contributions are \norders of magnitude smaller than the other contributions to the EOS of\nEIP (see, e.g., \\cite{YaSha,NSB1}).\n\n\\section{One-component plasma}\n\\label{sect:ii}\n\n\\begin{figure}\n\\begin{minipage}[t]{.48\\linewidth}\n \\includegraphics[width=\\linewidth]{iye_f09.ps}\n\\caption{Anharmonic contribution to the reduced free energy of an ion\nlattice as a function of the quantum parameter $\\eta=T_\\mathrm{p}\/T$ for\ntwo values of the Coulomb coupling parameter, $\\Gamma=175$ (upper\ncurves) and 1000 (lower curves).\nA comparison of different approximations (see text):\nWigner expansion (long-dashed lines); an approximation\nfrom \\cite{Iyetomi-OI} (IOI, dotted lines); the same\napproximation with the coefficients adjusted to\nRef.~\\cite{FaroukiHamaguchi} (short-dashed lines); and present\ninterpolation (\\ref{f_ah_fit}) (solid lines).\nThe points with errorbars show simulation results \nfrom \\cite{Iyetomi-OI} for $\\Gamma=1000$.\n\\label{fig:f_ah}\n}\n\\end{minipage}\n \\hfill\n\\begin{minipage}[t]{.48\\linewidth}\n \\includegraphics[width=\\linewidth]{iye_cv.ps}\n\\caption{Harmonic and anharmonic lattice contributions to the reduced heat\ncapacity at $\\Gamma=175$ and 1000.\nThe harmonic lattice contribution according to\nRef.~\\cite{Baiko-ea01} (dot-dashed lines) is compared to the\nmodel \\cite{Chabrier93} (long-short-dash lines) and to\nthe anharmonic correction in\ndifferent approximations (see text): \na derivative of the IOI fit \\cite{Iyetomi-OI} (dotted\nlines: original; short-dashed lines: improved; see text) and\na corresponding derivative of\n\\req{f_ah_fit} (solid lines). \n}\n\\label{fig:cv_ah}\n\\end{minipage}\n\\end{figure}\n\n\\subsection{Coulomb liquid}\n\nFor the reduced free energy of the ion-ion interaction\n$f_\\mathrm{ii}\\equiv {F_\\mathrm{ii}}\/{N_\\mathrm{i}k_B T}$\nin the liquid OCP at any values of $\\Gamma$ we use\nthe analytic approximation derived in Ref.~\\cite{PC00}.\nIn order to extend its applicability range from $T\\ggT_\\mathrm{p}$\nto $T\\simT_\\mathrm{p}$, we add the lowest-order\nquantum corrections to\nthe Helmholtz free energy\n\\cite{Hansen73}:\n$\n f_q^{(2)}=\\eta^2\/24.\n$\nThe next-order correction\n$\\propto\\hbar^4$ has been obtained in \\cite{HV75}.\nThese corrections have\nlimited applicability, because as soon as $\\eta$ becomes large,\nthe Wigner expansion diverges and\nthe plasma forms a quantum liquid, whose free energy\nis not known in an analytic form.\n\nA classical OCP freezes at temperature $T_\\mathrm{m}$ \ncorresponding to $\\Gamma=175$,\nbut in real plasmas $T_\\mathrm{m}$ is affected by electron polarization\n\\cite{PC00,Hama-Yukawa} and quantum effects\n(e.g., \\cite{NNN,Chabrier93,JonesCeperley96}).\nHence the liquid does not freeze, regardless of $T$, at\ndensities larger than the critical one. \nThe critical density values in the OCP correspond to\n$R_S\\approx 140$\\,--\\,160 for bosons and $R_S\\approx 90$\\,--\\,110\nfor fermions. In astrophysical applications, the appearance of\nquantum liquid can be important for hydrogen and helium\nplasmas (see, e.g., Ref.~\\cite{Chabrier93} for discussion).\n\n\\subsection{Coulomb crystal}\n\nThe reduced free energy of the Coulomb crystal is\n$\n f_\\mathrm{lat}\\equiv\nF_\\mathrm{lat}\/N_\\mathrm{i} k_B T = C_0\\Gamma + 1.5 u_1 \\eta + f_{\\mathrm{th}} +\n f_{\\mathrm{ah}} .\n$\nHere, the first term is the Madelung energy ($C_0\\approx-0.9$),\nthe second represents zero-point ion vibrations energy\n($u_1\\approx0.5$), $f_{\\mathrm{th}}$ is the thermal correction in\nthe harmonic approximation,\n and $f_{\\mathrm{ah}}$ is the anharmonic\ncorrection. We use the most accurate values of $C_0$, $u_1$, and\nanalytic\napproximations to $f_{\\mathrm{th}}$ for bcc and fcc Coulomb OCP\nlattices, which have been obtained in \\cite{Baiko-ea01}.\n\nFor the \\emph{classical anharmonic corrections}, 11\nfitting expressions were given in \\cite{FaroukiHamaguchi}. We have\nchosen one of them:\n$\n f_\\mathrm{ah}^{(0)}(\\Gamma) = a_1\/\\Gamma +a_2\/2\\Gamma^2 +\n a_3\/3\\Gamma^3,\n$\nwith $a_1=10.9$, $a_2=247$, and $a_3=1.765\\times10^5$, because\nthis choice is most consistent with the perturbation theory\n\\cite{NNN,Dubin}.\n\nIn applications one needs a continuous extension for the free\nenergy to $\\eta\\neq0$. \nWith the leading quantum anharmonic correction at small $\\eta$\none has \\cite{HV75}\n\\begin{equation}\n f_\\mathrm{ah} \\approx f_\\mathrm{ah}^{(0)}(\\Gamma)\n - ( 0.0018\/\\Gamma + 0.085\/\\Gamma^2)\\,\\eta^4.\n\\label{ah-sc}\n\\end{equation}\nAt $T\\to0$, the quantum anharmonic corrections \nwere studied in \\cite{Carr-ea,AlbersGubernatis,NNN},\nwhere an expansion in powers of $R_S^{-1\/2}$ was obtained,\nassuming $R_S\\gg1$. The leading\nterm of this expansion can be written in the form\n$\n f_{\\mathrm{ah},T\\to0} = -b_1\\eta^2\/\\Gamma.\n$\nIn the literature one finds different\nestimates for $b_1$; we use $b_1=0.12$ as an approximation\nconsistent with \\cite{Carr-ea,AlbersGubernatis}. \nFree and internal energies of\n\\emph{finite-temperature} quantum crystals were calculated using\nquantum Monte Carlo methods in \n\\cite{Iyetomi-OI,JonesCeperley96,Chabrier-DP}.\nIyetomi et al.\\ in Ref.~\\cite{Iyetomi-OI} (hereafter IOI)\nproposed\nan analytic expression\nfor the quantum anharmonic corrections.\nHowever, differences between the numerical results in\n\\cite{Iyetomi-OI,JonesCeperley96,Chabrier-DP}\nare comparable to the differences\nbetween the numerical results and the harmonic approximation. \nTherefore,\nfinite-temperature anharmonic corrections cannot be accurately determined\nfrom the listed results.\n\nIn order to reproduce the zero-temperature and classical limits,\nwe multiply $f_\\mathrm{ah}^{(0)}$ by exponential suppression factor\n$e^{-c_1\\eta^2}$ and add $f_{\\mathrm{ah},T\\to0}$.\nThen we have\n\\begin{equation}\n f_\\mathrm{ah} = f_\\mathrm{ah}^{(0)}(\\Gamma)\\,e^{-c_1\\eta^2}\n -b_1\\eta^2\/\\Gamma .\n\\label{f_ah_fit}\n\\end{equation}\nAccording to \\req{ah-sc},\nthe Taylor expansion coefficient at $\\eta^2\/\\Gamma$ equals zero.\nIn order to reproduce this property, we set \n$c_1=b_1\/a_1\\approx0.0112$.\nIn Fig.~\\ref{fig:f_ah}, the resulting approximation for the free\nenergy as a function of $\\eta$ (solid lines) is compared\n to the\nsemiclassical expression [Eq.~(\\ref{ah-sc}); long-dashed lines] and to\nthe elaborated IOI fitting formula:\nthe dotted lines correspond to the original coefficients\nof this fit, chosen by IOI so as to reproduce\nthe results of Ref.~\\cite{Dubin} at $\\eta\\to0$,\nwhile the short-dashed curves\nshow the same fit, but with the coefficients adjusted to\nmore recent results \\cite{FaroukiHamaguchi}\nat $\\eta\\to0$. The curves of the two latter types nearly coincide at\n$\\Gamma=1000$, but differ near the melting point $\\Gamma=175$.\n\nInterpolation (\\ref{f_ah_fit}) (unlike, for\nexample, Pad\\'e approximations \\cite{ChugunovBaiko}) does not produce an unphysical\nbehavior of thermodynamic functions. In particular, anharmonic\ncorrections to the heat capacity and entropy do not exceed\nthe reference (harmonic-lattice) values of these quantities\nat any $\\eta$.\nIn Fig.~\\ref{fig:cv_ah}, we\nshow the ion contributions to the reduced heat capacity\nof the ion lattice\n$c_{V,i}\\equiv C_{V,i}\/N_\\mathrm{i}k_B$, calculated through derivatives of\n\\req{f_ah_fit}\n(solid lines), and compare it to the contributions calculated\nthrough derivatives of the\nfitting formula in \\cite{Iyetomi-OI} (short-dashed and\ndotted lines). In the same figure we plot\nthe harmonic-crystal contribution to the reduced heat capacity:\nthe short-dash--long-dash line corresponds to the model of\nRef.~\\cite{Chabrier93} (which we adopted in \\cite{PC00}),\nwhile the dot-dashed line \ncorresponds to the most accurate formula \\cite{Baiko-ea01}.\n\nInterpolation \\req{f_ah_fit} should\nbe replaced by a more accurate formula in the future when\naccurate finite-temperature anharmonic quantum corrections become\navailable.\n\n\\section{Electron polarization}\n\\label{sect:ie}\n\n\\subsection{Coulomb liquid}\n\\label{sect:ie-liq}\n\nElectron polarization in Coulomb liquid was studied by\nperturbation \\cite{GalamHansen,YaSha} and HNC \n\\cite{C90,ChabAsh,CP98,PC00}\ntechniques. The results for $F_{ie}$\nare fitted by analytic expressions in \\cite{PC00}.\nThis fit is accurate within several\npercents, and it exactly\nrecovers the Debye-H\\\"uckel limit for EIP at $\\Gamma\\to0$ and\nthe Thomas-Fermi result\n\\cite{Salpeter61} at $\\Gamma\\gg1$ and $Z\\gg1$.\n\n\n\n\\subsection{Coulomb crystal}\n\\label{sect:ie-sol}\n\nCalculation of thermodynamic functions for a Coulomb crystal\nwith allowance for the electron polarization is a complex\nproblem. For classical ions, the simplest screening model\nconsists in replacing the Coulomb potential by the Yukawa\npotential. For instance,\nthere were molecular dynamics simulations of classical\nYukawa systems \n\\cite{Hama-Yukawa} and path-integral Monte Carlo simulations of\na quantum Yukawa crystal at $R_S=200$ \\cite{MilitzerGraham}. \nHowever, the Yukawa interaction\nreflects only the small-wavenumber asymptote of the electron\ndielectric function. A rigorous treatment would consist in\ncalculating the dynamical matrix and solving a corresponding\ndispersion relation for the phonon spectrum. The first-order\nperturbation approximation for the dynamical matrix of a\nclassical Coulomb solid with the polarization corrections was\ndeveloped in Ref.~\\cite{PollockHansen}. The phonon spectrum in\nsuch a quantum crystal has been calculated only in the harmonic\napproximation \\cite{Baiko02}, which has a restricted\napplicability: for example, it cannot reproduce\nthe known classical ($T\\gg T_\\mathrm{p}$) limit of the\nanharmonic $ie$ contribution\nto the heat capacity.\n\nA semiclassical perturbation approach was used in\n\\cite{PC00}. The results were fitted by\nthe analytic expression\n\\begin{equation}\n f_{ie} = -f_\\infty(x_\\mathrm{r})\\,\\Gamma\n \\left[ 1 + A(x_\\mathrm{r})\\,(Q(\\eta)\/\\Gamma)^s \\right].\n\\label{fitscr-sol}\n\\end{equation}\nIn the classical Coulomb crystal,\n$Q=1$,\n$\nf_\\infty(x) = b_1\\,\\sqrt{1+b_2\/x^2},\n$\nand\n$\nA(x) = (b_3+a_3 x^2 )\/( 1+b_4 x^2)$.\nParameters $s$ and $b_1$--$b_4$ depend on $Z$ and are chosen\nso as to fit the perturbational results; $a_3$ is a constant.\nAll the parameters weakly depend on the lattice type;\nthey are explicit in \\cite{PC00} for the bcc and fcc lattices.\n\nThe factor $Q(\\eta)$ in \\req{fitscr-sol} is designed\nto reproduce the suppression of the dependence of $F_{ie}$ on $T$\nat $T\\ll T_\\mathrm{p}$. For the classical solid, $Q(0)=1$. \nIn the quantum limit ($\\eta\\to\\infty$) $Q(\\eta) \\simeq\nq\\eta$, so that the ratio $Q\/\\Gamma$ in \\req{fitscr-sol}\nbecomes independent of $T$. The \nproportionality coefficient $q$ was found numerically\nin \\cite{PC00}; it equals 0.205 for the bcc lattice.\n\nThe form of $Q(\\eta)$ suggested in \\cite{PC00}\nassumed a too slow\ndecrease of the heat capacity ($C_{V,ie}\\propto \\eta^{-1}\\propto\nT$) at $\\eta\\to \\infty$, which signals the violation of the\nemployed semiclassical perturbation theory in the strong quantum\nlimit, related to the approximate form of the ion\nstructure factor used in the calculations, as discussed in\n\\cite{PC00,NSB1}. In order to fix this problem, we\nhave changed the form of $Q(\\eta)$ to\n\\begin{equation}\n Q(\\eta) = \\left[ { \\ln\\left(1+e^{(q\\eta)^2} \\right)\n }\\right]^{1\/2}\\,\\left[{\n \\ln\\left(e-(e-2)e^{-(q\\eta)^2} \\right)}\n \\right]^{-1\/2}.\n\\end{equation}\nThis form of $Q(\\eta)$ has the correct limits at $\\eta\\to 0$ and $\\eta\\to\n\\infty$ and is compatible with the numerical\nresults \\cite{PC00}. In addition, it eliminates the\nproblematic $ie$ contributions to the heat capacity and entropy\nat $\\eta\\gg1$. It can be\nimproved in the future when the polarization corrections for the\nquantum Coulomb crystal are accurately evaluated.\n\n\\section{Conclusions}\n\\label{sect:concl}\n\nWe have reviewed analytic approximations for the\nEOS of fully ionized electron-ion plasmas and described recent\nimprovements to the previously published approximations, taking\ninto account nonideality due to ion-ion, electron-electron, and\nelectron-ion interactions. The presented formulae are applicable\nin a wide range of plasma parameters, including the domains of\nnondegenerate and degenerate, nonrelativistic and relativistic\nelectrons, weakly and strongly coupled Coulomb liquids, classical\nand quantum Coulomb crystals.\n\nFor brevity we have considered plasmas composed of electrons and\none type of ions. Extension to the case where several types of\nions are present is provided by \n\\cite{PCR09,PCCDR09}.\n\nWe have made the Fortran code that realizes the analytical\napproximations for the free energy and its derivatives, described\nin this paper, freely available in the Internet\n\\footnote{http:\/\/www.ioffe.ru\/astro\/EIP\/}.\n\n\\begin{acknowledgement}\nThe work of A.Y.P.\\ was partially supported \nby the Rosnauka Grant NSh-2600.2008.2\nand the RFBR Grant 08-02-00837,\nand by CompStar, a Research Networking Programme of the\nEuropean Science Foundation. \n\\end{acknowledgement}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nThis paper is concerned with the following three\nmock theta functions of order 3 defined by Ramanujan,\n\\begin{align}\nf(q)&=\\sum_{n=0}^\\infty \\frac{q^{n^2}}{(-q;q)_n^2},\\\\[2pt]\n\\phi(q)&=\\sum_{n=0}^\\infty \\frac{q^{n^2}}{(-q^2;q^2)_n},\\\\[2pt]\n\\psi(q)&=\\sum_{n=1}^\\infty \\frac{q^{n^2}}{(q;q^2)_n}.\n\\end{align}\n\nMock theta functions have been extensively studied, see, for example,\n Andrews \\cite{Andrews-1989-283}, Fine \\cite[Chapters 2-3]{Fine-1988}, Gordon and McIntosh\n\\cite{Gordon-2010}, and Ono \\cite{Ono-2009}. These functions not\nonly have remarkable analytic properties, but also are closely\nconnected to the theory of partitions, see, for example, Agarwal\n\\cite{Agarwal-2004}, Andrews \\cite{Andrews-1966-2,\nAndrews-2005-4666}, Andrews and Garvan \\cite{ Andrews-1989-242},\nAndrews, Eriksson, Petrov, and Romik \\cite{Andrews-2007-545}, and\nChoi and Kim \\cite{Choi-2010-25}.\n\nIn this paper, we find two involutions on partitions that imply two\n partition identities for Ramanujan's third order mock theta\nfunctions $\\phi(-q)$ and $\\psi(-q)$.\nWe also give an involution for Fine's partition theorem on the mock theta function $f(q)$. These three partition identities lead to the following two\nidentities \\eqref{r-1} and \\eqref{r-2} of Ramanujan\n\\begin{align}\\label{r-1}\n&\\phi(-q)-2\\psi(-q)=f(q),\\\\\n&\\phi(-q)+2\\psi(-q)=\\frac{(q;q)_\\infty}{(-q;q)_\\infty^2},\\label{r-2}\n\\end{align}\nwhere we have adopted the standard notation\n\\begin{align}\n(a;q)_n&=(1-a)(1-aq)\\cdots(1-aq^{n-1}),\\\\\n(a;q)_\\infty&=\\prod_{n=0}^\\infty (1-aq^n),\\qquad |q|<1.\n\\end{align}\n The first proofs of \\eqref{r-1} and \\eqref{r-2} were given by Watson \\cite{Watson-1936}.\n Fine \\cite[p. 60]{Fine-1988} found another\n proof by using transformation formulas.\n\n\\medskip\n\n Andrews \\cite{Andrews-1966-64} defined the following functions as generalizations of Ramanujan's mock\n theta functions and he later found that these generalizations were already in Ramanujan's ``lost\" notebook \\cite{Andrews-1979-89},\n\\begin{align}\nf(\\alpha;q)&=\\sum_{n=0}^\\infty \\frac{q^{n^2-n}\\alpha^n}{(-q;q)_n(-\\alpha;q)_n},\\\\\n\\phi(\\alpha;q)&=\\sum_{n=0}^\\infty \\frac{q^{n^2}}{(-\\alpha q;q^2)_n},\\\\\n\\psi(\\alpha;q)&=\\sum_{n=1}^\\infty \\frac{q^{n^2}}{(\\alpha;q^2)_n}.\n\\end{align}\n\nWhen $\\alpha=q$, the above functions reduce to Ramanujan's mock\ntheta functions. Furthermore, Andrews showed these three functions\nturn out to be mock theta functions for $\\alpha=q^r$, where $r$ is any positive\ninteger. More importantly, Ramanujan's\nidentities \\eqref{r-1} and \\eqref{r-2} can be extended to\nthe functions $f(\\alpha;q)$, $\\phi(\\alpha;q)$ and $\\psi(\\alpha;q)$,\n\\begin{align}\\label{rg-1}\n\\phi(-\\alpha;-q)-(1+\\alpha q^{-1})\\psi(-\\alpha;-q)&=f(\\alpha;q),\\\\\n\\label{rg-2} \\phi(-\\alpha;-q)+(1+\\alpha\nq^{-1})\\psi(-\\alpha;-q)\n&=\\frac{(q;q)_\\infty}{(-q;q)_\\infty(-\\alpha;q)_\\infty},\n\\end{align}\nsee Andrews \\cite[p. 78, Eqs. (3a)--(3b)]{Andrews-1966-64}.\nClearly, the above identities \\eqref{rg-1} and \\eqref{rg-2}\nspecialize to \\eqref{r-1} and \\eqref{r-2} by setting $\\alpha=q$.\n\n\n\n\n\nThe connection between Ramanujan's third order mock theta function\nand the theory of partitions was first explored by Fine. In\n\\cite[p. 55, Chapter 3]{Fine-1988}, he derived the following\nidentity from his transformation formula:\n\\begin{equation}\\label{fine-1}\nf(q)=1+\\sum_{k\\geq\n1}\\frac{(-1)^{k-1}q^k}{(-q;q)_k}=1+\\frac{1}{(-q;q)_\\infty}\\sum_{k\\geq\n1}(-1)^{k-1}q^k(-q^{k+1};q)_\\infty.\n\\end{equation}\nIn fact, \\eqref{fine-1} can be easily established from the\ncombinatorial definition \\eqref{Int-f} of $f(q)$. The following partition identity\n for $f(q)$ can be deduced from \\eqref{fine-1}.\n\n\n \\begin{thm}[Fine]\\label{ParIden-1}Let\n$p_{do}(n)$ denote the number of partitions of $n$ into distinct\nparts with the smallest part being odd. Then\n \\begin{equation}\\label{pi-1}\n(-q;q)_\\infty f(q)=1+2\\sum_{n\\geq 1}p_{do}(n)q^n.\n\\end{equation}\n\\end{thm}\n\n\nWe obtain the following two partition identities\nfrom our involutions.\n\n\\begin{thm}\\label{add-1}We have\n\\begin{equation} (-q;q)\\phi(-q)=1+\\sum_{n=1}^\\infty\np_{do}(n)q^n+\\sum_{k=1}^\\infty (-1)^k q^{k^2}.\\label{add-1-e}\n\\end{equation}\n \\end{thm}\n\n \\begin{thm}\\label{add-2}We have\n\\begin{equation} 2(-q;q)\\psi(-q)=-\\sum_{n=1}^\\infty\np_{do}(n)q^n+\\sum_{k=1}^\\infty (-1)^k q^{k^2}.\\label{add-2-e}\n\\end{equation}\n \\end{thm}\n Since the generating function for $p_{do}(n)$ is easy to compute, it would be interesting to establish the\nabove relations as $q$-series identities without resort to\npartitions.\n\nIt can be seen that the above partition identities lead to\nRamanujan's identities \\eqref{r-1} and \\eqref{r-2}. It follows\nfrom \\eqref{add-1-e} and \\eqref{add-2-e} that\n\\begin{align*}\n&(-q;q)_\\infty \\phi(-q)-2(-q;q)_\\infty \\psi(-q)\\\\[2pt]\n&\\quad=1+\\sum_{n=1}^\\infty p_{do}(n)q^n+\\sum_{k=1}^\\infty (-1)^k\nq^{k^2}\\\\\n&\\quad \\quad \\quad +\\sum_{n=1}^\\infty\np_{do}(n)q^n-\\sum_{k=1}^\\infty (-1)^k q^{k^2}\\\\\n&\\quad =1+2\\sum_{n\\geq 1}p_{do}(n)q^n\\\\\n&\\quad =(-q;q)_\\infty f(q),\n\\end{align*}\nwhere the last equality is a consequence of \\eqref{pi-1}. So we\nobtain the identity \\eqref{r-1} by dividing both sides by\n$(-q;q)_\\infty$.\n\n\n\n\n\n\n In view of\n\\eqref{add-1-e} and \\eqref{add-2-e}, we find\n\\begin{align*}\n&(-q;q)_\\infty\\phi(-q)+2(-q;q)_\\infty \\psi(-q)\\\\\n&\\quad =1+\\sum_{n=1}^\\infty p_{do}(n)q^n+\\sum_{k=1}^\\infty (-1)^k\nq^{k^2}\\\\\n&\\quad \\quad \\quad -\\sum_{n=1}^\\infty\np_{do}(n)q^n+\\sum_{k=1}^\\infty (-1)^k q^{k^2}\\\\\n&\\quad =1+2\\sum_{k\\geq 1}(-1)^k\nq^{k^2}\\\\\n&\\quad =\\frac{(q;q)_\\infty}{(-q;q)_\\infty},\n\\end{align*}\nwhere the last equality follows from Gauss' identity\n\\begin{equation}\\label{Gauss}\n\\sum_{k=-\\infty }^\\infty\n(-1)^kq^{k^2}=\\frac{(q;q)_\\infty}{(-q;q)_\\infty}.\n\\end{equation}\nThis yields Ramanujan's identity \\eqref{r-2} after dividing both sides by $(-q;q)_\\infty$.\n\nIn fact, we can deduce two partition identities for $\\phi(-q)$ and $\\psi(-q)$ analogous to\nFine's identity for $f(q)$ by employing the following\npartition theorem of Bessenrodt and Pak\n\\cite{Bessenrodt-2004-1139} which extends a theorem of Fine in\n\\cite[Theorem 5]{Fine-1948}.\nIt is worth mentioning that there are other involutions which also\nimply this partition theorem, see, Berndt, Kim and Yee\n\\cite{Berndt-2010}, Chen and Liu \\cite{Chen}, and Yee\n\\cite{Yee-2010-a, Yee-2010-b}.\n\n\n\n\n\n\\begin{thm}[Bessenrodt and Pak]\\label{thm}Let $p^e_{do}(n)$ {\\rm(}$p^o_{do}(n)${\\rm)} denote the number of partitions of $n$\ninto even {\\rm(}odd\\,{\\rm)} distinct parts with the smallest part\nbeing odd. Then\n\\begin{equation}\n\\sum_{n=1}^\\infty [p^e_{do}(n)-p^o_{do}(n)]q^n=\\sum_{k=1}^\\infty\n(-1)^k q^{k^2}.\n\\end{equation}\n\\end{thm}\n\n\\medskip\n\nIn light of Theorem \\ref{thm}, we can restate Theorems\n\\ref{add-1} and \\ref{add-2} as follows.\n\n\\begin{thm}\\label{ParIden-2}\n We have\n\\begin{equation}\\label{pi-2}\n(-q;q)_\\infty \\phi(-q)=1+2\\sum_{n\\geq 1}p^e_{do}(n)q^n.\n\\end{equation}\n\\end{thm}\n\n\n\\begin{thm}\\label{ParIden-3}\nWe have\n\\begin{equation}\\label{pi-3}\n(-q;q)_\\infty \\psi(-q)=-\\sum_{n\\geq 1}p^o_{do}(n)q^n.\n\\end{equation}\n\\end{thm}\n\n\n\nThis paper is organized as follows. In Section 2, we provide an involution for Fine's theorem \\ref{ParIden-1}. In Sections\n3 and 4, we give two involutions that lead to partition\nidentities \\eqref{add-1-e} and \\eqref{add-2-e} for $\\phi(-q)$ and\n$\\psi(-q)$. Section 5 is devoted to partition\nidentities for $f(\\alpha q;q),\\phi(-\\alpha q;-q)$, and\n$\\psi(-\\alpha q;-q)$ based on our involutions, which imply Andrews' identities \\eqref{rg-1} and\n\\eqref{rg-2}.\n\n\n\\section{An involution for Fine's theorem}\n\n\nIn this section we give an involution for Fine's partition\ntheorem. Let $P$ denote the set of partitions, and let $D$\ndenote the set of partitions with distinct parts. The\n rank of a partition $\\lambda$, denoted by $r(\\lambda)$, is\ndefined as the largest part minus the number of parts, as\nintroduced by Dyson \\cite{Dyson-1944}. The empty partition is\nassumed to have rank zero. A pair of partitions $(\\lambda,\\mu)$ is\ncalled a bipartition of $n$ if $|\\lambda|+|\\mu|=n.$ Fine\n\\cite[p.49]{Fine-1988} found the following combinatorial\ninterpretation for $f(q)$:\n\\begin{equation}\\label{Int-f}\nf(q)=1+\\sum_{\\lambda \\in P}\n(-1)^{r(\\lambda)}q^{|\\lambda|},\n\\end{equation}\nfrom which we can construct an involution to prove Theorem\n\\ref{ParIden-1}.\n\n\\medskip\n\n {\\noindent \\it Proof of Theorem \\ref{ParIden-1}.} For a bipartition $(\\lambda,\\mu) \\in P\\times D$ , let $s(\\mu)$ denote\nthe smallest part of $\\mu$, $m(\\lambda)$ denote the number of\noccurrences of the largest part of $\\lambda$ and\n $l(\\lambda)$ denote the number of (positive) parts of $\\lambda$. Let $U$ be the set of\n two classes of bipartitions $(\\emptyset,\\mu)$ and bipartitions $(\\lambda,\\mu)$, where\n$\\lambda=(1,1,\\ldots,1)$ and $s(\\mu)>l(\\lambda)$, and let $V$ be\nthe set of\n bipartitions $(\\lambda,\\mu)\\in\nP\\times D$ of $n$ except for bipartitions in $U$. We shall\nconstruct an involution $\\Upsilon$ on the set $V$. The following\ntwo cases are considered.\n\n\\begin{itemize}\n\\item[(1)]If $s(\\mu)\\leq m(\\lambda)$, then delete the smallest\npart in $\\mu$ and add $1$ to each of the first $s(\\mu)$ parts\n$\\lambda_1,\\lambda_2,\\ldots, \\lambda_{s(\\mu)}$ of $\\lambda$.\n\\item[(2)]If $s(\\mu)> m(\\lambda)$, then subtract\n $1$ from each of the first $m(\\lambda)$ parts\n$\\lambda_1,\\lambda_2,\\ldots, \\lambda_{m(\\lambda)}$ of $\\lambda$\nand add a part of size $m(\\lambda)$ to $\\mu$.\n\\end{itemize}\n\nIt is easy to check that the above mapping is an involution.\nMoreover, $\\Upsilon$ changes the parity of the rank of $\\lambda$\nin $V$.\n\nLet $(\\lambda, \\mu)$ be a bipartition in $U$. It is easily seen\nthat if $l(\\lambda)$ is even, then $r(\\lambda)$ is odd. In this\ncase, we obtain a bipartition $(\\emptyset,\\nu)$ with $s(\\nu)$\nbeing even by moving all the parts of $\\lambda$ to $\\mu$ as a\nsingle part which cancels with $(\\lambda,\\mu)$. In the case that\n$l(\\lambda)$ is odd, we see that $r(\\lambda)$ is even. Thus we get\nbipartition $(\\emptyset,\\nu)$ with $s(\\nu)$ being odd by moving\nall the parts of $\\lambda$ to $\\mu$ as a single part. Now, we\nare left with two types of bipartitions $(\\emptyset,\\nu)$ such\nthat $s(\\nu)$ is odd, which correspond to the right side of\n\\eqref{pi-1}. This completes the proof. \\qed\n\n\nHere is an example. There are eight bipartitions\n$(\\lambda,\\mu)\\in P\\times D$ of $4$ with the\nrank of $\\lambda$ being even,\n$$((3),(1)),((2,1),(1)),((1,1,1),(1)),((1),(2,1)),((2,2),\\emptyset),\n((1),(3)),(\\emptyset,(3,1)),(\\emptyset,(4)).$$\n Meanwhile, there are\nsix bipartitions $(\\lambda,\\mu)\\in P\\times D$ of $4$ with\nthe rank of $\\lambda$ being odd,\n$$((4),\\emptyset),((3,1),\\emptyset),((2,1,1),\\emptyset),((2),(2)),\n((1,1),(2)),((1,1,1,1),\\emptyset),$$ and there is only one\npartition of $4$ into distinct parts with the smallest part being\nodd, i.e., $(3,1)$.\n\n\\noindent The involution $\\Upsilon$ gives the following\ncorrespondence:\n\\begin{align*}\n\\begin{array}{ccc}\n((3),(1))\\leftrightarrow ((4),\\emptyset),\n&((2,1),(1))\\leftrightarrow ((3,1),\\emptyset),\n&((1,1,1),(1))\\leftrightarrow ((2,1,1),\\emptyset),\\\\[6pt]\n((1),(2,1))\\leftrightarrow\n((2),(2)),&((2,2),\\emptyset)\\leftrightarrow ((1,1),(2)).\n\\end{array}\n\\end{align*}\n For the\nremaining four bipartitions $((1),(3))$, $(\\emptyset,(3,1))$,\n$(\\emptyset,(4))$, and $((1,1,1,1),\\emptyset)$, we can transform\n$((1),(3))$ to $(\\emptyset,(3,1))$ and transform\n$(\\emptyset,(4))$ to $((1,1,1,1),\\emptyset)$.\n\n\n\\section{A partition identity for $\\phi(-q)$}\n\nIn this section, we shall prove the partition identity for $\\phi(-q)$ as stated in Theorem \\ref{add-1}. Let us begin with an interpretation\nof $\\phi(-q)$ given by Fine \\cite[p.49]{Fine-1988}.\nLet $DO$ denote the set of partitions with\ndistinct odd parts. Fine showed that\n\\begin{equation}\\label{Int-phi}\n\\phi(-q)=1+\\sum_{\\lambda \\in\nDO}(-1)^{\\frac{\\lambda_1+1}{2}}q^{|\\lambda|}.\n\\end{equation}\n Note that Choi and Kim found another interpretation of\n $\\phi(q)$ in terms of $n$-color partitions\n \\cite[Theorem\n3.1]{Choi-2010-25}.\n\nHence we have\n\\begin{equation}\\label{add-3}\n(-q;q)_\\infty \\phi(-q)=\\sum_{\\mu \\in\nD}q^{|\\mu|}+\\sum_{(\\lambda,\\mu) \\in DO\\times\nD}(-1)^{\\frac{\\lambda_1+1}{2}}q^{|\\lambda|+|\\mu|}.\n\\end{equation}\n\nIn order to deal with the sum $\\sum_{k\\geq 1} (-1)^k q^{k^2}$ on\nthe right hand side of \\eqref{add-1-e}, special attention has to\nbe paid to certain\n bipartitions\n$Q_k=(2k-1,2k-3,\\ldots,1)$, which is a partition of $k^2$.\n\nLet $s_o(\\mu)$ ($s_e(\\mu)$) denote the smallest odd (even) part of\n$\\mu$, and let $W$ denote the set of bipartitions $(\\lambda, \\mu)$\nof $n$ in $DO\\times D$ except for those of the form $(Q_k,\n\\emptyset)$ and those bipartitions $(\\lambda,\\mu)=((1),\\mu)$ with\n$s_o(\\mu)+11$, then subtract $2$ from each of the first $c(\\lambda)$ parts $\\lambda_1,\\lambda_2,\\ldots,\\lambda_{c(\\lambda)}$ of $\\lambda$, and add a part of size $2c(\\lambda)$ to $\\mu$.\n\nIt is easy to see the above process is well defined and bipartitions which can not be paired by the\n involution are those bipartitions\n $(Q_k, \\mu)$ where $s_e(\\mu)>2k$. This is the task of the second part of the involution.\n\n\\noindent Part II of $\\Phi$. If $s_e(\\mu)> s_o(\\mu)+(2k-1)$, then\ndelete the smallest odd part of $\\mu$ and delete the part $2k-1$\nfrom $Q_k$. Then\n add an even part of size $s_o(\\mu)+(2k-1)$ to $\\mu$.\n\nIf $s_e(\\mu)\\leq s_o(\\mu)+(2k-1)$, then split the smallest even part of $\\mu$ into two parts, one is of size $2k+1$\nand the other is of size $s_e(\\mu)-(2k+1)$. Observe that $s_e(\\mu)-(2k+1)$ is smaller than $s_o(\\mu)$. Then add a part of size $2k+1$ to $Q_k$ and add a part of size $s_e(\\mu)-(2k+1)$\nto $\\mu$.\n\n So we have obtained an involution $\\Phi$. It is readily\nseen that this involution changes the parity of $(\\lambda_1+1)\/2$.\n\nFor example, when $n=7$, there are $6$ bipartitions\n$(\\lambda,\\mu)\\in DO\\times D$ with $\\lambda_1 \\equiv 1\\bmod{4}$,\n$$((5),(2)),((5,1),(1)),((1),(6)),((1),(4,2)),((1),(3,2,1)),((1),(5,1)).$$\nOn the other hand, there are $5$ bipartitions $(\\lambda,\\mu)\\in\nDO\\times D$ with $\\lambda_1 \\equiv 3\\bmod{4}$, that is,\n$$((7),\\emptyset),((3,1),(2,1)),((3,1),(3)),((3),(4)),((3),(3,1)),$$\nand there is only one partition of $7$ into distinct parts with\nthe smallest part being even, namely, $(5,2)$.\n\n The involution $\\Phi$ gives the following pairs\n of bipartitions:\n\\begin{align*}\n\\begin{array}{ccc}\n((5),(2))\\leftrightarrow ((7),\\emptyset),\n&((5,1),(1))\\leftrightarrow ((3,1),(2,1)),\n &((1),(6))\\leftrightarrow ((3,1),(3)),\\\\[6pt]\n((1),(4,2))\\leftrightarrow ((3),(4)),\n&((1),(3,2,1))\\leftrightarrow ((3),(3,1)).\n\\end{array}\n\\end{align*}\nFor the remaining bipartition $((1),(5,1))$, we can construct a partition into\ndistinct part with smallest part being even, that is, $(5,2)$.\n\n\n\n\n\\section{A partition identity for $\\psi(-q)$}\n\n\nThe aim of this section is to prove the partition identity\n\\eqref{add-2-e} for $\\psi(-q)$. There is also a combinatorial\ninterpretation of $\\psi(-q)$ given by Fine \\cite[p.49]{Fine-1988}.\nLet $OC$ denote the set of partitions of $n$ into odd parts\nwithout gaps. Fine \\cite[p.49]{Fine-1988} showed that\n\\begin{equation}\\label{Int-psi}\n\\psi(-q)=\\sum_{\\lambda \\in\nOC}(-1)^{l(\\lambda)}q^{|\\lambda|}.\n\\end{equation}\nNote that Agarwal \\cite{Agarwal-2004, Agarwal-2007} found two combinatorial interpretations for $\\psi(q)$ by using $q$-difference equations.\n\nLet $D^0$ denote the set of partitions with distinct parts where the zero\npart is allowed. So\n the number of partitions of $n$ in the\nset $D^0$ is twice the number of partitions of $n$ in the set\n$D$. Let $R$ denote the set of bipartitions $(\\lambda,\\mu)\\in\nOC\\times D^0$ except for those of the form $(Q_k, \\emptyset)$ and\nthose bipartitions $(\\lambda,\\mu)=((1),\\mu)$ with\n$s_e(\\mu)+1 r_p(\\lambda)$, then\ndelete one part of size $r_p(\\lambda)$ from $\\lambda$ and add it\nas a part to $\\mu$. On the other hand, if $s_o(\\mu)\\leq\nr_p(\\lambda)$, then move a part of size $s_o(\\mu)$ from $\\mu$ to\n$\\lambda$.\n\nThe above process is well defined and the bipartitions not covered by this case are those bipartitions $(Q_k, \\mu)$ for which $s_o(\\mu)>2k-1.$\n We continue to describe the second part\n of $\\Psi$.\n\n \\noindent\n Part II of $\\Psi$.\n Assume that $(Q_k,\\mu)$ is a bipartition such that $s_o(\\mu)>2k-1$.\nIf $\\mu$ has a zero part, then we get a bipartition\n$(Q_{k-1},\\mu^*)$ where $\\mu^*$ is obtained from $\\mu$ by adding a\npart of size $2k-1$ and deleting the zero part.\n\nIf $\\mu$ has no zero part and $s_o(\\mu)=2k+1$, then we get a\nbipartition $(Q_{k+1},\\mu^*)$ where $\\mu^*$ is obtained from $\\mu$\nby removing a part of size $2k+1$ and adding a zero part.\n\nIt can be seen that the above mapping is well defined except for those bipartitions $(Q_k, \\mu)$ such that $\\mu$ has\nno zero part and $s_o(\\mu)>2k+1$.\nIndeed, it is the object of the third part of\n$\\Psi$ to deal with these remaining bipartitions.\n\n\n\\noindent Part III of $\\Psi$.\nIf $s_o(\\mu)> s_e(\\mu)+(2k-1)$, then delete the smallest even part of $\\mu$ and delete the part $2k-1$ from $Q_k$,\n and add an odd part of size $s_e(\\mu)+(2k-1)$ to $\\mu$.\n\n\nIf $s_o(\\mu)\\leq s_e(\\mu)+(2k-1)$, then split the\nsmallest odd part of $\\mu$ into two parts, one is of size $2k+1$\nand the other is of size $s_o(\\mu)-(2k+1)$, which is less than\n$s_e(\\mu)$, add a part of size $2k+1$ to $Q_k$ and add a\npart of size $s_o(\\mu)-(2k+1)$ to $\\mu$.\n\n\nIt is routine to check that the map $\\Psi$ is\nan involution and it changes the parity of the length of $\\lambda$.\n\n\nFor example, when $n=4$ there are six\nbipartitions $(\\lambda,\\mu)\\in OC\\times D^0$\nsuch that $l(\\lambda)$ is odd,\n$$((1),(3)),((1,1,1),(1)),((1,1,1),(1,0)),((1),(2,1)),((1),(2,1,0)),((1),(3,0)).$$\nIn the other case, there are five bipartitions $(\\lambda,\\mu)\\in\nOC\\times D^0$ such that $l(\\lambda)$ is even,\n$$((3,1),(0)),((1,1,1,1),\\emptyset),((1,1,1,1),(0)),((1,1),(2)),((1,1),(2,0)),$$\nand there is one partition of $4$ with distinct parts such that the\nsmallest part is odd, i.e., $(3,1)$.\n\nThe involution $\\Psi$ is illustrated below: \\begin{align*}\n\\begin{array}{lll}\n((1),(3))\\leftrightarrow ((3,1),(0)),\n&((1,1,1),(1))\\leftrightarrow ((1,1,1,1),\\emptyset),\n \\ ((1),(2,1))\\leftrightarrow ((1,1),(2))\\\\[6pt]\n((1),(2,1,0))\\leftrightarrow ((1,1),(2,0)),&\n((1,1,1),(1,0))\\leftrightarrow ((1,1,1,1),(0)).\n\\end{array}\n\\end{align*}\nFor the remaining bipartition $((1),(3,0))$, we can form a partition into\ndistinct parts with smallest part being odd, that is, $(3,1)$.\n\nAs another example, the involution $\\Psi$ also gives the\nfollowing correspondence:\n$$((9,7,5,3,1),(16,15,8,6,2))\\leftrightarrow ((7,5,3,1),(16,15,11,8,6)).$$\n\n\n\n\n\n\n\n\\section{Andrews' generalizations}\n\nThis section is devoted to proofs of Andrews' identities\n\\eqref{rg-1} and \\eqref{rg-2}. First, we give combinatorial\ninterpretations for $f(\\alpha q;q), \\phi(-\\alpha q;-q),$ and\n$\\psi(-\\alpha q;-q)$ by extending the arguments of Fine. More\nprecisely, we have the following partition theoretic\ninterpretations.\n\n\\begin{thm}\\label{int-g}We have\n\\begin{align}\\label{Int-f-g}\nf(\\alpha q;q)&=1+\\sum_{\\lambda \\in P} (-1)^{r(\\lambda)}\n \\alpha^{\\lambda_1}q^{|\\lambda|},\\\\ \\label{Int-phi-g}\n \\phi(-\\alpha q;-q)&=1+\\sum_{\\lambda \\in\nDO}\n(-1)^{\\frac{\\lambda_1+1}{2}}\\alpha^{\\frac{\\lambda_1+1}{2}-l(\\lambda)}q^{|\\lambda|},\\\\\n\\label{Int-psi-g}\n \\psi(-\\alpha q;-q)&=\\sum_{\\lambda \\in\nOC}(-1)^{l(\\lambda)} \\alpha^{l(\\lambda)-\\frac{\\lambda_1+1}{2}}\nq^{|\\lambda|}.\n\\end{align}\n\\end{thm}\n\n\\pf Recall that\n$$f(\\alpha q;q)=\\sum_{n=0}^\\infty \\frac{q^{n^2}\\alpha^n}{(-q;q)_n(-\\alpha q;q)_n},$$\nit is easy to check that\n\\eqref{Int-f-g} follows from the\nDurfee square dissection of a\npartition $\\lambda \\in P$, see Figure 1.\n\n\\begin{figure}[h]\n\\begin{center}\n\\begin{picture}(220,160)\n\\put(20,140){\\line(1,0){60}} \\put(20,140){\\line(0,-1){60}}\n\\put(20,80){\\line(1,0){60}} \\put(80,140){\\line(0,-1){60}}\n\\put(85,140){\\line(1,0){100}} \\put(185,140){\\line(0,-1){20}}\n\\put(185,120){\\line(-1,0){20}} \\put(165,120){\\line(0,-1){20}}\n\\put(165,100){\\line(-1,0){40}} \\put(125,100){\\line(0,-1){20}}\n\\put(125,80){\\line(-1,0){40}} \\put(85,80){\\line(0,1){60}}\n\\put(20,75){\\line(0,-1){60}} \\put(20,75){\\line(1,0){60}}\n\\put(80,75){\\line(0,-1){20}} \\put(80,55){\\line(-1,0){20}}\n\\put(60,55){\\line(0,-1){20}} \\put(60,35){\\line(-1,0){20}}\n\\put(40,35){\\line(0,-1){20}} \\put(40,15){\\line(-1,0){20}}\n\\put(40,110){$\\alpha^n q^{n^2}$} \\put(22,145){$\\alpha$}\n\\put(72,145){$\\alpha$} \\put(60,145){$\\alpha$}\n\\put(40,145){$\\cdots$}\n\n\\put(87,145){$-\\alpha$} \\put(117,145){$\\cdots$}\n\\put(147,145){$-\\alpha$} \\put(167,145){$-\\alpha$} \\put(3,15){$-1$}\n\\put(3,65){$-1$} \\put(3,30){$-1$} \\put(11,40){$\\cdot$}\n\\put(11,45){$\\cdot$} \\put(11,50){$\\cdot$} \\put(95,110){$1\/(-\\alpha\nq;q)_n$} \\put(22,50){$\\frac{1}{(-q;q)_n}$}\n\\end{picture}\n\\caption{The Durfee square dissection.}\n\\end{center}\n\\end{figure}\n\n\nFrom the definition of $\\phi(\\alpha;q)$, we see that\n$$\\phi(-\\alpha q;-q)=\\sum_{n=0}^\\infty\n\\frac{(-1)^nq^{n^2}}{(-\\alpha q^2;q^2)_n}.$$ The term\n$(-1)^nq^{n^2}$ corresponds to a partition $\\pi$ of the form\n$Q_n=(2n-1,2n-3,\\ldots,3,1)$, which has weight\n$(-1)^{(\\pi_1+1)\/2}$. Moreover, $1\/(-\\alpha q^2;q^2)_n$\n is the generating function for partitions $\\sigma$\nwith at most $n$ even parts and with no odd parts. The weight of such a partition is endowed with weight $(-\\alpha)^{\\sigma_1\/2}$.\n\n Define\n $\\lambda=\\pi+\\sigma=(\\pi_1+\\sigma_1, \\pi_2+\\sigma_2, \\ldots)$. We see that $\\lambda \\in DO$, namely, $\\lambda$ is a partition into distinct odd parts. Now, the weight of $\\lambda$ equals $(-1)^{(\\lambda_1+1)\/2}\\alpha^{(\\lambda_1\n+1)\/2-l(\\lambda)}$. So\n \\eqref{Int-phi-g} has been verified.\n\nFor the combinatorial interpretation for $\\psi(-\\alpha q;-q)$, we\nnote that\n$$\\psi(-\\alpha q;-q)=\\sum_{n=1}^\\infty \\frac{(-1)^nq^{n^2}}{(-\\alpha q;q^2)_n}.$$\n The summand can be expanded as follows\n\\begin{align*}\n\\frac{(-1)^nq^{n^2}}{(-\\alpha q;q^2)_n}&=\\frac{-q}{1+\\alpha\nq}\\frac{-q^3}{1+\\alpha q^3}\\cdots \\frac{-q^{2n-1}}{1+\\alpha\nq^{2n-1}}\\\\\n&=(-q+\\alpha q^{1+1}-\\alpha^2 q^{1+1+1}\\cdots)(-q^3+\\alpha\nq^{3+3}-\\alpha^2 q^{3+3+3}+\\cdots)\\\\\n&\\qquad \\cdots (-q^{2n-1}+\\alpha q^{2(2n-1)}-\\alpha^2\nq^{3(2n-1)}\\cdots).\n\\end{align*}\n It follows that the summand $(-1)^nq^{n^2}\/(-\\alpha q;q^2)_n$ is the generating function of\npartitions $\\lambda$ in $OC$ with the largest part not exceeding\n$2n-1$ and with weight $\n(-1)^{l(\\lambda)}\\alpha^{l(\\lambda)-(\\lambda_1+1)\/2}$. This proves\n(\\ref{Int-psi-g}). \\qed\n\n\n\nWe can extend Fine's partition theorem to Andrews' function\n$f(\\alpha q;q)$. It can be seen that the involution $\\Upsilon$ in\nSection 2 preserves the quantity $\\lambda_1+l(\\mu)$. Therefore,\nfrom \\eqref{Int-f-g} we deduce the following partition theorem.\n\n \\begin{thm}\\label{ParIden-1-g}Let $P_{do}$ denote the set of\n partitions into distinct parts with the smallest part being odd.\n Then\n \\begin{equation}\\label{pi-1-g}\n (-\\alpha q;q)_\\infty f(\\alpha q;q)=1+2\\sum_{\\nu \\in P_{do}}\\alpha^{l(\\nu)}q^{|\\nu|}.\n \\end{equation}\n\\end{thm}\n\n\nNext, we give a generalization of Theorem \\ref{add-1} to\n$\\phi(-\\alpha q;-q)$. By the combinatorial interpretation\n\\eqref{Int-phi-g}, we find that\n\\begin{align}\n(-\\alpha q;q)_\\infty\\phi(-\\alpha q;-q)&=(-\\alpha q;q)_\\infty+\\sum_{(\\lambda,\\mu)\n\\in DO\\times D}(-1)^{\\frac{\\lambda_1+1}{2}}\\alpha^{\\frac{\\lambda_1+1}{2}-l(\\lambda)\n+l(\\mu)}q^{|\\lambda|+|\\mu|}.\n\\end{align}\nMoreover, we observe that the involution $\\Phi$ in Section 3\npreserves the quantity of\n\\[ l(\\mu)-l(\\lambda)+\\frac{\\lambda_1+1}{2}.\\] Hence we have\n\\begin{equation}\\label{temp-3}\n\\sum_{(\\lambda,\\mu) \\in DO\\times D}(-1)^{\\frac{\\lambda_1+1}{2}}\n\\alpha^{\\frac{\\lambda_1+1}{2}-l(\\lambda)+l(\\mu)}q^{|\\lambda|+|\\mu|}\n=-\\sum_{\\nu \\in P_{de}}\\alpha^{l(\\nu)}q^{|\\nu|}+\\sum_{k=1}^\\infty (-1)^k q^{k^2},\n\\end{equation}\nwhere $P_{de}$ denotes the set of partitions into distinct parts\nwith the smallest part being even. On the other hand,\n\\begin{equation}\\label{temp-4}\n(-\\alpha q;q)_\\infty=1+\\sum_{\\nu \\in P_{de}}\\alpha^{l(\\nu)}q^{|\\nu|}+\\sum_{\\nu \\in P_{do}}\\alpha^{l(\\nu)}q^{|\\nu|},\n\\end{equation}\nTherefore, from \\eqref{temp-3} and \\eqref{temp-4} we deduce the following partition identity for\n$\\phi(-\\alpha q;-q)$.\n\n\\begin{thm}\\label{ParIden-2-g}We have\n\\begin{equation}\\label{pi-2-g}\n(-\\alpha q;q)_\\infty\\phi(-\\alpha q;-q)=1+\\sum_{\\nu \\in P_{do}}\\alpha^{l(\\nu)}q^{|\\nu|}+\\sum_{k=1}^\\infty (-1)^k q^{k^2}.\n\\end{equation}\n\\end{thm}\n\nWe now proceed to give a generalization of Theorem \\ref{add-2} to\n$\\psi(-\\alpha q;-q)$. By the combinatorial interpretation\n\\eqref{Int-psi-g}, we obtain that\n\\begin{equation}\n(-\\alpha;q)_\\infty\\psi(-\\alpha q;-q)=\\sum_{(\\lambda,\\mu) \\in\nDO\\times D^0} (-1)^{l(\\lambda)}\\alpha^{l(\\lambda)-\n\\frac{\\lambda_1+1}{2}+l(\\mu)}q^{|\\lambda|+|\\mu|}.\n\\end{equation}\n\nIt can be verified that the involution $\\Psi$ in Section 4 preserves the quantity of\n\\[ l(\\lambda)-\\frac{\\lambda_1+1}{2}+l(\\mu)\\]\n and it changes the parity of $l(\\lambda)$.\nSo we get the following partition theorem.\n\n\\begin{thm}\\label{ParIden-3-g}We have\n\\begin{equation}\\label{pi-3-g}\n(-\\alpha;q)_\\infty\\psi(-\\alpha q;-q)=-\\sum_{\\nu \\in P_{do}}\\alpha^{l(\\nu)}q^{|\\nu|}+\\sum_{k=1}^\\infty (-1)^k q^{k^2}.\n\\end{equation}\n\\end{thm}\n\nBased on the above partition theorems for $f(\\alpha\nq;q),\\,\\phi(-\\alpha q;-q)$ and $\\psi(-\\alpha q;-q)$, we can deduce\nAndrews' generalizations of Ramanujan's identities. More\nprecisely, it follows from \\eqref{pi-2-g} and \\eqref{pi-3-g} that\n\\begin{align*}\n&(-\\alpha q;q)_\\infty\\phi(-\\alpha q;-q)-(-\\alpha;q)_\\infty\\psi(-\\alpha q;-q)\\\\[2pt]\n&\\quad =1+\\sum_{\\nu \\in P_{do}}\\alpha^{l(\\nu)}q^{|\\nu|}+\\sum_{k=1}^\\infty (-1)^k q^{k^2}\\\\\n&\\quad \\qquad +\\sum_{\\nu \\in P_{do}}\\alpha^{l(\\nu)}q^{|\\nu|}-\\sum_{k=1}^\\infty (-1)^k q^{k^2}\\\\\n&\\quad =1+2\\sum_{\\nu \\in P_{do}}\\alpha^{l(\\nu)}q^{|\\nu|}\\\\\n&\\quad = (-\\alpha q;q)_\\infty f(\\alpha q;q),\n\\end{align*}\nwhere the last equality is a consequence of identity\n\\eqref{pi-1-g}. Dividing both sides by $(-\\alpha q ;q)_\\infty$ yields\n\\[\\phi(-\\alpha q;-q)-(1+\\alpha)\\psi(-\\alpha q;-q)=f(\\alpha q;q).\n\\]\nHence we deduce the identity \\eqref{rg-1} by replacing $\\alpha\nq$ by $\\alpha$.\n\n\n\n\nAccording to \\eqref{pi-2-g} and \\eqref{pi-3-g}, we have\n\\begin{align*}\n&(-\\alpha q;q)_\\infty\\phi(\\alpha q;-q)+(-\\alpha;q)_\\infty\\psi(\\alpha q;-q)\\\\[2pt]\n&\\quad =1+\\sum_{\\nu \\in P_{do}}\\alpha^{l(\\nu)}q^{|\\nu|}+\\sum_{k=1}^\\infty (-1)^k q^{k^2}\\\\\n&\\qquad \\quad -\\sum_{\\nu \\in P_{do}}\\alpha^{l(\\nu)}q^{|\\nu|}+\\sum_{k=1}^\\infty (-1)^k q^{k^2}\\\\\n&\\quad =1+2\\sum_{k=1}^\\infty (-1)^k q^{k^2}\\\\\n&\\quad =\\frac{(q;q)_\\infty}{(-q;q)_\\infty},\n\\end{align*}\nwhere the last equality follows from Gauss' identity\n\\eqref{Gauss}. Dividing both sides by $(-\\alpha q ;q)_\\infty$, we\nobtain\n\\[\\phi(-\\alpha q;-q)+(1+\\alpha)\\psi(-\\alpha q;-q)=\\frac{(q;q)_\\infty}{(-q;q)_\\infty(-\\alpha q;q)_\\infty},\n\\]\nwe arrive at the identity \\eqref{rg-2} by replacing\n$\\alpha q$ by $\\alpha$.\n\n\nTo conclude, we note that our approach can be\n viewed as combinatorial proofs of Andrews' identities in the forms multiplied by the factor $(-\\alpha q ;q)_\\infty$.\n\n\n\\noindent{\\bf Acknowledgments.} This work was supported by the 973\nProject, the PCSIRT Project and the Doctoral Program Fund of the Ministry of Education, the\nNational Science Foundation of China.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Mode expansion on the hyperceylinder}\n\\label{sec:apA}\n\nWe define in this appendix the inner product for the modes (\\ref{eq:uklm}).\nConsider the 2-form $\\omega$ on the space of solutions of the Klein-Gordon equation in spherical coordinate (\\ref{eq:KGhyp}), associated with the hypersurface composed by two hypercylinders of radii $\\varrho_1$ and $\\varrho_2$, with a bottom cover at $t=t_1$,\n\\bea\n\\omega(u_{k,l,m},u_{k',l',m'})&=& \\int_{t_1}^{\\infty} \\xd t \\int \\xd \\Omega \\, \\sqrt{g^{(3)}_r(t,\\Omega)} \\, \\left( \\overline{u_{l,m,n}}(t,r, \\Omega) \\stackrel{\\leftrightarrow}{\\partial_r} u_{k',l',m'}(t,r, \\Omega)\\right)\\bigg|^{r=\\varrho_2}_{r=\\varrho_1} \\nonumber\\\\\n&+& \\int_{\\varrho_1}^{\\varrho_2} \\xd r \\, \\int \\xd \\Omega \\, \\sqrt{g^{(3)}_t(r,\\Omega)} \\,\n \\left( \\overline{u_{k,l,m}}(t,r, \\Omega) \\stackrel{\\leftrightarrow}{\\partial_t} u_{k',l',m'}(t,r, \\Omega) \\right)\\bigg|_{t=t_1},\n\\label{eq:ap01}\n\\eea\nwhere $g^{(3)}_r(t,\\Omega)$ and $g^{(3)}_t(r,\\Omega)$ are the 3-metrics induced of the hypercylinder of radius $r$ and the disk at time $t$ respectively; also we adopted the following notation,\n\\be\n\\left( \\overline{u_{k,l,m}}(t,r, \\Omega) \\stackrel{\\leftrightarrow}{\\partial_t} u_{k',l',m'}(t,r, \\Omega) \\right) = \\overline{u_{k,l,m}}(t,r, \\Omega) \\, \\frac{\\xd}{\\xd t} u_{k',l',m'}(t,r, \\Omega) - u_{k',l',m'}(t,r, \\Omega) \\, \\frac{\\xd}{\\xd t} \\overline{u_{k,l,m}}(t,r, \\Omega).\n\\ee\nThe substitution in (\\ref{eq:ap01}) the expression (\\ref{eq:uklm}) of the modes $u_{k,l,m}$ gives\n\\bea\n\\omega(u_{k,l,m},u_{k',l',m'})&=& R^2 \\int_{t_1}^{\\infty} \\xd t \\int \\xd \\Omega \\, \\overline{Y_l^m(\\Omega)} \\, Y_{l'}^{m'}(\\Omega) \\, t \\, \\overline{H_{\\nu}}(k t) \\, H_{\\nu}(k' t) \\, \\varrho_2^2 \\nonumber\\\\\n&\\times & \\left( (\\overline{c_1} \\, j_l(kr) + \\overline{c_2} \\, n_l(kr)) \\stackrel{\\leftrightarrow}{\\partial_r} (c_1 j_{l'}(k' r) + c_2 n_{l'}(k' r)) \\right)\\bigg|_{r=\\varrho_2} \n\\nonumber\\\\\n&-& R^2 \\int_{t_1}^{\\infty} \\xd t \\int \\xd \\Omega \\, \\overline{Y_l^m(\\Omega)} \\, Y_{l'}^{m'}(\\Omega) \\, t \\, \\overline{H_{\\nu}}(k t) \\, H_{\\nu}(k' t) \\, \\varrho_1^2 \\nonumber\\\\\n&\\times & \\left( (\\overline{c_1} \\, j_l(kr) + \\overline{c_2} \\, n_l(kr)) \\stackrel{\\leftrightarrow}{\\partial_r} (c_1 j_{l'}(k' r) + c_2 n_{l'}(k' r)) \\right)\\bigg|_{r=\\varrho_1}\n\\nonumber\\\\\n&+& R^2 \\int_{\\varrho_1}^{\\varrho_2} \\xd r \\, r^2 \\, \\int \\xd \\Omega \\, \\overline{Y_l^m(\\Omega)} \\, Y_{l'}^{m'}(\\Omega) \n(\\overline{c_1} \\, j_l(kr) + \\overline{c_2} \\, n_l(kr)) (c_1 j_{l'}(k' r) + c_2 n_{l'}(k' r))\n\\nonumber\\\\\n&\\times & \nt_1 \\left( \\overline{H_{\\nu}}(k t) \\stackrel{\\leftrightarrow}{\\partial_t} H_{\\nu}(k' t)\\right)\\bigg|_{t=t_1}.\n\\label{eq:ap1}\n\\eea\nThe integration in the angular variables gives $\\delta_{l,l'} \\delta_{m,m'}$. We then notice that \n\\begin{multline}\n\\int_{\\varrho_1}^{\\varrho_2} \\xd r \\, r^2 \\,(\\overline{c_1} \\, j_l(kr) + \\overline{c_2} \\, n_l(kr)) (c_1 j_l(k' r) + c_2 n_l(k' r)) = \\\\\n\\frac{r^2}{k^2-k'^2} \\left( (\\overline{c_1} \\, j_l(kr) + \\overline{c_2} \\, n_l(kr)) \\stackrel{\\leftrightarrow}{\\partial_r} (c_1 j_l(k' r) + c_2 n_l(k' r)) \\right)\\bigg|_{r=\\varrho_1}^{r=\\varrho_2},\n\\end{multline}\nwhere we used the formula 5.11.8 of \\cite{Wat:bessel}. We also can rewrite the last line in (\\ref{eq:ap1}) as\n\\bea\nt_1 \\left( \\overline{H_{\\nu}}(k t) \\stackrel{\\leftrightarrow}{\\partial_t} H_{\\nu}(k' t)\\right)\\bigg|_{t=t_1} &=&\n(k^2-k'^2) \\int^{t_1} \\xd t \\, t\\, \\overline{H_{\\nu}}(k t) \\, H_{\\nu}(k' t), \\nonumber\\\\\n&=& (k^2-k'^2) \\, F \\left[\\, \\overline{H_{\\nu}}, H_{\\nu}, k,k' \\right](t_1),\n\\eea\nwhere we have introduced the notation\n\\be\nF \\left[\\, \\overline{H_{\\nu}}, H_{\\nu}, k,k' \\right](t_1) = \n\\frac{t_1}{k^2-k'^2} \\left( k' \\, \\overline{H_{\\nu}}(k t) \\, H_{\\nu-1}(k' t) - k \\, \\overline{H_{\\nu-1}}(k t) \\, H_{\\nu}(k' t) \\right).\n\\ee\nThe integral of product of Hankel functions can be written as\n\\be\n\\int_{t_1}^{\\infty} \\xd t \\, t \\, \\overline{H_{\\nu}}(k t) \\, H_{\\nu}(k' t) = \\frac{2}{\\sqrt{kk'}} \\delta(k-k') - F \\left[\\, \\overline{H_{\\nu}}, H_{\\nu}, k,k' \\right](t_1).\n\\ee\nCombining all this in (\\ref{eq:ap1}) yields\n\\bea\n\\omega(u_{k,l,m},u_{k',l',m'})&=& \\delta_{l,l'} \\, \\delta_{m,m'} \\, \\frac{2 R^2 }{\\sqrt{kk'}} \\, \\delta(k-k') \\, r^2 \\left( (\\overline{c_1} \\, j_l(kr) + \\overline{c_2} \\, n_l(kr)) \\stackrel{\\leftrightarrow}{\\partial_r} (c_1 j_l(k' r) + c_2 n_l(k' r)) \\right) \\bigg|^{r=\\varrho_2}_{r=\\varrho_1}, \\nonumber\\\\\n&=& 0.\n\\eea\nTherefore we can conclude that the following structure\n\\bea\n\\omega(u_{k,l,m},u_{k',l',m'}) &=&\nR^2 \\int \\xd \\Omega \\, \\overline{Y_l^m(\\Omega)} \\, Y_{l'}^{m'}(\\Omega) \\left( \\int_{t_1}^{\\infty} \\xd t \\, t \\, \\overline{H_{\\nu}}(k t) \\, H_{\\nu}(k' t) \\, r^2 \\right. \\nonumber\\\\\n&\\times&\n\\left( (\\overline{c_1} \\, j_l(kr) + \\overline{c_2} \\, n_l(kr)) \\stackrel{\\leftrightarrow}{\\partial_r} (c_1 j_{l'}(k' r) + c_2 n_{l'}(k' r)) \\right)\n\\nonumber\\\\\n&+& \\int^r \\xd r \\, r^2 \\, (\\overline{c_1} \\, j_l(kr) + \\overline{c_2} \\, n_l(kr)) \\, (c_1 j_{l'}(k' r) + c_2 n_{l'}(k' r)) \\nonumber\\\\\n&\\times& \\left. t_1 \\left( \\overline{H_{\\nu}}(k t) \\stackrel{\\leftrightarrow}{\\partial_t} H_{\\nu}(k' t)\\right)\\bigg|_{t=t_1} \\right),\n\\eea\nexpressed in therms of the modes $u_{k,l,m}$ as\n\\bea\n\\omega(u_{k,l,m},u_{k',l',m'}) &=& \\int_{t_1}^{\\infty} \\xd t \\int \\xd \\Omega \\, \\sqrt{g^{(3)}_r(t,\\Omega)} \\, \\left( \\overline{u_{l,m,n}}(t,r, \\Omega) \\stackrel{\\leftrightarrow}{\\partial_r} u_{k',l',m'}(t,r, \\Omega)\\right) \\nonumber\\\\\n&+& \\int^{r} \\xd r \\, \\int \\xd \\Omega \\, \\sqrt{g^{(3)}_t(r,\\Omega)} \\,\n \\left( \\overline{u_{k,l,m}}(t,r, \\Omega) \\stackrel{\\leftrightarrow}{\\partial_t} u_{k',l',m'}(t,r, \\Omega) \\right)\\bigg|_{t=t_1},\n\\eea\nis independent of $t_1$ and $r$. The above expression defines the conserved inner product on the space of solutions of the Klein-Gordon equation.\n\n\nWith the result just obtained we can expand the field configuration $\\varphi(t,\\Omega)$ defined on the hypercylinder according to the formula\n\\be\n\\varphi(t,\\Omega) = \\int_{- \\infty}^{\\infty} \\xd k \\sum_{l,m} \\, Y_l^m(\\Omega) \\, |t|^{3\/2} \\, {\\mathscr H}_{\\nu}(kt) \\, \\varphi_{l,m}(k),\n\\label{eq:a1}\n\\ee\nand the inverse relation for the modes $\\varphi_{l,m}$ are expressed as\n\\be\n\\varphi_{l,m}(k) = \\int \\xd t \\, \\xd \\Omega \\, Y_l^{-m}(\\Omega) \\, |t|^{-1\/2} \\, \\overline{{\\mathscr H}_{\\nu}(kt)} \\, \\frac{|k|}{4} \\, \\varphi(t, \\Omega).\n\\label{eq:a2}\n\\ee\nThe appearance of the modulus of $t$ and $k$ in (\\ref{eq:a1}) and (\\ref{eq:a2}) guarantees the reality of the modes $\\varphi(t,\\Omega)$.\n\n\n\\section{On the expression of the Feynman propagator in the hypercylinder geometry}\n\\label{sec:apB}\n\nIn this appendix we prove the equivalence of the propagator (\\ref{eq:GreenFhyp}) derived in the hypercylinder geometry with the Feynman propagator obtained in the time-interval setting of Section \\ref{sec:asymampfree1}. \nThe first step consists in summing over $l$ and $m$ in (\\ref{eq:GreenFhyp}) according to the following relation (see formulas (B.98) and (B.100) of \\cite{Mes:qm1}),\n\\be\n\\frac{e^{\\im |k| |\\underline{x} - \\underline{x}'|}}{4 \\pi |\\underline{x} - \\underline{x}'|} = \\im |k| \\sum_{l,m }Y_l^m(\\Omega_z) \\overline{Y_l^m(\\Omega_{z'})} \\left( \\theta(r-r') j_l(|k| r') h_l(|k| r) + \\theta(r'-r) j_l(|k| r) h_l(|k| r') \\right).\n\\ee \nSo, we obtain for $G_F$ the form\n\\be\nG_F(x,x') = \\frac{1}{16 \\pi R^2 |\\underline{x} - \\underline{x}'|} \\int_{-\\infty}^{\\infty} \\xd k \\, |k| \\, e^{\\im |k| |\\underline{x} - \\underline{x}'|} \\, (tt')^{3\/2} {\\mathscr H}_{\\nu}(k t) \\overline{{\\mathscr H}_{\\nu}(kt') }.\n\\label{eq:apB1}\n\\ee\nThen, we rewrite the exponential as\n\\be\ne^{\\im |k| |\\underline{x}-\\underline{x}'|} = \n\\lim_{\\epsilon \\rightarrow 0^+} - \\frac{\\im}{\\pi} \\int_{-\\infty}^{\\infty} \\xd q \\, q \\, \\frac{e^{\\im q |\\underline{x}-\\underline{x}'| }}{q^2 - k^2 - \\im \\epsilon}.\n\\ee\nWe substitute in (\\ref{eq:apB1}), invert the order of the integrals and perform before the integral in $\\xd k$,\n\\bea\nG_F(x,x')\n&=& \\frac{(tt')^{3\/2}}{16 \\pi R^2} \n\\frac{1}{ |\\underline{x}-\\underline{x}'|}\n\\int_{-\\infty}^{\\infty} \\xd q \\, q \\,e^{\\im q |\\underline{x}-\\underline{x}'| }\n\\left(\n\\theta(t'-t) \\, \\overline{{\\mathscr H}_{\\nu}}(|q| t') \\, {\\mathscr H}_{\\nu}(|q| t) +\n\\theta(t-t') \\, \\overline{{\\mathscr H}_{\\nu}}(|q| t) \\, {\\mathscr H}_{\\nu}(|q| t') \n\\right), \\nonumber\\\\\n&=& \\im \\frac{(tt')^{3\/2}}{8 \\pi R^2} \n\\int_{0}^{\\infty} \\xd q \\, q \\, \\frac{\\sin \\left( q |\\underline{x}-\\underline{x}'| \\right)}{ |\\underline{x}-\\underline{x}'|} \n\\left(\n\\theta(t'-t) \\, \\overline{{\\mathscr H}_{\\nu}}(q t') \\, {\\mathscr H}_{\\nu}(q t) +\n\\theta(t-t') \\, \\overline{{\\mathscr H}_{\\nu}}(q t) \\, {\\mathscr H}_{\\nu}(q t') \n\\right), \\nonumber\\\\\n\\eea\nwhich coincides with (\\ref{eq:GF1}) since $\\overline{{\\mathscr H}_{\\nu}}(x) \\, {\\mathscr H}_{\\nu}(y) = \\overline{H_{\\nu}}(x) \\, H_{\\nu}(y)$ due to the relation (\\ref{eq:modHankel}). This concludes the proof.\n\n\n\\section{Classical theory}\n\\label{sec:classical}\n\nWe begin in this section by studying the classical theory of a real massive minimally coupled scalar field in de Sitter spacetime. The action in a spacetime region $M$ is given by\n\\be\nS_M(\\phi) = \\frac{1}{2} \\int_M \\xd^4 x \\sqrt{-g}\\left( g^{\\mu \\nu} \\partial_{\\mu} \\phi \\, \\partial_{\\nu} \\phi - m^2 \\phi^2 \\right),\n\\label{eq:action0}\n\\ee\nwhere we use the notation $\\partial_{\\mu}= \\partial \/ \\partial x^{\\mu}$, and $g \\equiv \\det g_{\\mu \\nu}$ denotes the determinant of the de Sitter metric. Via an integration by parts, the action of a classical solution $\\phi_{cl}$ of the equation of motion obtained from (\\ref{eq:action0}) reduces to a boundary term,\n\\be\nS_{M}(\\phi_{cl}) = \\frac{1}{2} \\int_{\\partial M} \\xd ^3 s \\sqrt{g^{(3)}} \\, \\left( \\phi_{cl} \\, \\partial_n \\phi_{cl} \\right),\n\\label{eq:actcl}\n\\ee\nwhere $s$ indicates generic three dimensional coordinates on the boundary $\\partial M$ of the region $M$, $\\partial_n$ is the normal outward derivative to $\\partial M$, namely $\\partial_n = n^{\\mu} \\partial_{\\mu}$ with $n^{\\mu}$ the normal to the surface and $g^{(3)}$ is the determinant of the 3-metric induced on $\\partial M$. In the following subsections two different regions M will be considered. First we will be interested in a region with spacelike boundaries specified by constant values of the de Sitter time (defined below). Then, the second region we will look at is a region with timelike boundaries determined by constant values of the radial distance from the origin of de Sitter spacetime. Our main goal will be to provide an expression for both the action (\\ref{eq:actcl}) and the classical field $\\phi_{cl}$ in terms of the boundary field configurations $\\varphi$,\n\\be\n\\phi(x) \\big|_{x \\in \\partial M} = \\varphi (s).\n\\ee \n\n\n\\subsection{Region with spacelike boundary}\n\nThe first geometry we will consider is appropriately described in terms of a coordinate system $(t,\\underline{x})$, where the de Sitter metric takes the form\n\\bea\n\\xd s^2 = \\frac{R^2}{t^2} \\left( \\xd t^2 - \\xd \\underline{x}^2 \\right),\n\\label{eq:dSmetric}\n\\eea\nwhere $t \\in (0, \\infty)$, $\\underline{x} \\in \\mathbb{R}^3$ and $R$ denotes the inverse of the Hubble constant. This coordinate system covers the half of de Sitter space, the remaining half can be included by extending the domain of the conformal time $t$ to negative values.\nWe consider a region $M$ of de Sitter spacetime bounded by the disjoint union of two hypersurfaces of constant conformal time $t$, namely $\\Sigma_1 = \\{ (t, \\underline{x}) : t= t_1 \\}$ and $\\Sigma_2 = \\{ (t, \\underline{x}) : t= t_2 \\}$, with $t_1 < t_2$. We denote this spacetime region, $M= [t_1, t_2] \\times \\mathbb{R}^3$, simply by $[t_1, t_2]$. \n\nThe action of a real massive minimally coupled scalar field $\\phi$ in this region $[t_1,t_2]$ is given by\n\\bea\nS_{[t_1, t_2]}(\\phi) = \\frac{1}{2} \\int_{t_1}^{t_2} \\xd t \\int_{{\\mathbb R}^3} \\xd^3 \\underline{x} \\, \\frac{R^2}{t^2} \\left((\\partial_t \\phi)^2 - \\sum_i (\\partial_{i} \\phi)^2 - m^2 \\phi^2\\right).\n\\label{eq:action}\n\\eea\nThe Klein-Gordon equation satisfied by the field $\\phi$ has the form\n\\be\n\\left[ \\frac{t^2}{R^2} \\left(\\partial_t^2 - \\Delta_{\\underline{x}} \\right) -\\frac{2t}{R^2} \\partial_t +m^2 \\right]\\phi(t, \\underline{x}) = 0,\n\\ee\nwhere $\\Delta_{\\underline{x}}$ is the Laplacian operator in the coordinates $\\underline{x}$. This equation can be solved by the method of separation of variables and the general solution can by written as \\cite{Schomblond:1976xc}\n\\be\n\\phi(t, \\underline{x}) =\\int \\frac{\\xd ^3 \\underline{k}}{(2 \\pi)^{3\/2}} \\left( v_k(t) \\, e^{\\im \\underline{k} \\cdot \\underline{x}} + \\overline{v_k}(t) \\, e^{-\\im \\underline{k} \\cdot \\underline{x}} \\right),\n\\label{eq:classsol}\n\\ee\nwhere \n\\be\nv_k(t)= t^{3\/2} \\left(c_{1}(k) J_{\\nu}(kt) + c_{2}(k) Y_{\\nu}(kt) \\right),\n\\ee\nwhere $k=|\\underline{k}|$, $J_{\\nu}(z)$ and $Y_{\\nu}(z)$ are the Bessel functions of the first and second kind respectively, with index $\\nu = \\sqrt{\\frac{9}{4} - (m R)^2}$, and $c_{1}(k)$ and $c_{2}(k)$ are two coefficients. In order for the classical solution (\\ref{eq:classsol}) to be bounded in the spacetime region $M$ the components of the 3-vector $\\underline{k}$ have to be real. Consequently the modulus $k$ is a non negative quantity, $k \\geq 0$.\n\nIt will be convenient to express the classical solution (\\ref{eq:classsol}) in a different form,\n\\be\n\\phi(t, \\underline{x}) = \\left( J_{\\nu}(k t) \\, \\varphi_J \\right)(\\underline{x}) + \\left( Y_{\\nu}(k t) \\, \\varphi_Y \\right)(\\underline{x}).\n\\ee\nIn this expression the Bessel functions $J_{\\nu}$ and $Y_{\\nu}$ represent operators acting on the field configurations $\\varphi_J$ and $\\varphi_Y$ respectively. The relation between these field configurations and the boundary field configurations, indicated by $\\varphi_1$ and $\\varphi_2$ on the hypersurfaces $\\Sigma_1$ and $\\Sigma_2$ respectively, namely\n\\be\n\\varphi_1(\\underline{x}) \\defeq \\phi(t_1,\\underline{x}) \\qquad \\hbox{and} \\qquad \\varphi_2(\\underline{x}) \\defeq \\phi(t_2,\\underline{x}),\n\\ee\nis given by the matrix operator equation\n\\be\n \\begin{pmatrix}\\varphi_1 \\\\ \\varphi_2\\end{pmatrix}\n =\\begin{pmatrix}J_{\\nu}(k t_1) & Y_{\\nu}(k t_1)\\\\ J_{\\nu}(k t_2) & Y_{\\nu}(k t_2)\\end{pmatrix}\n \\begin{pmatrix}\\varphi_J \\\\ \\varphi_Y\\end{pmatrix}\n\\ee\nInverting this equation we obtain for the field $\\phi$ the following dependence on the boundary field configurations,\n\\be\n\\phi(t, \\underline{x}) = \\left( \\frac{\\delta_k( t, t_2)}{\\delta_k ( t_1, t_2)} \\, \\varphi_1 \\right)(\\underline{x}) + \\left( \\frac{\\delta_k ( t_1, t)}{\\delta_k ( t_1, t_2)} \\, \\varphi_2 \\right) (\\underline{x}),\n\\label{eq:boundconfig}\n\\ee\nwhere the quotients have to be understood as operators acting on a Fourier expansion of the boundary configurations $\\varphi_1$ and $\\varphi_2$, and the operator $\\delta_k$ is defined as\n\\be\n\\delta_k( z, \\hat{z}) \\defeq z^{3\/2} \\, \\hat{z}^{3\/2} \\left[ J_{\\nu}(k z) \\, Y_{\\nu}(k \\hat{z}) - Y_{\\nu}(k z) \\, J_{\\nu}(k \\hat{z})\\right].\n\\label{eq:delta2}\n\\ee\nThe expression (\\ref{eq:boundconfig}) allows the evaluation of the action (\\ref{eq:action}) for a classical solution of the Klein-Gordon equation in terms of the boundary field configurations $\\varphi_1$ and $\\varphi_2$. The result is\n\\be\nS_{[t_1, t_2]}(\\phi)\n= \\frac{1}{2} \\int \\xd^3 \\underline{x} \\, \n\\begin{pmatrix}\\varphi_1 & \\varphi_2 \\end{pmatrix} W_{[t_1, t_2]} \n\\begin{pmatrix} \\varphi_1 \\\\ \\varphi_2 \\end{pmatrix},\n\\label{eq:actionbound}\n\\ee\nwhere the $W_{[t_1, t_2]}$ is a 2x2 matrix with elements $W_{[t_1, t_2]}^{(i,j)}, (i,j=1,2),$ given by\n\\bea\nW_{[t_1, t_2]}^{(1,1)} &=& - \\frac{R^2}{t_1^2} \\left( \\frac{3}{2 t_1} + k \\frac{J_{\\nu}'(k t_1) \\, Y_{\\nu}(k t_2) - Y_{\\nu}'(k t_1) \\, J_{\\nu}(k t_2)}{J_{\\nu}(k t_1) \\, Y_{\\nu}(k t_2) - Y_{\\nu}(k t_1) \\, J_{\\nu}(k t_2)}\\right), \\label{eq:W11} \\\\\nW_{[t_1, t_2]}^{(1,2)} &=& W_{[t_1, t_2]}^{(2,1)} = - \\frac{2 R^2}{\\pi \\delta_k (t_1, t_2)}, \\label{eq:W12} \\\\\nW_{[t_1, t_2]}^{(2,2)} &=& \\frac{R^2}{t_2^2} \\left( \\frac{3}{2 t_2} + k \\frac{J_{\\nu}(k t_1) \\, Y_{\\nu}'(k t_2) - Y_{\\nu}(k t_1) \\, J_{\\nu}'(k t_2)}{J_{\\nu}(k t_1) \\, Y_{\\nu}(k t_2) - Y_{\\nu}(k t_1) \\, J_{\\nu}(k t_2)}\\right),\\label{eq:W22}\n\\eea\nwhere a prime indicates the derivative with respect to the argument. These matrix elements have to be understood as operators acting on the boundary field configurations.\n\n\n\\subsection{Region with timelike boundary}\n\\label{sec:hyp}\n\nThe second geometry we are interested in is conveniently described in terms of spherical coordinates in space, defined by three parameters: $r \\in [0, \\infty)$, $\\theta \\in [0, \\pi)$ and $\\varphi \\in [0, 2 \\pi)$. The de Sitter metric (\\ref{eq:dSmetric}) in this coordinate system takes the form\n\\be\n\\xd s^2 = \\frac{R^2}{t^2} \\left( \\xd t^2 - \\xd r^2 - r^2 \\xd \\vartheta^2 -r^2 \\sin^2 \\vartheta \\, \\xd \\varphi^2 \\right).\n\\label{eq:dSmetric2}\n\\ee\nIt will be useful in the following to adopt $\\Omega$ as a collective notation for $\\theta$ and $\\varphi$. \nTwo different spacetime regions will be considered: The first region is bounded by one hypersurface of radius $r$, denoted by $\\Sigma_{\\varrho}= \\{(t,r,\\Omega) : r =\\varrho \\}$. In analogy with the case of Minkowski spacetime, we refer to the hypersurface $\\Sigma_{\\varrho}$ as the hypercylinder of radius $\\varrho$. The second region is the spacetime region in between two hypercylinders of different radii $\\Sigma_{\\varrho}$ and $\\Sigma_{\\hat{\\varrho}}$. Both these regions have timelike boundary, in contrast to the region $[t_1,t_2]$ of the previous subsection. Moreover, the region enclosed by one hypercylinder has a more exotic property: its boundary is not the disjoint union of two disconnected hypersurfaces, it is completely connected.\n\nThe Klein-Gordon equation in the metric (\\ref{eq:dSmetric2}) reads\n\\be\n\\left( \\frac{t^2}{R^2} \\left[ \\partial_t^2 - \\Delta_r - \\Delta_{\\Omega}\\right] -\\frac{2 t}{R^2} \\partial_t + m^2 \\right)\\phi(t,r, \\Omega) = 0,\n\\label{eq:KGhyp}\n\\ee\nwhere\n\\be\n\\Delta_r= \\frac{1}{r^2} \\, \\partial_r (r^2 \\, \\partial_r), \\qquad \\mbox{and} \\qquad\n\\Delta_{\\Omega} =\\frac{1}{r^2 \\, \\sin \\vartheta} \\, \\partial_{\\vartheta} (\\sin \\vartheta \\, \\partial_{\\vartheta}) +\\frac{1}{r^2 \\, \\sin^2 \\vartheta} \\, \\partial_{\\varphi}^2.\n\\ee\nThe bounded solutions of (\\ref{eq:KGhyp}) in the region bounded by one or two hypercylinders can be expanded as follows\n\\be\n\\phi(t,r,\\Omega) = \\int_{-\\infty}^{\\infty} \\xd k \\, \\sum_{l=0}^{\\infty} \\sum_{m=-l}^l \\left( a_{k,l,m} \\, u_{k,l,m}(t,r,\\Omega) + c.c. \\right),\n\\label{eq:classsolhyp}\n\\ee\nwhere $a_{k,l,m}$ are coefficients and with $u_{k,l,m}$ we denote the unnormalized modes\n\\be\nu_{k,l,m}(t,r,\\Omega) = t^{3\/2} \\, {\\mathscr H}_{\\nu}(k t) \\, Y_l^m(\\Omega) \\left( c_1 (k)j_l(k r) + c_2(k) n_l(k r) \\right).\n\\label{eq:uklm}\n\\ee\nThe coefficients $c_1$ and $c_2$ are in general different from those introduced in the previous section. $Y_l^m$ are the spherical harmonics satisfying the equation\n\\be\n\\left( \\Delta_{\\Omega} Y_l^m \\right) (\\Omega) = -\\frac{l(l+1)}{r^2} Y_l^m(\\Omega).\n\\ee\nIn the modes (\\ref{eq:uklm}), $j_l$ and $n_l$ denote the spherical Bessel functions of the first and second kind respectively, solutions of the equation\n\\be\n\\left( \\Delta_r j_l\\right) (k r) = \\left( \\frac{l(l+1)}{r^2} -k^2 \\right) j_l(kr),\n\\label{eq:radialKG}\n\\ee\nand the same equation is satisfied by $n_l$. \nNotice that for the spacetime region enclosed by the hypercylinder, where the origin ($r=0$) is part of the region, the coefficient $c_2$ in (\\ref{eq:uklm}) will be zero and the radial component of the modes $u_{k,l,m}$ will reduce to the spherical Bessel function of the fist kind, $j_l$. The reason lies in the singular character of $n_l$ in the origin, whereas $j_l$ remains finite \\cite{AbSt:handbook}. On the other hand, the spacetime region bounded by two hypercylinders does not contain the origin, and both $j_l$ and $n_l$ will appear in the modes $u_{k,l,m}$. \n\nFinally, the function ${\\mathscr H}_{\\nu}$ in (\\ref{eq:uklm}) is proportional to the Bessel functions of the third kind of order $\\nu$, $H_{\\nu}$, as called Hankel function\\footnote{Working with ${\\mathscr H}_{\\nu}$ instead of $H_{\\nu}$ turns out to be more convenient due to the following property,\n\\be\n{\\mathscr H}_{\\nu} ( - kt)= \\overline{{\\mathscr H}_{\\nu}} (kt).\n\\label{eq:modHankel}\n\\ee\nThis relation follows from the analytic continuation of the Hankel function \\cite{Wat:bessel} and will be used in many occasions in the rest of paper.},\n\\be\n{\\mathscr H}_{\\nu} (kt)= e^{\\im \\nu \\pi\/2} H_{\\nu}(kt).\n\\ee \nIn order for the Hankel function to be bounded $k$ must be real. \n\n\nConsider the region with the hypercylinder of radius $\\varrho$ as boundary. In the following we will refer to this region simply by $\\varrho$.\\footnote{The symbol $\\varrho$ should not be confused with the symbol $\\rho$ denoting the amplitude in the subsequent sections.} The classical solution of (\\ref{eq:KGhyp}) matching the boundary field configuration $\\varphi$ on the hypercylinder, i.e. for $r= \\varrho$, can be written as\n\\be\n\\phi(t,r,\\Omega) = \\left( \\frac{j_l(k r)}{j_l(r \\varrho)} \\, \\varphi \\right) (t, \\Omega),\n\\label{eq:hypbound}\n\\ee\nwhere the quotient of spherical Bessel functions as to be understood as an operator. The action (\\ref{eq:actcl}) associated with the region $\\varrho$ takes the form\n\\be\nS_{\\varrho} (\\phi)= -\\frac{1}{2} \\int \\xd t \\, \\xd \\Omega \\, \\frac{R^2}{t^2} \\, \\varrho^2 \\, \\varphi(t, \\Omega) \\left( k \\,\\frac{j_l'(k \\varrho)}{j_l(k \\varrho)} \\, \\varphi \\right) (t, \\Omega),\n\\label{eq:actbound2}\n\\ee\nwhere the prime indicates the derivative with respect to the argument.\n\nWe now turn to the spacetime region bounded by two hypercylinders of different radii, $\\varrho_1$ and $\\varrho_2$, indicated by $[\\varrho_1, \\varrho_2]$.\nLet $\\varphi_1$ and $\\varphi_2$ denote the boundary field configurations on $r=\\varrho_1$ and $r=\\varrho_2$ respectively. The classical solution of the Klein-Gordon equation reducing to these field configurations on the boundary of $[\\varrho_1, \\varrho_2]$ is relate to $\\varphi_1$ and $\\varphi_2$ via\n\\be\n\\phi(t,r,\\Omega) = \\left( \\frac{\\Delta_k(r, \\varrho_2)}{\\Delta_k(\\varrho_1, \\varrho_2)} \\varphi_1\\right) (t, \\Omega) +\n\\left( \\frac{\\Delta_k(\\varrho_1, r)}{\\Delta_k(\\varrho_1, \\varrho_2)} \\varphi_2 \\right) (t, \\Omega),\n\\label{eq:boundconfighyp}\n\\ee\nwhere the function $\\Delta_k$ is to be understood as an operator defined by\n\\be\n\\Delta_k(\\varrho_1, \\varrho_2) = j_l(k \\varrho_1)n_l(k \\varrho_2) - n_l(k \\varrho_1)j_l(k \\varrho_2).\n\\ee\nThe action (\\ref{eq:actcl}) of the field (\\ref{eq:boundconfighyp}) is then\n\\be\nS_{[\\varrho_1, \\varrho_2]}(\\phi) = \\frac{1}{2} \\int \\xd t \\, \\xd \\Omega \\, \n\\begin{pmatrix}\\varphi_1 & \\varphi_2 \\end{pmatrix} {\\cal W}_{[\\varrho_1, \\varrho_2]} \n\\begin{pmatrix} \\varphi_1 \\\\ \\varphi_2 \\end{pmatrix},\n\\label{eq:actbound3}\n\\ee\nwhere the ${\\cal W}_{[\\varrho_1, \\varrho_2]}$ is a 2x2 matrix with elements ${\\cal W}_{[\\varrho_1, \\varrho_2]} ^{(i,j)}, (i,j=1,2),$ given by\n\\bea\n{\\cal W}_{[\\varrho_1, \\varrho_2]}^{(1,1)} &=& \\frac{R^2}{t^2} \\, \\varrho_1^2 \\, \\frac{k \\, \\sigma_k(\\varrho_2, \\varrho_1)}{\\Delta_k(\\varrho_1, \\varrho_2)},\n\\label{eq:Whyp1}\\\\\n{\\cal W}_{[\\varrho_1, \\varrho_2]}^{(1,2)} &=& {\\cal W}_{[\\varrho_1, \\varrho_2]}^{(2,1)} = - \\frac{ R^2}{t^2} \\frac{1}{ k \\, \\Delta_k (\\varrho_1, \\varrho_2)},\n\\label{eq:Whyp2}\\\\\n{\\cal W}_{[\\varrho_1, \\varrho_2]}^{(2,2)} &=& \\frac{R^2}{t^2} \\, \\varrho_2^2 \\, \\frac{k \\, \\sigma_k(\\varrho_1, \\varrho_2)}{\\Delta_k(\\varrho_1, \\varrho_2)}.\n\\label{eq:Whyp3}\n\\eea\nThe function $\\sigma_k$ is to be understood as an operator defined as\n\\be\n\\sigma_k(\\varrho_1, \\varrho_2)= j_l(k \\varrho_1)n_l'(k \\varrho_2) - n_l(k \\varrho_1)j_l'(k \\varrho_2).\n\\ee\nThe expressions of the action of a classical solution of the equation of motion in terms of the boundary field configurations for the different spacetime regions considered, (\\ref{eq:actionbound}), (\\ref{eq:actbound2}) and (\\ref{eq:actbound3}), will be an important ingredient for the computation of the quantum field propagator, as will be clear in the next section.\n\n\\section{Summary and outlook}\n\\label{sec:con}\n\nLet us summarize the results we have obtained. We have applied the GBF of quantum field theory to study a massive scalar field in de Sitter spacetime. Inspired by previous results obtained in Minkowski spacetime \\cite{CoOe:spsmatrix,CoOe:smatrixgbf}, two different quantization schemes, associated with different spacetime regions, have been implemented and compared: in the first and more traditional one the region considered is of a time-interval type, namely the boundary of the region is the disjoint union of two equal-time hypersurfaces; in the second scheme, the scalar field is quantized in a region that incorporates key nonstandard features of the GBF, namely the hypercylinder region bounded by one connected and timelike hypersurface. After constructing all the relevant algebraic structures for the free theory, i.e., the Hilbert space, the field propagator and the amplitude associated with the regions in question, we consider the interacting theory, starting with the case of the interaction with a source field and then using functional derivative techniques to treat the general interacting theory. We then show the existence of an isomorphism between the Hilbert spaces defined in the two schemes and provide the explicit correspondence between the multiparticle states defined on the equal-time hypersurfaces with those defined on the hypercylinder.\n\nThis work was first motivated by its obvious relevance for the GBF, in order to improve our understanding of its technical and conceptual aspects. Indeed the results presented here constitute the first application of the GBF to a field theory on a curved space. Furthermore, in the particular coordinate systems chosen the de Sitter metric is conformal to the Minkowski metric and we were able to show how all the relevant structures of the GBF in de Sitter space reduce to those computed in Minkowski space in the appropriate asymptotic limit. Such a correspondence not only constitutes a requirement the GBF in de Sitter has to satisfy, but also provides a useful tool to compare the vacuum states defined on the different hypersurfaces considered. What emerges from this is that the field theory of a massive scalar field in the de Sitter metric tends asymptotically to the field theory of a \\emph{massless} scalar field in the Minkowski metric. Based on these observations, we can extend to de Sitter space some of the conclusions discussed for the field theory in Minkowski space: In particular the crossing symmetry of the $S$-matrix. As explained in the paper, the asymptotic amplitude for the hypercylinder geometry may be interpreted as the $S$-matrix for the scalar field, and because of the connectedness of the boundary hypersurface no a priori distinction exists between incoming and outgoing states. The usual notion of crossing symmetry becomes consequently implicit in the hypercylinder geometry.\n\nSo, the GBF of the field theory in the hypercylinder geometry provides a new perspective on the quantum dynamics in de Sitter space. We have indeed obtained a new representation for the de Sitter invariant Feynman propagator in the hypercylinder region, (\\ref{eq:GreenFhyp}), and the corresponding spatially asymptotic boundary condition (\\ref{eq:bcGreenFhyp}) for the non-homogeneous Klein-Gordon equation. Moreover, the propagator (\\ref{eq:GreenFhyp}) is of the Hadamard form, implying that the vacuum state (\\ref{eq:vachyp}) corresponds to the Bunch-Davies vacuum \\cite{Bunch:1978yq} defined on the hypercylinder. Hence, apart from its significance for the development of the GBF, our result represents a contribution to the development of the Schr\\\"odinger representation for quantum fields in curved spacetime. Besides the representation of the Bunch-Davies vacuum, the definition and use of coherent states in the two settings considered here, in the Schr\\\"odinger representation, not known previously, constitute a novelty of our paper.\n\nAs a possible application of the results presented here, we mention the study of the response of the detector moving in the radial direction, i.e. from one hypercylinder to another one of different radius, and the comparison with the well known thermal bath of radiation perceived by an observer moving with proper time $t$, \\cite{BiDa:qfcs}.\n\nThe implementation of the GBF quantization scheme for more general spacetime regions and boundaries represents a line of future research; in particular compact regions, whose boundary hypersurface includes timelike as well as spacelike parts, will be the main focus. The field theory defined in such kind of regions will provide a fully local description of the quantum dynamics, and it will be interesting the study how a particle concept (associated with the boundary of the region) could be implemented and compare it with the standard notion of particle. A possible direction along this line is the use of a different coordinate system to describe the de Sitter spacetime \\cite{BiDa:qfcs}.\n\nWe conclude by mentioning a new quantization scheme for the GBF recently proposed by Oeckl in \\cite{Oe:newGBF}. The ability to recover the results presented here may constitute an important test for Oeckl's proposal.\n\n\n\n\\section{Region with spacelike boundary - Free theory}\n\\label{sec:free}\n\n\\subsection{Field propagator}\n\\label{sec:freefieldpropagator}\n\nWe evaluate the field propagator associated with the spacetime region $[t_1, t_2]$ of Section \\ref{sec:classical}. Substituting the expression (\\ref{eq:actionbound}) of the classical action of the scalar field in terms of the boundary field configurations $\\varphi_1$ and $\\varphi_2$ in (\\ref{eq:propcl}) leads to \n\\be\nZ_{[t_1, t_2],0} (\\varphi_1, \\varphi_2) = N_{[t_1, t_2],0} \\, \\exp \\left( \\frac{\\im}{2} \\int \\xd^3 \\underline{x} \\, \n\\begin{pmatrix}\\varphi_1 & \\varphi_2 \\end{pmatrix} W_{[t_1, t_2]} \n\\begin{pmatrix} \\varphi_1 \\\\ \\varphi_2 \\end{pmatrix}\\right),\n\\label{eq:propt}\n\\ee\nwhere the normalization factor is determined by the gluing properties of the field propagator (an additional subscript 0 has been written in the field propagator and the normalization factor in order to indicate that these quantities refer to the free theory).\nThe proof of the consistency of the definition (\\ref{eq:propt}) is provided by the composition rule satisfied by the field propagator: The evolution in the variable $t$ from $t_1$ to $t_2$ and subsequently from $t_2$ to $t_3$ must equal the direct evolution from $t_1$ to $t_3$. This composed evolution can be expressed in terms of the following equation,\n\\be\nZ_{[t_1, t_3],0} (\\varphi_1, \\varphi_3) = \\int \\xD \\varphi_2 \\, Z_{[t_1, t_2],0} (\\varphi_1, \\varphi_2) \\, Z_{[t_2, t_3],0} (\\varphi_2, \\varphi_3). \n\\label{eq:compprop}\n\\ee \nThe propagator (\\ref{eq:propt}) satisfies such composition if the normalization factors corresponding to the three regions are related by\n\\be\nN_{[t_1, t_3],0} = N_{[t_1, t_2],0} \\, N_{[t_2, t_3],0} \\int \\xD \\varphi_2 \\exp \\left( \\frac{\\im}{2} \\int \\xd ^3 \\underline{x} \\, \\varphi_2(\\underline{x}) \\left( \\frac{2 R^2}{ \\pi} \\frac{\\delta_k(t_1,t_3)}{\\delta_k(t_2,t_3)\\delta_k(t_2,t_3)} \\varphi_2 \\right) (\\underline{x})\\right).\n\\ee\nThe solution turns out to be\n\\be\nN_{[t_1, t_2],0} \n= \\det \\left( - \\frac{\\im R^2}{ \\pi^2 \\delta_k(t_1,t_2)} \\right)^{1\/2}.\n\\label{eq:normft}\n\\ee\nWe are now in the position to check the unitarity of the quantum evolution implemented by the field propagator. Indeed, in this context the unitarity is translated in the following condition \\cite{Oe:GBQFT,Oe:KGtl},\n\\be\n\\int \\xD \\varphi_2 \\, \\overline{Z_{[t_1, t_2]} (\\varphi_1, \\varphi_2)} \\, Z_{[t_1, t_2]} (\\varphi_1', \\varphi_2) = \\delta(\\varphi_1 - \\varphi_1').\n\\label{eq:unitarity}\n\\ee\nFor the free field propagator (\\ref{eq:propt}) we have\n\\begin{align}\n& \\int \\xD \\varphi_2 \\, \\overline{Z_{[t_1, t_2],0} (\\varphi_1, \\varphi_2)} \\, Z_{[t_1, t_2],0} (\\varphi_1', \\varphi_2) \\nonumber\\\\\n& = |N_{[t_1, t_2],0}|^2 \\int \\xD \\varphi_2 \\exp \\left( \\im \\int \\xd ^3 \\underline{x} \\, \\varphi_2 \\, \\frac{2 R^2}{ \\pi \\delta_k(t_1,t_2)}(\\varphi_1- \\varphi_1') \\right) \\nonumber\\\\\n& \\times \\exp \\left( \\im \\int \\xd ^3 \\underline{x} \\, (\\varphi_1 + \\varphi_1')\n\\left[ \n- \\frac{R^2}{t_1^2} \\left( \\frac{3}{2 t_1} + k \\frac{J_{\\nu}'(k t_1) \\, Y_{\\nu}(k t_2) - Y_{\\nu}'(k t_1) \\, J_{\\nu}(k t_2)}{J_{\\nu}(k t_1) \\, Y_{\\nu}(k t_2) - Y_{\\nu}(k t_1) \\, J_{\\nu}(k t_2)}\\right)\n\\right] \n(\\varphi_1 - \\varphi_1') \\right), \\nonumber\\\\\n&=|N_{[t_1, t_2],0}|^2 \\, \\det \\left( - \\frac{2 \\im R^2}{ \\pi^2 \\delta_k(t_1,t_2)} \\right)^{1\/2} \\, \\delta(\\varphi_1 - \\varphi_1').\n\\end{align}\nUsing (\\ref{eq:normft}), the product of the first two terms on the right-hand side above equals 1, and consequently the condition (\\ref{eq:unitarity}) is verified. Therefore we conclude that the field propagator (\\ref{eq:propt}) implements a unitary quantum evolution in the variable $t$ for the free scalar field in de Sitter space. This was shown in \\cite{CoOe:unit} in a general setting, but we nevertheless present the detailed steps here and verify them independently.\n\nA well known property of the de Sitter space is that in the limit in which the curvature goes to zero de Sitter space tends to Minkowski space. Recalling that the Ricci scalar is proportional to $R^{-2}$ \\cite{BiDa:qfcs}, we can recover the Minkowski metric from the de Sitter metric (\\ref{eq:dSmetric}) by considering the limits $R \\rightarrow \\infty$ and $t \\rightarrow \\infty$ in such a way that $R\/t \\rightarrow 1$,\n\\be\n\\lim_{\\stackrel{R \\rightarrow \\infty}{t \\rightarrow \\infty}} \\frac{R}{t} = 1.\n\\label{eq:dStoM}\n\\ee\nWe now show that the free field propagator (\\ref{eq:propt}) reduces to the free field propagator in Minkowski space evaluated in \\cite{CoOe:smatrixgbf}. The Bessel functions of the first and second kind have the following asymptotic expansions for large values of their argument \\cite{AbSt:handbook},\n\\bea\nJ_{\\nu}(x) &=& \\sqrt{\\frac{2}{\\pi x}} \\left( \\cos \\left(x- \\nu \\frac{\\pi}{2} - \\frac{\\pi}{4} \\right) + o(x^{-1})\\right), \\label{eq:Jasympt}\\\\\nY_{\\nu}(x) &=& \\sqrt{\\frac{2}{\\pi x}} \\left( \\sin \\left(x- \\nu \\frac{\\pi}{2} - \\frac{\\pi}{4} \\right) + o(x^{-1})\\right). \\label{eq:Yasympt}\n\\eea\nTaking the limits defined by (\\ref{eq:dStoM}) and using the expansions (\\ref{eq:Jasympt}) and (\\ref{eq:Yasympt}), the matrix elements $W_{[t_1,t_2]}^{(i,j)}$ (\\ref{eq:W11},\\ref{eq:W12},\\ref{eq:W22}) reduce to\n\\bea\nW_{[t_1, t_2]}^{(1,1)} &=& k \\, \\frac{\\cos k(t_2 -t_1)}{\\sin k(t_2 -t_1)}, \\\\\nW_{[t_1, t_2]}^{(1,2)} &=& W_{[t_1, t_2]}^{(2,1)} = - \\frac{k}{\\sin k(t_2 -t_1)}, \\\\\nW_{[t_1, t_2]}^{(2,2)} &=& k \\, \\frac{\\cos k(t_2 -t_1)}{\\sin k(t_2 -t_1)}.\n\\eea\nThese matrix elements equal those derived in Minkowski space (see \\cite{Oe:timelike,Oe:KGtl,CoOe:smatrixgbf}) providing the identification $k = \\omega$ holds, where $\\omega$ is the operator $\\sqrt{- \\Delta_{\\underline{x}} +m^2}$. Moreover, it is easy to show that the normalization factor (\\ref{eq:normft}) reduces to the one in flat spacetime. Therefore we conclude that the free field propagator in de Sitter space (\\ref{eq:propt}) coincides with the one in Minkowski space in the limit defined by (\\ref{eq:dStoM}).\n\n\\subsection{Vacuum state}\n\\label{sec:vacuum}\n\nWe compute the vacuum state on the hypersurface of constant conformal time $t$. The starting point is the following Gaussian ansatz for the vacuum wave function,\n\\be\n\\psi_{t,0}(\\varphi)=C_{t} \\exp\\left(-\\frac{1}{2}\\int \\xd ^3 \\underline{x} \\, \\varphi(\\underline{x})(A_{t} \\varphi)(\\underline{x})\\right) ,\n\\ee\nwhere $C_{t}$ is a normalization factor and $A_{t}$ denotes a family of operators indexed by ${t}$. The explicit form of the operator $A_{t}$ has been given in \\cite{Co:vacuum}, and the form of the vacuum wave function results to be \n\\be\n \\psi_{t,0}(\\varphi) = C_{t} \\exp\\left(\\frac{\\im}{2}\\int \\xd^3 \\underline{x} \\, \\frac{R^2}{t^2} \\, \\varphi(\\underline{x})\\left[ k \\frac{H_{ \\nu}'(k t)}{H_{\\nu}(k t)} + \\frac{3}{2 t}\\right] \\varphi(\\underline{x})\\right).\n\\label{eq:vac}\n\\ee \nThe requirement that the vacuum state is normalized to 1 fixes the normalization factor $C_{t}$ up to a phase,\n\\be\n|C_{t}|^{-2} = \\int \\xD \\varphi \\exp \\left( - \\frac{1}{2} \\int \\xd ^3 x \\, \\varphi(x) \\frac{4 R^2}{\\pi t^3 |H_{\\nu}(k t)|^2} \\varphi(x) \\right) = \\det \\left(\\frac{2 R^2}{\\pi^2 t^3 |H_{\\nu}(k t)|^2} \\right)^{-1\/2}.\n\\ee\nThe phase of the normalization factor is fixed by the relation between two vacuum wave functions defined on hypersurfaces of different conformal time. In particular the vacuum wave function (\\ref{eq:vac}) satisfies the identity\n\\be\n\\psi_{t_2,0}(\\varphi_2) = \\int \\xD \\varphi_1 \\, \\psi_{t_1,0}(\\varphi_1) \\, Z_{[t_1,t_2],0}(\\varphi_1, \\varphi_2),\n\\ee\nwhich implies the following identity for the normalization factors\n\\bea\nC_{t_2} &=& C_{t_1} \\, N_{[t_1,t_2],0} \\, \\int \\xD \\varphi_1 \\exp \\left( - \\frac{1}{2} \\int \\xd ^3 x \\, \\varphi_1 \\left[- \\frac{2 \\im R^2}{\\pi \\delta_k(t_1,t_2)} \\, \\frac{t_2^{3\/2} \\,H_{\\nu}(k t_2)}{ t_1^{3\/2} \\, H_{\\nu}(k t_1)} \\right] \\varphi_1 \\right), \\nonumber\\\\\n&=& C_{t_1} \\, N_{[t_1,t_2],0} \\, \\det \\left( - \\frac{ \\im R^2}{\\pi^2 \\delta_k(t_1,t_2)} \\, \\frac{t_2^{3\/2} \\,H_{\\nu}(k t_2)}{ t_1^{3\/2} \\, H_{\\nu}(k t_1)}\\right)^{-1\/2}.\n\\eea\nSubstituting $N_{[t_1,t_2],0}$ with its expression (\\ref{eq:normft}), we arrive at\n\\be\nC_{t_2} = C_{t_1} \\, \\det \\left( \\frac{t_2^{3\/2} \\,H_{\\nu}(k t_2)}{ t_1^{3\/2} \\, H_{\\nu}(k t_1)}\\right)^{-1\/2}.\n\\ee\nThis suggest the following solution for the normalization factor $C_t$,\n\\be\nC_t = \\det \\left(\\frac{\\sqrt{2} R}{\\pi t^{3\/2} \\, H_{\\nu}(k t)} \\right)^{1\/2}.\n\\ee\nThe asymptotic limit, according to the prescription (\\ref{eq:dStoM}), of the vacuum wave function (\\ref{eq:vac}) coincides (up to a phase factor) with the vacuum wave function defined on equal time hyperplanes in Minkowski space, namely\n\\be\n\\psi_{t,0}(\\varphi)= \\det \\left(\\frac{k \\, e^{- \\im 2 k t } }{\\pi } \\right)^{1\/4} \\exp\\left(-\\frac{1}{2}\\int \\xd ^3 \\underline{x} \\, \\varphi(\\underline{x})(k \\varphi)(\\underline{x})\\right),\n\\ee \nproviding the identification $k= \\omega$ holds (see \\cite{Oe:timelike,Oe:KGtl,CoOe:smatrixgbf}).\n\n\\subsection{Coherent states}\n\nIn previous works \\cite{CoOe:spsmatrix,CoOe:smatrixgbf,CoOe:smatrix2d,Co:dS} coherent states have been an essential tool for the computation of the asymptotic amplitudes. We follow here the same approach of these works, and define coherent states for the Klein-Gordon field \nin the Schr\\\"odinger representation on the hypersurface of constant $t$, in terms of a complex function $\\eta$ by the expression\n\\be\n\\psi_{t, \\eta}(\\varphi) = K_{t, \\eta} \\, \\exp \\left( \\int \\frac{\\xd ^3 \\underline{x} \\, \\xd^3 \\underline{k}}{(2 \\pi)^3} \\, \\eta(\\underline{k}) \\, e^{\\im \\underline{k}\\cdot \\underline{x}} \\varphi(\\underline{x}) \\right) \\psi_{t,0}(\\varphi),\n\\ee\nwhere the normalization factor $K_{t, \\eta}$ is given by\n\\be\nK_{t, \\eta} = \\exp \\left( - \\frac{\\pi}{8} \\int \\frac{ \\xd^3 \\underline{k}}{(2 \\pi)^3} \\, \\frac{t^3}{R^2} |H_{\\nu}(k t)|^2 \\left( \\eta(\\underline{k}) \\eta(-\\underline{k}) + |\\eta(\\underline{k})|^2\\right) \\right).\n\\ee\nThe inner product of two coherent states defined by the complex functions $\\eta_1$ and $\\eta_2$ results to be\n\\be\n\\langle \\psi_{t, \\eta_2} | \\psi_{t, \\eta_1} \\rangle = \\exp \\left( - \\frac{\\pi}{8} \\int \\frac{ \\xd^3 \\underline{k}}{(2 \\pi)^3} \\, \\frac{t^3}{R^2} |H_{\\nu}(k t)|^2 \\left( |\\eta_1(\\underline{k})|^2 + |\\eta_2(\\underline{k})|^2 - 2 \\overline{\\eta_2(\\underline{k})} \\, \\eta_1(\\underline{k}) \\right) \\right).\n\\ee\nThe coherent states satisfy the following completeness relation\n\\be\nD^{-1} \\int \\xd \\eta \\, \\xd \\overline{\\eta} \\, | \\psi_{t, \\eta} \\rangle \\langle \\psi_{t, \\eta}| = I,\n\\ee\nwith $I$ being the identity operator and the constant $D$ is given by\n\\be\nD= \\int \\xd \\eta \\, \\xd \\overline{\\eta} \\, \\exp \\left( - \\frac{\\pi}{4} \\int \\frac{ \\xd^3 \\underline{k}}{(2 \\pi)^3} \\, \\frac{t^3}{R^2} |H_{\\nu}(k t)|^2 |\\eta(\\underline{k})|^2 \\right).\n\\ee\nThe characteristic property of coherent states is to remain coherent under the action of the free field propagator,\n\\be\n\\psi_{t_2,\\eta_2}(\\varphi_2) = \\int \\xD \\varphi_1 \\, \\psi_{t_1,\\eta_1}(\\varphi_1) \\, Z_{[t_1,t_2],0}(\\varphi_1, \\varphi_2),\n\\ee\nThis equation yields the following relation for the complex functions $\\eta_1$ and $\\eta_2$ defined on the hypersurfaces $t=t_1$ and $t=t_2$ respectively,\n\\be\n\\eta_2(\\underline{k}) = \\frac{t_1^{3\/2} H_{\\nu}(k t_1)}{t_2^{3\/2} H_{\\nu}(k t_2)} \\, \\eta_1(\\underline{k}).\n\\ee \nHence, the product $\\xi(\\underline{k}) = t^{3\/2} H_{\\nu}(k t) \\, \\eta(\\underline{k})$ is preserved under free evolution in the variable $t$. It will be useful to define the interaction representation in terms of the function $\\xi$. The coherent state defined as\n\\be\n\\psi_{t, \\xi}(\\varphi) = K_{t, \\xi} \\, \\exp \\left( \\int \\frac{\\xd ^3 \\underline{x} \\, \\xd^3 \\underline{k}}{(2 \\pi)^3} \\, \\frac{\\xi(\\underline{k})}{t^{3\/2} H_{\\nu}(k t)} \\, e^{\\im \\underline{k}\\cdot \\underline{x}} \\varphi(\\underline{x}) \\right) \\psi_{t,0}(\\varphi),\n\\label{eq:cohstint}\n\\ee\nis invariant under free evolution. We will adopt (\\ref{eq:cohstint}) as the interaction representation for coherent states. The normalization factor in (\\ref{eq:cohstint}) is equal to\n\\be\nK_{t, \\xi} = \\exp \\left( - \\frac{\\pi}{8 R^2} \\int \\frac{\\xd^3 \\underline{k}}{(2 \\pi)^3} \\left( \\frac{\\overline{H_{\\nu}(k t)}}{H_{\\nu}(k t)} \\xi(\\underline{k}) \\xi(-\\underline{k}) + |\\xi(\\underline{k})|^2 \\right) \\right).\n\\label{eq:normcoh}\n\\ee\nCoherent states can be expanded in terms of multiparticle states as\n\\be\n\\psi_{t, \\xi}(\\varphi) = \\exp \\left( - \\frac{\\pi}{8 R^2} \\int \\frac{ \\xd^3 \\underline{k}}{(2 \\pi)^3} \\, |\\xi(\\underline{k})|^2 \\right) \\sum_{n=0}^{\\infty} \\frac{1}{n!} \\int \\xd ^3 \\underline{k}_1 \\cdots \\int \\xd ^3 \\underline{k}_n \\, \\xi(\\underline{k}_1) \\cdots \\xi(\\underline{k}_n) \\, \\psi_{t,\\underline{k}_1, \\dots, \\underline{k}_n}(\\varphi),\n\\label{eq:expmulpar}\n\\ee\nwhere $\\psi_{t,\\underline{k}_1, \\cdots, \\underline{k}_n}$ denotes the $n$-particle state in the interaction picture with quantum numbers $\\underline{k}_1, \\dots, \\underline{k}_n$.\nThe inner product between a coherent state and a multiparticle state then results in being\n\\be\n\\langle \\psi_{t,\\underline{k}_1, \\cdots, \\underline{k}_n} |\\psi_{t, \\xi} \\rangle = \n\\exp \\left( - \\frac{\\pi}{8 R^2} \\int \\frac{ \\xd^3 \\underline{k}}{(2 \\pi)^3} \\, |\\xi(\\underline{k})|^2 \\right)\n\\xi(\\underline{k}_1) \\cdots \\xi(\\underline{k}_n) \\left(\\frac{\\pi}{4 R^2}\\right)^n.\n\\label{eq:mulpar}\n\\ee\n\n\\subsection{Asymptotic amplitude}\n\nWe compute in this section the amplitude (\\ref{eq:ampl}) associated with the spacetime region $[t_1,t_2]$ in the case of the free theory. Considering the state given by the tensor product of two coherent states in the interaction representation defined at times $t_1$ and $t_2$ by the complex functions $\\xi_1$ and $\\xi_2$ respectively, such that $\\psi_{t_1, \\xi_1} \\otimes \\overline{\\psi_{t_2, \\xi_2}} \\in {\\cal H}_{t_1} \\otimes {\\cal H}_{t_2}^*$, where we denote with ${\\cal H}_{t_1}$ the Hilbert space associated with the hypersurface $t=t_1$ and with ${\\cal H}_{t_2}$ the Hilbert space associated with the hypersurface $t=t_2$ with the same orientation. The amplitude for the non interacting theory is obtained from (\\ref{eq:ampl}) and can be interpreted as the transition amplitude from the coherent state $\\psi_{t_1, \\xi_1}$ to the coherent state $\\psi_{t_2, \\xi_2}$,\n\\be\n\\rho_{[t_1,t_2],0}(\\psi_{t_1, \\xi_1} \\otimes \\overline{\\psi_{t_2, \\xi_2}}) = \\int \\xD \\varphi_1 \\, \\xD \\varphi_2 \\, \\overline{\\psi_{t_2, \\xi_2} (\\varphi_2)} \\, \\psi_{t_1, \\xi_1}(\\varphi_1) \\, Z_{[t_1,t_2],0} (\\varphi_1, \\varphi_2).\n\\label{eq:freeampl0}\n\\ee\nThis amplitude is by construction independent of the initial and final times $t_1$ and $t_2$, and reduces to the inner product between the coherent states $\\psi_{t_1, \\xi_1}$ and $\\psi_{t_2, \\xi_2}$,\n\\be\n\\rho_{[t_1,t_2],0}(\\psi_{t_1, \\xi_1} \\otimes \\overline{\\psi_{t_2, \\xi_2}})\n= \\exp \\left( - \\frac{\\pi}{8} \\int \\frac{\\xd^3 \\underline{k}}{(2 \\pi)^3} \\, \\frac{1}{R^2} \\left( |\\xi_1(\\underline{k})|^2 + |\\xi_2(\\underline{k})|^2 - 2 \\overline{\\xi_2( \\underline{k})} \\, \\xi_1(\\underline{k}) \\right) \\right) \n= \\langle \\psi_{\\xi_2} | \\psi_{ \\xi_1} \\rangle.\n\\label{eq:freeampl}\n\\ee\nWe can then trivially take the limit for asymptotic values of the times $t_1$ and $t_2$ and interpret (\\ref{eq:freeampl}) as the elements of the S-matrix ${\\cal S}_0$ of the free theory,\n\\be\n\\langle \\psi_{\\xi_2}| {\\cal S}_0 | \\psi_{\\xi_1} \\rangle = \\langle \\psi_{\\xi_2} | \\psi_{ \\xi_1} \\rangle.\n\\ee\n\\section{Region with timelike boundary - Free theory}\n\\label{sec:hypfree}\n\nIn this and the next section, we develop the quantum theory for a scalar field in the hypercylinder region defined in Section \\ref{sec:hyp}. We will closely follow the treatment of Sections \\ref{sec:free} and \\ref{sec:int}.\n\n\\subsection{Field propagators}\n\\label{sec:hypfreefieldpropagator}\n\nWe start by evaluating the field propagator $Z_{[\\varrho_1,\\varrho_2],0}$ of the free theory associated with the region $[\\varrho_1,\\varrho_2]$, namely the region bounded by the hypercylinders of radii $\\varrho_1$ and $\\varrho_2$. The expression of $Z_{[\\varrho_1,\\varrho_2],0}$ results from the substitution of the action (\\ref{eq:actbound3}) in (\\ref{eq:propcl}),\n\\be\nZ_{[\\varrho_1, \\varrho_2],0}(\\varphi_1, \\varphi_2) = N_{[\\varrho_1, \\varrho_2],0} \\exp \\left( \\frac{\\im}{2} \\int \\xd t \\, \\xd \\Omega \\, \n\\begin{pmatrix}\\varphi_1 & \\varphi_2 \\end{pmatrix} {\\cal W}_{[\\varrho_1, \\varrho_2]} \n\\begin{pmatrix} \\varphi_1 \\\\ \\varphi_2 \\end{pmatrix} \\right),\n\\label{eq:prophyp2}\n\\ee\nwhere the ${\\cal W}_{[\\varrho_1, \\varrho_2]}$ is a 2x2 matrix with elements given by (\\ref{eq:Whyp1}), (\\ref{eq:Whyp2}) and (\\ref{eq:Whyp3}). The propagator $Z_{[\\varrho_1, \\varrho_2],0}$ must satisfy a composition rule analogue to (\\ref{eq:compprop}),\n\\be\nZ_{[\\varrho_1, \\varrho_3],0}(\\varphi_1, \\varphi_3) = \\int \\xD \\varphi_2 \\, Z_{[\\varrho_1, \\varrho_2],0}(\\varphi_1, \\varphi_2) \\, Z_{[\\varrho_2, \\varrho_3],0}(\\varphi_2, \\varphi_3).\n\\ee\nAs in Section \\ref{sec:freefieldpropagator}, this relation fixes the normalization factor appearing in (\\ref{eq:prophyp2}),\n\\be\nN_{[\\varrho_1, \\varrho_2],0} = \\det \\left( - \\frac{\\im}{2 \\pi}\\frac{ R^2}{t^2} \\frac{1}{ k \\, \\Delta_k (\\varrho_1, \\varrho_2)}\\right)^{1\/2}.\n\\label{eq:normfhyp2}\n\\ee\nThe quantum evolution implemented by $Z_{[\\varrho_1, \\varrho_2],0}$ turns out to be unitary since the following relation holds,\n\\be\n\\int \\xD \\varphi_2 \\, \\overline{Z_{[\\varrho_1, \\varrho_2],0} (\\varphi_1, \\varphi_2)} \\, Z_{[\\varrho_1, \\varrho_2],0} (\\varphi_1', \\varphi_2) = \\delta(\\varphi_1 - \\varphi_1'),\n\\label{eq:unitarityhyp}\n\\ee\nas can be easily checked using (\\ref{eq:normfhyp2}). We now turn to the free field propagator associated with the region enclosed by the hypercylinder of radius $\\varrho$. The expression (\\ref{eq:propcl}) with the action (\\ref{eq:actbound2}) gives\n\\be\nZ_{\\varrho,0} (\\varphi)= N_{\\varrho,0} \\exp \\left( -\\frac{1}{2} \\int \\xd t \\, \\xd \\Omega \\, \\varphi(t, \\Omega) \\, \\im \\frac{R^2}{t^2} \\, \\varrho^2 \\,k \\frac{j_l'(k \\varrho)}{j_l(k \\varrho)} \\, \\varphi (t, \\Omega)\\right).\n\\label{eq:prophyp1}\n\\ee\nThe composition rule satisfied by $Z_{\\varrho,0}$ involves the propagator (\\ref{eq:prophyp2}),\n\\be\nZ_{\\varrho_1,0}(\\varphi_1) = \\int \\xD \\varphi_2 \\, Z_{\\varrho_2,0}(\\varphi_2) \\, Z_{[\\varrho_1, \\varrho_2],0}(\\varphi_1, \\varphi_2).\n\\ee\nThis equation translates into the following equality for the normalization factors,\n\\be\nN_{\\varrho_1,0} = N_{\\varrho_2,0} \\, N_{[\\varrho_1,\\varrho_2],0} \\int \\xD \\varphi_2 \\, \\exp \\left( \\frac{\\im}{2} \\int \\xd t \\, \\xd \\Omega \\, \\frac{R^2}{t^2} \\, \\varphi_2(t, \\Omega) \\left( \\frac{1}{k \\, \\Delta_k(\\varrho_1,\\varrho_2)} \\frac{j_l(k \\varrho_1)}{j_l(k \\varrho_2)} \\, \\varphi_2 \\right) (t, \\Omega) \\right).\n\\ee\nUsing (\\ref{eq:normfhyp2}), we find the solution,\n\\be\nN_{\\varrho_1,0} =\\det \\left( \\frac{1}{ j_l(k \\varrho_1)}\\right)^{1\/2}.\n\\label{eq:normfhyp1}\n\\ee\nWe conclude this subsection by looking at the asymptotic limit of the field propagators (\\ref{eq:prophyp2}) and (\\ref{eq:prophyp1}). We expect to recover the corresponding field propagators defined in Minkowski spacetime where the notion of the hypercylinder was originally introduced, in particular we refer to Section IV.A.2 of \\cite{CoOe:smatrixgbf}. The comparison between de Sitter propagators (\\ref{eq:prophyp2}) and (\\ref{eq:prophyp1}), and Minkowski propagators (74) and (75) of \\cite{CoOe:smatrixgbf}, manifests only one significant difference: The presence of the quotient $\\frac{R^2}{t^2}$ in the expression of the former propagators. We notice that it is precisely this quotient that get canceled in the limit (\\ref{eq:dStoM}), provided that $k$ in (\\ref{eq:prophyp2}) and (\\ref{eq:prophyp1}) equals $\\sqrt{E^2-m^2}$ (for $E^2>m^2$) appearing in the propagators (74) and (75) of \\cite{CoOe:smatrixgbf}. This identification is consequently valid only for positive $k$. An alternative would be to replace $k$ with $|k|$. Such change will receives its justification in the next section, where we introduce the vacuum state.\n\n\n\\subsection{Vacuum state}\n\\label{sec:vacuumhyp}\n\nThe Gaussian ansatz for the vacuum wave function associated with the hypercylinder of radius $\\varrho$ reads\n\\be\n\\psi_{\\varrho,0}(\\varphi)=C_{\\varrho} \\exp\\left(-\\frac{1}{2}\\int \\xd t \\, \\xd \\Omega \\, \\varphi(t, \\Omega)(A_{\\varrho} \\varphi)(t, \\Omega)\\right) ,\n\\ee\nwhere $C_{\\varrho}$ is the normalization factor and $A_{\\varrho}$ denotes a family of operators indexed by ${\\varrho}$. Applying the same techniques developed in \\cite{Co:vacuum}, we obtain the following form for the vacuum wave function,\n\\be\n \\psi_{\\varrho,0}(\\varphi) = C_{\\varrho} \\exp\\left(\\frac{\\im}{2}\\int \\xd t \\, \\xd \\Omega \\, \\frac{R^2}{t^2} \\varrho^2 \\, \\varphi(t, \\Omega) |k| \\frac{h_{l}'(|k| \\varrho)}{h_{l}(|k| \\varrho)} \\varphi(t, \\Omega)\\right),\n\\label{eq:vachyp}\n\\ee \nwhere $h_l$ is the spherical Bessel function of the third kind, $h_l = j_l + \\im n_l$, and the appearance of the modulus of $k$ guarantees the normalizability of the vacuum state, as can be seen from the analytic continuation of the spherical Bessel functions (see in particular formulas 10.1.34-35 of \\cite{AbSt:handbook}). Based on these properties and the observation at the end of the previous section concerning the asymptotic limit of the field propagator, in the rest of paper the spherical Bessel functions will depend on $|k|$ and not $k$.\n$C_{\\varrho}$ is the normalization factor given by (up to a phase)\n\\bea\n|C_{\\varrho}|^{-2} = \\int \\xD \\varphi \\, \\exp \\left( - \\frac{1}{2} \\int \\xd t \\, \\xd \\Omega \\, \\varphi(t, \\Omega) \\, \\frac{R^2}{t^2} \\frac{1}{|k| \\, |h_l(|k| \\varrho)|^2} \\, \\varphi(t, \\Omega)\\right) = \\det \\left( \\frac{R^2}{2 \\pi \\, t^2 \\, |k| \\, |h_l(|k| \\varrho)|^2}\\right)^{-1\/2}.\n\\eea\nAs in Section \\ref{sec:vacuum}, the phase of $C_{\\varrho}$ is fixed by the invariance of the vacuum state under free evolution,\n\\be\n\\psi_{\\varrho_2,0}(\\varphi_2) = \\int \\xD \\varphi_1 \\, \\psi_{\\varrho_1,0}(\\varphi_1) \\, Z_{[\\varrho_1,\\varrho_2],0}(\\varphi_1, \\varphi_2),\n\\ee\nwhich implies the following identity for the normalization factors\n\\bea\nC_{\\varrho_2} &=& C_{\\varrho_1} \\, N_{[\\varrho_1,\\varrho_2],0} \\, \\int \\xD \\varphi_1 \\exp \\left( - \\frac{1}{2} \\int \\xd t \\, \\xd \\Omega \\, \\frac{R^2}{t^2} \\, \\varphi_1 \\left[- \\frac{\\im}{|k| \\Delta_k(\\varrho_1,\\varrho_2)} \\, \\frac{h_{l}(|k| \\varrho_2)}{ h_{l}(|k| \\varrho_1)} \\right] \\varphi_1 \\right), \\nonumber\\\\\n&=& C_{\\varrho_1} \\, \\det \\left( \\frac{h_{l}(|k| \\varrho_2)}{ h_{l}(|k| \\varrho_1)} \\right)^{-1\/2},\n\\eea\nwhere (\\ref{eq:normfhyp2}) has been used. It can be easily verify that the above equation admits the solution\n\\be\nC_{\\varrho} = \\det \\left( \\frac{R}{t \\sqrt{2 \\pi \\, |k|} \\, h_l(|k| \\varrho)}\\right)^{1\/2}.\n\\ee\nThe asymptotic limit of (\\ref{eq:vachyp}) coincides with the vacuum wave function defined on the hypercylinder in Minkowski spacetime if $|k|=\\sqrt{E^2-m^2}$ (for $E^2>m^2$), see (79) and (80) of \\cite{CoOe:smatrixgbf}. \n\nNow that we have at our disposal both their asymptotic limit, we can compare the vacuum state defined on the hypersurface of constant time $t$, (\\ref{eq:vac}), with the one on the hypercylinder of radius $\\varrho$, (\\ref{eq:vachyp}). This amounts to compare the corresponding asymptotic Minkowski vacuum wave functions, which are related at spatial and temporal infinity as explained in \\cite{CoOe:smatrixgbf}. We then conclude that the same relation holds for the vacua (\\ref{eq:vac}) and (\\ref{eq:vachyp}).\n\nThe next step is the definition of coherent states on the hypercylinder.\n\n\n\\subsection{Coherent states}\n\nThe wave function of a coherent state defined on the hypercylinder of radius $\\varrho$, in Schr\\\"odinger representation, is parametrized by a complex function $\\eta$,\n\\be\n\\psi_{\\varrho, \\eta}(\\varphi) = K_{\\varrho, \\eta} \\, \\exp \\left( \\int \\xd t \\, \\xd \\Omega \\, \\xd k \\sum_{l,m} \\eta_{l,m}(k) \\, \n|t|^{-1\/2} \\overline{{\\mathscr H}_{\\nu}(kt) } Y_{l}^{-m}(\\Omega) \\, \\frac{|k|}{4} \n\\varphi(t, \\Omega) \\right) \\psi_{\\varrho,0}(\\varphi),\n\\ee\nwhere the integration in $k$ and the sum over the indexes $l$ and $m$ are defined as in (\\ref{eq:classsolhyp}). The normalization factor $K_{\\varrho, \\eta}$ results to be\n\\bea\nK_{\\varrho, \\eta}\n&=& \\exp \\left(- \\frac{1}{16 \\, R^2} \\int\n \\xd k \\sum_{l,m} \\, k^2 |h_l(|k| \\varrho)|^2 \\left[ |\\eta_{l,m}(k)|^2 - \\eta_{l,m}(k) \\eta_{l,-m}(-k) \\right] \\right).\n\\eea\nThe inner product of two coherent states defined by the complex functions $\\eta$ and $\\eta'$ results to be\n\\be\n\\langle \\psi_{\\varrho, \\eta'} | \\psi_{\\varrho, \\eta} \\rangle = \\exp \\left(- \\frac{1}{16 \\, R^2} \\int\n \\xd k \\sum_{l,m} \\, k^2 |h_l(|k| \\varrho)|^2 \\left[ |\\eta_{l,m}(k)|^2 + |\\eta_{l,m}'(k)|^2 - 2 \\eta_{l,m}(k) \\overline{\\eta_{l,m}'(k)} \\right] \\right).\n\\ee\nThe completeness relation satisfied by the coherent states satisfy can be written as\n\\be\nD^{-1} \\int \\xd \\eta \\, \\xd \\overline{\\eta} \\, | \\psi_{\\varrho, \\eta} \\rangle \\langle \\psi_{\\varrho, \\eta}| = I,\n\\label{eq:complrelhyp}\n\\ee\nwith $I$ being the identity operator and the constant $D$ is given by\n\\be\nD= \\int \\xd \\eta \\, \\xd \\overline{\\eta} \\, \\exp \\left( - \\frac{1}{8 R^2} \\int \\xd k \\sum_{l,m} \\, k^2 \\, |h_{l}(|k| \\varrho)|^2 \\, |\\eta_{l,m}(k)|^2 \\right).\n\\ee\n\nTwo coherent states parametrized by the complex functions $\\eta_1$ and $\\eta_2$, defined respectively on the hypercylinders of radii $\\varrho_1$ and $\\varrho_2$ are related by\n\\be\n\\psi_{\\varrho_2,\\eta_2}(\\varphi_2) = \\int \\xD \\varphi_1 \\, \\psi_{\\varrho_1,\\eta_1}(\\varphi_1) \\, Z_{[\\varrho_1,\\varrho_2],0}(\\varphi_1, \\varphi_2).\n\\ee\nThis equality holds if the complex functions $\\eta_1$ and $\\eta_2$ satisfy\n\\be\n\\eta_{1,l,m}(k) = \\eta_{2,l,m}(k) \\frac{h_{l}(|k| \\varrho_2)}{h_{l}(|k| \\varrho_1)}.\n\\ee\nAs a consequence, the function $\\xi_{l,m}(k) = \\eta_{l,m}(k) h_{l}(|k| \\varrho)$ appears to be independent of the radius $\\varrho$, since it is preserved under free evolution. The interaction picture for coherent states will then be defined in terms of $\\xi$, namely\n\\be\n\\psi_{\\varrho, \\xi}(\\varphi) = K_{\\varrho, \\xi} \\, \\exp \\left( \\int \\xd t \\, \\xd \\Omega \\, \\xd k \\sum_{l,m} \\xi_{l,m}(k) \\frac{ t^{-1\/2} \\overline{{\\mathscr H}_{\\nu}(kt) } Y_{l}^{-m}(\\Omega)}{h_l(|k| \\varrho)} \\, \\varphi(t, \\Omega) \\right) \\psi_{\\varrho,0}(\\varphi).\n\\ee\nThe normalization factor $K_{\\varrho, \\xi}$ is equal to\n\\be\nK_{\\varrho, \\xi} = \\exp \\left( - \\frac{1}{16 R^2} \\int \\xd k \\sum_{l,m} \\, k^2 \\left( |\\xi_{l,m}(k)|^2 - \\frac{\\overline{h_{l}(|k| \\varrho)}}{h_{l}(|k| \\varrho)} \\xi_{l,m}(k) \\xi_{l,-m}(-k) \\right) \\right).\n\\label{eq:normcohhyp}\n\\ee\nThe completeness relation (\\ref{eq:complrelhyp}) in the interaction picture reads\n\\be\n\\tilde{D}^{-1} \\int \\xd \\xi \\, \\xd \\overline{\\xi} \\, | \\psi_{\\varrho, \\xi} \\rangle \\langle \\psi_{\\varrho, \\xi}| = I,\n\\label{eq:complrelhyp2}\n\\ee\nwhere the constant $\\tilde{D}$ is \n\\be\n\\tilde{D}= \\int \\xd \\xi \\, \\xd \\overline{\\xi} \\, \\exp \\left( - \\frac{1}{8 R^2} \\int \\xd k \\sum_{l,m} \\, k^2 \\, |\\xi_{l,m}(k)|^2 \\right).\n\\label{eq:Dtilde}\n\\ee\n\nThe expansion of a coherent state in terms of multiparticle states reads\n\\bea\n\\psi_{\\varrho, \\xi}(\\varphi) &=& \\exp \\left( - \\frac{1}{16 R^2} \\int \\xd k \\sum_{l,m} k^2 \\, |\\xi_{l,m}(k)|^2 \\right) \\sum_{n=0}^{\\infty} \\frac{1}{n!} \\nonumber\\\\\n& \\times & \\int \\xd k_1 \\sum_{l_1,m_1} \\cdots \\int \\xd k_n \\sum_{l_n,m_n} \\xi_{l_1,m_1}(k_1) \\cdots \\xi_{l_n,m_n}(k_n) \\, \\psi_{\\varrho, (k_1,l_1,m_1), \\dots, (k_n,l_n,m_n)} (\\varphi).\n\\label{eq:expmulparhyp}\n\\eea\nThe state with $n$ particles with quantum numbers $(k_1,l_1,m_1), \\dots, (k_n,l_n,m_n)$ has been denoted by $\\psi_{\\varrho, (k_1,l_1,m_1), \\dots, (k_n,l_n,m_n)}$ in the interaction picture. The inner product between a coherent state and an $n$-particle state is\n\\bea\n\\langle \\psi_{\\varrho, (k_1,l_1,m_1), \\dots, (k_n,l_n,m_n)} |\\psi_{\\varrho, \\xi} \\rangle &=& \n\\exp \\left( - \\frac{1}{16 R^2} \\int \\xd k \\sum_{l,m} k^2 \\, |\\xi_{l,m}(k)|^2 \\right) \\nonumber\\\\\n& \\times &\n\\xi_{l_1,m_1}(k_1) \\cdots \\xi_{l_n,m_n}(k_n) \\frac{k_1^{2}}{4 R^{2}} \\cdots \\frac{k_n^{2}}{4 R^{2}}.\n\\label{eq:mulparhyp}\n\\eea\n\n\n\\subsection{Asymptotic amplitude}\n\\label{sec:asyamplfree}\n\nWe now have all the ingredients to evaluate the amplitude associated with the spacetime region enclosed by the hypercylinder of radius $\\varrho$, for the coherent state $\\psi_{\\varrho, \\xi}$ defined on the hypercylinder, in the case of the free theory. According to (\\ref{eq:ampl}), this amplitude is\n\\bea\n\\rho_{\\varrho,0}(\\psi_{\\varrho, \\xi} ) &=& \\int \\xD \\varphi \\, \\psi_{\\varrho, \\xi}(\\varphi) \\, Z_{\\varrho,0} (\\varphi), \\nonumber\\\\\n&=& \\exp \\left( - \\frac{1}{16 R^2} \\int \\xd k \\sum_{l,m} \\, k^2 \\left[ |\\xi_{l,m}(k)|^2 + \\xi_{l,m}(k) \\, \\xi_{l,-m}(-k) \\right] \\right).\n\\label{eq:freeamplhyp}\n\\eea\nThis amplitude is manifestly independent of the radius $\\varrho$ of the hypercylinder.\n\n\n\\section{Region with timelike boundary - Interacting theory}\n\\label{sec:hypint}\n\nThe study of the interacting Klein-Gordon theory in the spacetime region bounded by the hypercylinder will parallel the analysis performed in Section \\ref{sec:int}. We turn first to the theory describing the interaction with an external source field and we then conclude the section by the treatment of the general interacting theory.\n\n\\subsection{Theory with source}\n\nThe action of a scalar field interacting with a real source field $\\mu$ has the form\n\\be\nS_{\\varrho,\\mu} (\\phi)= S_{\\varrho,0} (\\phi) + \\int \\xd ^4 x \\sqrt{-g(x)} \\, \\mu(x) \\, \\phi(x),\n\\ee\nwhere $S_{\\varrho,0}$ is the action of the free theory (\\ref{eq:actbound2}). The source field $\\mu$ is assumed to vanish outside the spacetime region $\\varrho$, namely $\\mu(t,r,\\Omega)\\big|_{r \\geq \\varrho} =0$.\n\n\\subsubsection{Field propagator}\n\nThe field propagator can be expressed in terms of the field propagator of the free theory,\n\\be\nZ_{\\varrho,\\mu}(\\varphi) = \\frac{N_{\\varrho,\\mu}}{N_{\\varrho,0}} \\, Z_{\\varrho,0}(\\varphi) \\, \\exp \\left( \\im \\int \\xd t \\, \\xd \\Omega \\, M_l(t,\\Omega) \\, \\frac{1}{j_l(|k| \\varrho)} \\, \\varphi(t, \\Omega)\\right),\n\\label{eq:propsrchyp2}\n\\ee\nwhere we used the expression of the field $\\phi$ in terms of the boundary field configuration $\\varphi$, given by (\\ref{eq:hypbound}), and we introduced the quantity\n\\be\nM_l(t, \\Omega) = \\int_0^{\\infty} \\xd r \\, \\sqrt{-g(t,r,\\Omega)} \\, j_l(|k| r) \\, \\mu(t,r,\\Omega).\n\\ee\nThe quotient of normalization factor appearing in the right hand side of (\\ref{eq:propsrchyp2}) can be rewritten as follows,\n\\be\n\\frac{N_{\\varrho,\\mu}}{N_{\\varrho,0}} = \\exp \\left( \\frac{\\im}{2} \\int \\xd ^4 x \\sqrt{-g(x)} \\, \\alpha(x) \\, \\mu(x)\\right),\n\\ee\nwhere the function $\\alpha$ is a solution of the inhomogeneous Klein-Gordon equation\n\\be\n\\left( \\frac{t^2}{R^2} \\left[ \\partial_t^2 - \\Delta_r - \\Delta_{\\Omega}\\right] -\\frac{2 t}{R^2} \\partial_t + m^2 \\right)\\alpha(t,r, \\Omega) = \\mu(t,r, \\Omega),\n\\label{eq:inKGhyp}\n\\ee\nwith vanishing boundary condition,\n\\be\n\\alpha \\big|_{r= \\varrho} = 0.\n\\label{eq:bchyp}\n\\ee\nWe express the solution of (\\ref{eq:inKGhyp}) with the boundary condition (\\ref{eq:bchyp}) in terms of the Green function $G$ of the Klein-Gordon equation,\n\\be\nG(x,x') = \\frac{1}{2 R^2} \\int \\xd k \\sum_{l,m} k^2 \\, (tt')^{3\/2} {\\mathscr H}_{\\nu}(k t) \\overline{{\\mathscr H}_{\\nu}(kt') } Y_l^{m}(\\Omega) \\overline{Y_l^{m}(\\Omega')} \\, g_{l}(k,r,r'),\n\\ee\nwhere the function $g_l$ is given by\n\\be\ng_l(k,r,r') = \\theta(r-r')[ j_l(|k| r) \\, n_l(|k| r')- n_l(|k| r) \\, j_l(|k| r')] - j_l(|k| r) \\, n_l(|k| r')+ j_l(|k| r) \\, \\frac{n_l(|k| \\varrho)}{j_l(|k| \\varrho)}\\, j_l(|k| r').\n\\ee\nThe function $\\alpha$ can then be written as\n\\be\n\\alpha(x) = \\int \\xd ^4 x' \\sqrt{-g(x')} \\, G(x,x') \\, \\mu(x').\n\\ee\nBy substituting this result in (\\ref{eq:propsrchyp2}), the field propagator takes the form\n\\be\nZ_{\\varrho,\\mu}(\\varphi) = Z_{\\varrho,0}(\\varphi) \\exp \\left( \\im \\int \\xd t \\, \\xd \\Omega \\, M_l(t,\\Omega) \\frac{1}{j_l(|k| \\varrho)} \\varphi(t, \\Omega)\\right) \\exp \\left( \\frac{\\im}{2} \\int \\xd ^4 x \\, \\xd ^4 x'\\sqrt{g(x) g(x')} \\, \\mu(x) G(x,x') \\mu(x')\\right).\n\\label{eq:propsrchyp}\n\\ee\n\n\\subsection{Asymptotic amplitude}\n\nThe amplitude $\\rho_{\\varrho,\\mu}$ associated with the spacetime region $\\varrho$, for the coherent state $\\psi_{\\varrho,\\xi}$ defined on the boundary of this region is\n\\be\n\\rho_{\\varrho,\\mu}(\\psi_{\\varrho,\\xi}) = \\int \\xD \\varphi \\, \\psi_{\\varrho,\\xi}(\\varphi) \\, Z_{\\varrho,\\mu}(\\varphi).\n\\label{eq:amplsrchyp1}\n\\ee\nIntroducing the function $\\tilde{\\xi}$, given by\n\\be\n\\tilde{\\xi}(t, \\Omega) \\defeq \\int \\xd k \\sum_{l} \\left( \\sum_m \\xi_{l,m}(k) \\frac{ t^{-1\/2} \\overline{{\\mathscr H}_{\\nu}(kt)} Y_{l}^{-m}(\\Omega)}{h_l(|k| \\varrho)} + \\frac{\\im}{j_l(|k| \\varrho)} \\, M_l(t,\\Omega) \\right),\n\\ee\nwe can define the new coherent state $\\psi_{\\varrho, \\tilde{\\xi}}$. The amplitude (\\ref{eq:amplsrchyp1}) can then be expressed in terms of the free amplitude (\\ref{eq:freeamplhyp}) for the new coherent state defined by the function $\\tilde{\\xi}$, \n\\be\n\\rho_{\\varrho,\\mu}(\\psi_{\\varrho,\\xi}) = \\rho_{\\varrho,0}(\\psi_{\\varrho,\\tilde{\\xi}}) \\, \\frac{K_{\\varrho,\\xi}}{K_{\\varrho,\\tilde{\\xi}}} \\, \n\\exp \\left( \\frac{\\im}{2} \\int \\xd ^4 x \\, \\xd ^4 x' \\sqrt{g(x) g(x')}\\, \\mu(x) \\, G(x,x') \\, \\mu(x')\\right).\n\\ee\nSubstituting the expressions (\\ref{eq:freeamplhyp}) and (\\ref{eq:normcohhyp}) we arrive at\n\\begin{multline}\n\\rho_{\\varrho,\\mu}(\\psi_{\\varrho,\\xi}) = \\rho_{\\varrho,0}(\\psi_{\\varrho,\\xi}) \\, \\exp \\left( \\im \\int \\xd ^4 x \\sqrt{-g(x)} \\, \\mu(x) \\, \\hat{\\xi}(x)\\right) \\\\ \n\\times \\exp \\left( \\frac{\\im}{2} \\int \\xd ^4 x \\, \\xd ^4 x' \\sqrt{g(x) g(x')} \\, \\mu(x) [G(x,x') + \\beta(x,x')] \\mu(x') \n\\right),\n\\label{eq:amplsrchyp3}\n\\end{multline}\nwhere the function $\\hat{\\xi}$ is the complex classical solution of the Klein-Gordon equation parametrized by $\\xi$,\n\\be\n\\hat{\\xi}(x) = - \\frac{1}{8 R^2} \\int \\xd k \\sum_{l,m} j_l(|k| r) \\, k^2 \\left( \\xi_{l,m}(k) \\, \\overline{Y_l^m(\\Omega)} \\, |t|^{3\/2} \\overline{{\\mathscr H}_{\\nu}}(k t) - \\xi_{l,-m}(-k) \\, Y_l^m(\\Omega) \\, |t|^{3\/2} {\\mathscr H}_{\\nu}(k t) \\right),\n\\label{eq:hatxihyp}\n\\ee\nimplying a one-to-one correspondence between coherent states parametrized by functions $\\xi$ and complex solutions $\\hat{\\xi}$.\nThe function $\\beta$ in the right hand side of (\\ref{eq:amplsrchyp3}) is equal to\n\\be\n\\beta(x,x') = \\frac{\\im}{4 R^2} \\int \\xd k \\sum_{l,m} k^2 \\, (tt')^{3\/2} {\\mathscr H}_{\\nu}(k t) \\overline{{\\mathscr H}_{\\nu}(kt') } Y_l^m(\\Omega) \\overline{Y_l^m(\\Omega')} \\, j_l(|k| r) \\frac{h_l(|k| \\varrho)}{j_l(|k| \\varrho)} j_l(|k| r').\n\\ee\nWe denote by $G_F$ the sum of the Green $G$ and the function $\\beta$ that appears in the last exponential of (\\ref{eq:amplsrchyp3}),\n\\begin{flalign}\nG_F(x,x') \\defeq& \\, \\, G(x,x') + \\beta(x,x'),& & \\nonumber\\\\\n=& \\, \\, \n\\frac{\\im}{4 R^2} \\int \\xd k \\sum_{l,m} k^2 \\, (tt')^{3\/2} {\\mathscr H}_{\\nu}(k t) \\overline{{\\mathscr H}_{\\nu}(kt') } \\, Y_l^m(\\Omega) \\overline{Y_l^m(\\Omega')} & & \\nonumber\n\\end{flalign}\n\\vspace{-8mm}\n\\begin{flalign}\n& & \\times \\left( \\theta(r-r') \\, j_l(|k| r') \\, h_l(|k| r) + \\theta(r'-r) \\, j_l(|k| r) \\, h_l(|k| r') \\right) &.\n\\label{eq:GreenFhyp}\n\\end{flalign}\nThis function solves the inhomogeneous Klein-Gordon equation, \n\\be\n\\left( \\frac{t^2}{R^2} \\left[ \\partial_t^2 - \\Delta_r - \\Delta_{\\Omega}\\right] -\\frac{2 t}{R^2} \\partial_t + m^2 \\right) G(t,r,\\Omega,t',r',\\Omega') = (-g(t,r,\\Omega))^{-1\/2} \\delta(t-t')\\delta(r-r')\\delta(\\Omega-\\Omega'),\n\\ee\nwith the boundary condition\n\\be\nG(t,r,\\Omega,t',r',\\Omega')\\bigg|_{r = \\varrho} = \\frac{\\im}{4 R^2} \\int_0^{\\infty} \\xd k \\sum_{l,m} k^2 \\, (tt')^{3\/2} {\\mathscr H}_{\\nu}(k t) \\overline{{\\mathscr H}_{\\nu}(kt') } \\, Y_l^m(\\Omega) \\overline{Y_l^m(\\Omega')} j_l(|k| r') h_l(|k| \\varrho).\n\\label{eq:bcGreenFhyp}\n\\ee\nThe equivalence of (\\ref{eq:GreenFhyp}) with the Feynman propagator $G_F$ obtained in Section \\ref{sec:asymampfree1} is shown in Appendix \\ref{sec:apB}.\nFinally, we write the amplitude $\\rho_{\\varrho,\\mu}$ as\n\\begin{multline}\n\\rho_{\\varrho,\\mu}(\\psi_{\\varrho,\\xi}) = \\rho_{\\varrho,0}(\\psi_{\\varrho,\\xi}) \\, \\exp \\left( \\im \\int \\xd ^4 x \\sqrt{-g(x)} \\, \\mu(x) \\, \\hat{\\xi}(x)\\right) \\\\ \n\\times \\exp \\left( \\frac{\\im}{2} \\int \\xd ^4 x \\, \\xd ^4 x' \\sqrt{g(x) g(x')} \\, \\mu(x) \\,G_F(x,x')\\, \\mu(x') \n\\right).\n\\label{eq:amplsrchyp4}\n\\end{multline}\nBased on the result of Section \\ref{sec:asyamplfree}, no dependence on the radius $\\varrho$ of the hypercylinder appears in the amplitude $\\rho_{\\varrho,\\mu}$. Consequently, expression (\\ref{eq:amplsrchyp4}) coincides with the asymptotic amplitude for $\\varrho \\rightarrow \\infty$.\n\n\n\n\n\\subsection{General interaction}\n\\label{sec:genint2}\n\nThe asymptotic amplitude in the presence of a general interaction can be computed by means of the same functional derivatives techniques used in Section \\ref{sec:genint1}. In the present case, we require that the potential $V$ defining the interaction vanishes outside the hypercylinder region, \n\\be\nV((t,r,\\Omega), \\phi(t,r, \\Omega)) = 0, \\qquad \\text{if $r \\geq \\varrho$}.\n\\ee\nThe computation of the amplitude for a generic $\\psi_{\\varrho}$ state on the hypercylinder gives the following result\n\\be\n\\rho_{\\varrho, V}(\\psi_{\\varrho}) = \\exp \\left(\\im \\int \\xd^4 x \\sqrt{-g(x)} \\, V\\left(x, -\\im \\frac{\\delta}{\\delta \\mu(x)} \\right) \\right)\n\\rho_{\\varrho, \\mu}(\\psi_{\\varrho}) \\bigg|_{\\mu=0}.\n\\label{eq:genintasyamp2}\n\\ee\nThe amplitude in the presence of a source interaction, $\\rho_{\\varrho, \\mu}$, does not depend on the radius of the hypercylinder. We can then lift the restriction on $V$, and interpret (\\ref{eq:genintasyamp2}) as the asymptotic amplitude for the general interacting theory.\n\\section{Region with spacelike boundary - Interacting theory}\n\\label{sec:int}\n\nWe now turn to the study of the interacting theory. We will start in the next section by considering the interaction of the scalar field with a source field confined inside the spacetime region of interest. The amplitude derived in this case will then be used in the subsequent section to express the amplitude for a general interaction by means of functional derivative techniques, following the same approach as in \\cite{CoOe:smatrixgbf}\n\n\\subsection{Theory with source}\n\nConsider the interaction of the scalar field with a real source field $\\mu$ described by the action\n\\be\nS_{[t_1,t_2],\\mu}(\\phi) = S_{[t_1,t_2],0}(\\phi) + \\int \\xd ^4 x \\, \\sqrt{-g(x)} \\, \\mu(x) \\phi(x),\n\\label{eq:actionsrc}\n\\ee\nwhere $S_{[t_1,t_2],0}$ is the free action (\\ref{eq:action}). We assume that the source field $\\mu$ vanishes outside the spacetime region $[t_1,t_2]$. \n\n\\subsubsection{Field propagator}\n\nThe field propagator corresponding to the action (\\ref{eq:actionsrc}) can be evaluate with the same technique applied in Section \\ref{sec:qt}, namely shift the integration variable of the path integral by a classical solution of the free theory matching the boundary configurations $\\varphi_1$ and $\\varphi_2$ on $t=t_1$ and $t=t_2$ respectively. The field propagator then results to be expressed in terms of the free field propagator (\\ref{eq:propt}),\n\\be\nZ_{[t_1,t_2],\\mu}(\\varphi_1, \\varphi_2) = \\frac{N_{[t_1,t_2],\\mu}}{N_{[t_1,t_2],0}} \\, Z_{[t_1,t_2],0}(\\varphi_1, \\varphi_2) \\, \\exp \\left(\\im \\int \\xd ^3 \\underline{x} \\left( \\mu_1(\\underline{x}) \\varphi_1(\\underline{x}) + \\mu_2(\\underline{x}) \\varphi_2(\\underline{x}) \\right) \\right),\n\\label{eq:propmu}\n\\ee\nwhere the subscript $\\mu$ has been added to the quantity referring to the interacting theory. The quantity $\\mu_1$ and $\\mu_2$ in the right-hand side of (\\ref{eq:propmu}) are defined as\n\\bea\n\\mu_1(\\underline{x}) &\\defeq& \\int_{t_1}^{t_2} \\xd t \\, \\sqrt{-g(t, \\underline{x})} \\, \\frac{\\delta_k(t, t_2)}{\\delta_k(t_1, t_2)} \\, \\mu(t, \\underline{x}), \\\\\n\\mu_2(\\underline{x}) &\\defeq& \\int_{t_1}^{t_2} \\xd t \\, \\sqrt{-g(t, \\underline{x})} \\, \\frac{\\delta_k(t_1, t)}{\\delta_k(t_1, t_2)} \\, \\mu(t, \\underline{x}).\n\\eea\nThe normalization factor $N_{[t_1,t_2],\\mu}$ is formally equal to\n\\be\nN_{[t_1,t_2],\\mu} = \\int_{\\phi|_{t_1}=\\phi|_{t_2}=0} \\xD \\phi \\, e^{\\im S_{[t_1,t_2],\\mu} (\\phi)}.\n\\label{eq:Nmu}\n\\ee\nAgain we evaluate the integral by shifting of the integration variable. In this case we shift $\\phi$ by the function $\\alpha$, solution of the inhomogeneous Klein-Gordon equation\n\\be\n\\left[ \\frac{t^2}{R^2} \\left(\\partial_t^2 - \\Delta_{\\underline{x}} \\right) -\\frac{2t}{R^2} \\partial_t +m^2 \\right] \\alpha(t, \\underline{x}) =\\mu(t, \\underline{x}),\n\\ee\nwith vanishing boundary conditions,\n\\be\n\\alpha |_{t=t_1} = \\alpha |_{t=t_2}=0.\n\\ee\nThe function $\\alpha$ results to be\n\\be\n\\alpha(x) = \\int_{[t_1,t_2]} \\xd^4 x' \\, \\sqrt{-g(x')} \\, G(x,x')\\, \\mu(x'),\n\\ee\nwhere $G(x,x')$ is the Green function of the Klein-Gordon equation, with vanishing boundary conditions, given by\n\\be\nG(x,x') = \\frac{\\pi}{2 R^2} \\int \\frac{\\xd ^3 \\underline{k}}{(2 \\pi)^3} \\left( \\frac{ \\delta_k(t,t_1)\\delta_k(t',t_2)}{\\delta_k(t_1,t_2)} - \\theta(t-t') \\, \\delta_k(t,t') \\right) e^{\\im \\underline{k}\\cdot(\\underline{x}-\\underline{x}')},\n\\label{eq:green}\n\\ee\nwhere $\\theta(t)$ is the step function\n\\be\n\\theta(t) = \\left\\{ \\begin{matrix} 1 & \\mbox{if} \\ \\ t>0, \\\\ 0 & \\mbox{if} \\ \\ t<0. \\end{matrix} \\right.\n\\ee\nFinally the noramlization factor (\\ref{eq:Nmu}) can be written as\n\\be\nN_{[t_1,t_2],\\mu} = N_{[t_1,t_2],0} \\, \\exp \\left( \\frac{\\im}{2} \\int_{[t_1,t_2]^2} \\xd^4 x \\, \\xd^4 x' \\, \\sqrt{g(x')g(x)} \\, \\mu(x) \\, G(x,x')\\, \\mu(x') \\right).\n\\label{eq:Nmu2}\n\\ee\nSubstituting in (\\ref{eq:propmu}), we obtain for the field propagator the expression\n\\begin{align}\nZ_{[t_1,t_2],\\mu}(\\varphi_1, \\varphi_2) =& \\, Z_{[t_1,t_2],0}(\\varphi_1, \\varphi_2) \\, \\exp \\left(\\im \\int \\xd ^3 \\underline{x} \\left( \\mu_1(\\underline{x}) \\varphi_1(\\underline{x}) + \\mu_2(\\underline{x}) \\varphi_2(\\underline{x}) \\right) \\right) \\nonumber\\\\\n& \\times \\exp \\left( \\frac{\\im}{2} \\int_{[t_1,t_2]^2} \\xd^4 x \\, \\xd^4 x' \\, \\sqrt{g(x')g(x)} \\, \\mu(x) \\, G(x,x')\\, \\mu(x') \\right).\n\\label{eq:propmu2}\n\\end{align}\nIt can be shown that the propagator (\\ref{eq:propmu2}) satisfies the composition rule analogue to (\\ref{eq:compprop}),\n\\be\nZ_{[t_1, t_3],\\mu} (\\varphi_1, \\varphi_3) = \\int \\xD \\varphi_2 \\, Z_{[t_1, t_2],\\mu} (\\varphi_1, \\varphi_2) \\, Z_{[t_2, t_3],\\mu} (\\varphi_2, \\varphi_3). \n\\ee\n\nThe unitarity condition (\\ref{eq:unitarity}) for the field propagator (\\ref{eq:propmu}) reads\n\\bea\n \\int \\xD \\varphi_2 \\, \\overline{Z_{[t_1, t_2],\\mu} (\\varphi_1, \\varphi_2)} \\, Z_{[t_1, t_2],\\mu} (\\varphi_1', \\varphi_2)\n &=& \\left| \\frac{N_{[t_1,t_2],\\mu}}{N_{[t_1,t_2],0}}\\right|^2 \\int \\xD \\varphi_2 \\, \\overline{Z_{[t_1, t_2],0} (\\varphi_1, \\varphi_2)} \\, Z_{[t_1, t_2],0} (\\varphi_1', \\varphi_2) \\nonumber\\\\\n&& \\times \\exp \\left(\\im \\int \\xd ^3 \\underline{x} \\, \\mu_1(\\underline{x}) (\\varphi_1(\\underline{x}) - \\varphi_1'(\\underline{x})) \\right), \\nonumber\\\\\n\\eea\nWe notice that the quotient of the normalization factors has modulus one because of the reality of the source field $\\mu$ and the Green function (\\ref{eq:green}) appearing in the exponential in right hand side of (\\ref{eq:Nmu2}). Recalling the result obtained in Section (\\ref{sec:freefieldpropagator}), the integral in $\\varphi_2$ gives\n\\be\n\\int \\xD \\varphi_2 \\, \\overline{Z_{[t_1, t_2],\\mu} (\\varphi_1, \\varphi_2)} \\, Z_{[t_1, t_2],\\mu} (\\varphi_1', \\varphi_2)\n= \\exp \\left(\\im \\int \\xd ^3 \\underline{x} \\, \\mu_1(\\underline{x}) (\\varphi_1(\\underline{x}) - \\varphi_1'(\\underline{x})) \\right) \\delta(\\varphi_1 - \\varphi_1') = \\delta(\\varphi_1 - \\varphi_1').\n\\ee\nWe conclude that the quantum evolution implemented by the field propagator in the presence of a source field is unitary.\n\n\n\\subsubsection{Asymptotic amplitude}\n\\label{sec:asymampfree1}\n\nThe amplitude $\\rho_{[t_1,t_2],\\mu}$ associated with the transition from the coherent state $\\psi_{t_1, \\xi_1}$ to the coherent state $\\psi_{t_2, \\xi_2}$\nis\n\\be\n\\rho_{[t_1,t_2],\\mu}(\\psi_{t_1, \\xi_1} \\otimes \\overline{\\psi_{t_2, \\xi_2}}) = \\int \\xD \\varphi_1 \\, \\xD \\varphi_2 \\, \\overline{\\psi_{t_2, \\xi_2} (\\varphi_2)} \\, \\psi_{t_1, \\xi_1}(\\varphi_1) \\, Z_{[t_1,t_2],\\mu} (\\varphi_1, \\varphi_2).\n\\label{eq:rhosrc0}\n\\ee\nUsing the expression (\\ref{eq:propmu}) of the field propagator and introducing two new complex functions $\\tilde{\\xi}_1$ and $\\tilde{\\xi}_2$ defined as\n\\bea\n\\tilde{\\xi}_1 (\\underline{k}) &\\defeq& \\xi_1(\\underline{k}) + \\im \\, t_1^{3\/2} H_{\\nu}(k t_1) \\int \\xd ^3 \\underline{x} \\, e^{-\\im \\underline{k} \\cdot \\underline{x}}\\, \\mu_1(\\underline{x}), \\\\\n\\tilde{\\xi}_2 (\\underline{k}) &\\defeq& \\xi_2(\\underline{k}) - \\im \\, t_2^{3\/2} H_{\\nu}(k t_2) \\int \\xd ^3 \\underline{x} \\,e^{\\im \\underline{k} \\cdot \\underline{x}}\\, \\mu_2(\\underline{x}),\n\\eea\nwe can rewrite (\\ref{eq:rhosrc0}) in terms of the free amplitude (\\ref{eq:freeampl0}) for the coherent states defined by the functions $\\tilde{\\xi}_1$ and $\\tilde{\\xi}_2$ in the form\n\\begin{align}\n\\rho_{[t_1,t_2],\\mu}(\\psi_{t_1, \\xi_1} \\otimes \\overline{\\psi_{t_2, \\xi_2}}) = \n\\rho_{[t_1,t_2],0}(\\psi_{t_1, \\tilde{\\xi}_1} \\otimes \\overline{\\psi_{t_2, \\tilde{\\xi}_2}})\n \\, \\frac{ N_{[t_1,t_2],\\mu} K_{t_1, \\xi_1} \\overline{K_{t_2, \\xi_2}} }{ N_{[t_1,t_2],0} K_{t_1, \\tilde{\\xi}_1} \\overline{K_{t_2, \\tilde{\\xi}_2}} }.\n\\end{align}\nWith the expressions (\\ref{eq:freeampl}), (\\ref{eq:Nmu2}) and (\\ref{eq:normcoh}), we finally arrive at the following result,\n\\begin{multline}\n\\rho_{[t_1,t_2],\\mu}(\\psi_{t_1, \\xi_1} \\otimes \\overline{\\psi_{t_2, \\xi_2}}) = \n\\rho_{[t_1,t_2],0}(\\psi_{t_1, \\xi_1} \\otimes \\overline{\\psi_{t_2, \\xi_2}})\n\\exp \\left( \\frac{\\im}{2} \\int_{[t_1,t_2]^2} \\xd^4 x \\, \\xd^4 x' \\, \\sqrt{g(x')g(x)} \\, \\mu(x) \\, G(x,x')\\, \\mu(x') \\right)\\\\\n \\times\n\\exp \\left( \\im \\int_{[t_1,t_2]} \\xd ^4 x \\, \\sqrt{-g(x)} \\, \\mu(x)\\, \\hat{\\xi}(x) + \\frac{\\im}{2} \\int_{[t_1,t_2]^2} \\xd ^4 x \\, \\xd^4 x' \\, \\sqrt{g(x')g(x)} \\, \\mu(x) \\, \\beta(x,x')\\, \\mu(x') \\right),\n\\label{eq:rhosrc2}\n\\end{multline}\nwhere the complex function $\\hat{\\xi}$ is the complex classical solution of the Klein-Gordon equation determined by $\\xi_1$ and $\\xi_2$,\n\\be\n\\hat{\\xi}(x) = \\frac{\\pi}{4 R^2} \\int \\frac{\\xd ^3 \\underline{k}}{(2 \\pi)^3} \\left(\\xi_1(\\underline{k}) \\, t^{3\/2} \\, \\overline{H_{\\nu}(k t)} \\, e^{\\im \\underline{k} \\cdot \\underline{x}}+ \\overline{\\xi_2}(\\underline{k}) \\, t^{3\/2} \\, H_{\\nu}(k t) \\, e^{-\\im \\underline{k} \\cdot \\underline{x}}\\right).\n\\label{eq:hatxi}\n\\ee\nThis equation establishes a one-to-one correspondence between pairs of coherent states parametrized by pairs of functions $(\\xi_1,\\xi_2)$ and complex solutions $\\hat{\\xi}$ of the Klein-Gordon equation.\nThe function $\\beta$ in the right hand side of (\\ref{eq:rhosrc2}) is defined as\n\\begin{align}\n\\beta(x,x') = & \\, \n\\frac{ \\pi}{2 R^2 } \\int \\frac{\\xd ^3 \\underline{k} }{(2 \\pi)^3} \\left(- \\frac{ \\delta_k(t,t_1)\\delta_k(t',t_2)}{\\delta_k(t_1,t_2)} -\\frac{1}{2} \\delta_k(t',t)+ \\frac{\\im}{2}(t' t)^{3\/2} \\left[ J_{\\nu}(k t')J_{\\nu}(k t) + Y_{\\nu}(k t')Y_{\\nu}(k t) \\right] \\right) \\nonumber\\\\\n& \\times e^{\\im \\underline{k} \\cdot (\\underline{x}- \\underline{x}')}.\n\\end{align}\nThe combination of the Green function $G$ with the function $\\beta$ according to (\\ref{eq:rhosrc2}) gives\n\\bea\nG_F(x,x') &\\defeq& G(x,x') + \\beta(x,x'), \\nonumber\\\\ \n&=& \\frac{\\im \\pi}{4 R^2 } \\int \\frac{\\xd ^3 \\underline{k} }{(2 \\pi)^3} \\left( 2 \\im \\theta(t-t') \\, \\delta_k(t,t') + \\im \\delta_k(t',t)+ (t' t)^{3\/2} \\left[ J_{\\nu}(k t')J_{\\nu}(k t) + Y_{\\nu}(k t')Y_{\\nu}(k t) \\right] \\right)\\nonumber\\\\\n&\\times& e^{\\im \\underline{k} \\cdot (\\underline{x}- \\underline{x}')} , \\nonumber\\\\\n&=& \\frac{\\im \\pi}{4 R^2 } \\int \\frac{\\xd ^3 \\underline{k} }{(2 \\pi)^3} \\, (t't)^{3\/2} \\left( \\theta(t-t') \\, \\overline{H_{\\nu}(k t)} \\, H_{\\nu}(k t') + \\theta(t'-t) \\, H_{\\nu}(k t) \\, \\overline{H_{\\nu}(k t')} \\right)e^{\\im \\underline{k} \\cdot (\\underline{x}- \\underline{x}')}.\n\\label{eq:Fprop0}\n\\eea\nThis expression can be related with the Feynman propagator for the scalar field in Minkowski spacetime via the asymptotic limit defined by (\\ref{eq:dStoM}). The limit of (\\ref{eq:Fprop0}),\n\\be\nG_F(x,x') = \\im \\int \\frac{\\xd ^3 \\underline{k} }{(2 \\pi)^3 2k}\\, \\left( \\theta(t-t') \\, e^{\\im k (t'-t)} + \\theta(t'-t) \\, e^{\\im k(t -t')} \\right)e^{\\im \\underline{k} \\cdot (\\underline{x}- \\underline{x}')},\n\\ee\nresults to be equal to the epxression of the Feynman propagator for a massless scalar field in Minkowski spacetime.\nIt is possible to evaluate the integral in (\\ref{eq:Fprop0}) performing the integration in the angular components of the vector $\\underline{k}$\n\\be\nG_F(x,x') = \\im \\frac{(tt')^{3\/2}}{8 \\pi R^2} \n\\int_{0}^{\\infty} \\xd k \\, k \\, \\frac{\\sin \\left( k |\\underline{x}-\\underline{x}'| \\right)}{ |\\underline{x}-\\underline{x}'|} \n\\left(\n\\theta(t'-t) \\, \\overline{{H}_{\\nu}}(k t') \\, {H}_{\\nu}(k t) +\n\\theta(t-t') \\, \\overline{{H}_{\\nu}}(k t) \\, {H}_{\\nu}(k t') \n\\right),\n\\label{eq:GF1}\n\\ee\nand then using the relation 6.672.3 of \\cite{GR:tables} to obtain\n\\be\nG_F(x,x')= - \\frac{1}{ 8 \\pi R^2} \\frac{1}{\\cos(\\nu \\pi)} \\frac{1}{\\sqrt{1-p(x,x')^2}} P_{\\nu - 1\/2}^1 (-p(x,x') + \\im 0), \\nonumber\\\\\n\\ee\nwhere $P$ is the associated Legendre function, and $p(x,x')$ is the de Sitter invariant distance given by\n\\be\np(x,x')= \\frac{t^2+t'^2-|\\underline{x}- \\underline{x}'|^2}{2 t't}.\n\\ee\nFinally, the function $G_F$ can be expressed in terms of the hypergeometric function $F$,\n\\be\nG_F(x,x') = - \\frac{\\im}{ 16 \\pi R^2} \\frac{\\left( \\frac{1}{4} - \\nu^2 \\right)}{\\cos(\\nu \\pi)} \\, F \\left( -\\nu + \\frac{3}{2}, \\nu +\\frac{3}{2}; 2 ; \\frac{1+p(x,x')- \\im 0}{2}\\right).\n\\label{eq:Fprop}\n\\ee\nThis form of the function $G_F$ coincides with the expression of the Feynman propagator computed in \\cite{Schomblond:1976xc} (see formula (A-16) of \\cite{Schomblond:1976xc}).\nThe Feynman propagator (\\ref{eq:Fprop}) satisfies the inhomogeneous Klein-Gordon equation,\n\\be\n\\left[ \\frac{t^2}{R^2} \\left(\\partial_t^2 - \\Delta_{\\underline{x}} \\right) -\\frac{2t}{R^2} \\partial_t +m^2 \\right] G_F(t,\\underline{x},t',\\underline{x}') = (-g(t, \\underline{x}))^{-1\/2} \\delta(t-t') \\delta^3(\\underline{x}-\\underline{x}'),\n\\ee\nwith the boundary conditions\n\\bea\nG_F(t,\\underline{x},t',\\underline{x}')\\big|_{t=t_1} &=& \\frac{\\im \\pi}{ 4 R^2} \\int \\frac{\\xd ^3 \\underline{k} }{(2 \\pi)^3} \\, (t't_1)^{3\/2} \\, \\overline{H_{\\nu}(k t')} \\, H_{\\nu}(k t_1) \\, e^{\\im \\underline{k} \\cdot (\\underline{x}- \\underline{x}')}, \\\\\nG_F(t,\\underline{x},t',\\underline{x}')\\big|_{t=t_2} &=& \\frac{\\im \\pi}{ 4 R^2} \\int \\frac{\\xd ^3 \\underline{k} }{(2 \\pi)^3} \\, (t't_2)^{3\/2} \\, \\overline{H_{\\nu}(k t_2)} \\, H_{\\nu}(k t') \\, e^{\\im \\underline{k} \\cdot (\\underline{x}- \\underline{x}')}.\n\\eea\nFinally, the amplitude in the theory with the source field results to be\n\\begin{multline}\n\\rho_{[t_1,t_2],\\mu}(\\psi_{t_1, \\xi_1} \\otimes \\overline{\\psi_{t_2, \\xi_2}}) = \\langle \\psi_{ \\xi_2} |\\psi_{ \\xi_1} \\rangle \\exp \\left( \\im \\int_{[t_1,t_2]} \\xd ^4 x \\, \\sqrt{-g(x)} \\, \\mu(x)\\, \\hat{\\xi}(x) \\right) \\\\ \n\\times \\exp \\left(\\frac{\\im}{2} \\int_{[t_1,t_2]^2} \\xd ^4 x \\, \\xd ^4 x' \\, \\sqrt{g(x)g(x')} \\, \\mu(x)\\, G_F(x,x') \\, \\mu(x') \\right). \n\\label{eq:amplsrc}\n\\end{multline}\nThis expression is independent of the times $t_1$ and $t_2$ as long as the source field $\\mu$ vanishes outside the region $[t_1,t_2]$. Therefore the limit for asymptotic values of the times $t_1$ and $t_2$ is trivial and we can then remove the restriction on the support of $\\mu$. Finally we interpret (\\ref{eq:amplsrc}) as the elements of the S-matrix for the theory with the source interaction,\n\\begin{multline}\n\\langle \\psi_{\\xi_2}| {\\cal S}_{\\mu} | \\psi_{\\xi_1} \\rangle = \\langle \\psi_{\\xi_2} | {\\cal S}_{0} |\\psi_{ \\xi_1} \\rangle \\exp \\left( \\im\\int_{[t_1,t_2]} \\xd ^4 x \\, \\sqrt{-g(x)} \\, \\mu(x)\\, \\hat{\\xi}(x) \\right) \\\\ \n\\times \\exp \\left(\\frac{\\im}{2} \\int_{[t_1,t_2]^2} \\xd ^4 x \\, \\xd ^4 x' \\, \\sqrt{g(x)g(x')} \\, \\mu(x)\\, G_F(x,x') \\, \\mu(x') \\right). \n\\label{eq:amplsrc1}\n\\end{multline}\n\n\\subsection{General interaction}\n\\label{sec:genint1}\n\nConsider the action of a scalar field in the presence of an arbitrary potential $V$ vanishing outside the spacetime region $[t_1,t_2]$,\n\\be\n S_{[t_1,t_2],V}(\\phi)=S_{[t_1,t_2],0}(\\phi)+\\int_{[t_1,t_2]} \\xd^4 x\\sqrt{-g(x)}\\, V(x,\\phi(x)).\n\\ee\nFollowing \\cite{CoOe:smatrixgbf} we use the technique of functional derivatives to rewrite the exponential of $\\im$ times this action,\n\\be\n\\exp \\left( \\im S_{[t_1,t_2],V}(\\phi)\\right) = \\exp \\left(\\im \\int_{[t_1,t_2]} \\xd^4 x\\sqrt{-g(x)} \\, V\\left(x, -\\im \\frac{\\delta}{\\delta \\mu(x)} \\right) \\right) \\exp \\left( \\im S_{[t_1,t_2],\\mu}(\\phi)\\right) \\bigg|_{\\mu=0},\n\\label{eq:fd}\n\\ee\nwhere $S_{[t_1,t_2],\\mu}$ denotes the action for the theory with the source interaction (\\ref{eq:actionsrc}). We can now perform all the calculations of the previous section by moving the functional derivative term in (\\ref{eq:fd}) to the front. Then, the amplitude in the presence of the potential $V$ can be immediately written as\n\\be\n\\rho_{[t_1,t_2],V}(\\psi_{t_1, \\xi_1} \\otimes \\overline{\\psi_{t_2, \\xi_2}}) = \\exp \\left(\\im \\int_{[t_1,t_2]} \\xd^4 x\\sqrt{-g(x)} \\, V\\left(x, -\\im \\frac{\\delta}{\\delta \\mu(x)} \\right) \\right) \\rho_{[t_1,t_2],\\mu}(\\psi_{t_1, \\xi_1} \\otimes \\overline{\\psi_{t_2, \\xi_2}})\\bigg|_{\\mu=0}.\n\\label{eq:genintasyamp1}\n\\ee\nWe notice first that the use of coherent states to compute the asymptotic amplitude is not indispensable and the same expression (\\ref{eq:genintasyamp1}) results for generic states $\\psi_{t_1,1} \\otimes \\overline{\\psi_{t_2,2}}$. Next, by recalling the independence from times $t_1$ and $t_2$ of the amplitude in the presence of a source interaction, expression (\\ref{eq:genintasyamp1}) also does not depend on $t_1$ and $t_2$. This allows us to remove the restriction on the potential $V$. The $S$-matrix elements can then written as\n\\be\n\\langle \\psi_{2}| {\\cal S}_{V} | \\psi_{1} \\rangle = \\exp \\left(\\im \\int \\xd^4 x \\sqrt{-g(x)} \\, V\\left(x, -\\im \\frac{\\delta}{\\delta \\mu(x)} \\right) \\right) \\langle \\psi_{2}| {\\cal S}_{\\mu} | \\psi_{1} \\rangle \\bigg|_{\\mu=0}.\n\\ee\n\n\\section{Introduction}\n\nThe present paper is devoted to the study of the quantization of a real massive scalar field in de Sitter spacetime and to derive the S-matrix for a general interaction within the framework of the general boundary formulation (GBF) of quantum field theory (QFT). The GBF has recently emerged as a new powerful tool to describe the dynamics of quantum fields. The key idea on which the GBF is based resides on a generalization of the notion of amplitudes. To fully appreciate the novelties introduced by the GBF, it would be useful to quickly review how evolution and transition amplitudes are usually described.\nIn standard Minkowski-based quantum field theory, transition amplitudes are expressed as the scalar product between a state defined on an initial equal-time hyperplane, say $\\xi$, evolved up to a final equal-time hyperplane by the evolution operator $U(t_f,t_i)$, and a state defined on the final hyperplane, $\\eta$, namely in Dirac notation\n\\be\n\\langle \\eta, t_f | U(t_f,t_i) |\\xi, t_i \\rangle.\n\\label{eq:standardTA}\n\\ee\nThe above expression is interpreted as the probability amplitude for the transition from the state $\\xi$ to the state $\\eta$ in the time interval ($t_f - t_i$). This represents the standard picture of dynamics understood as evolution from an initial state to a final one. The physical process described by (\\ref{eq:standardTA}) involves the non compact region of Minkowski spacetime bounded by the two disconnected spacelike hyperplanes defined by the times $t_i$ and $t_f$ respectively. \nNow, if we are interested in the study of process involving a spacetime region naturally bounded by, say, one connected hypersurface containing timelike parts, the application of the above formalism turns out to be problematic: Indeed it is the standard notion of evolution that appears to be questionable since in this case the distinction between initial and final states is missing due to the connectedness of the boundary and a more general notion of evolution is needed.\nOther problems may appear in the usual S-matrix technique to calculate probabilities for scattering processes, in which one usually assumes that the interaction vanishes at asymptotic times, so that the initial and final state, now defined on temporal asymptotic hyperplanes, belong to the state space of the corresponding free theory. \nBut how can we define the S-matrix if the interaction never vanishes or if for some reasons no temporal asymptotic states exist (as will be the case in anti-de Sitter spacetime)? \nMoreover, from an experimental point of view, infinitely distant regions are inaccessible, and real experiments always take place in finite regions of spacetime. This observation opens the question of the implementability of a fully local description of the dynamics of quantum fields, namely a description involving only compact spacetime regions\\footnote{In \\citep{ColRov:localpar} a notion of local particle states, understood as quanta of a local field operator, was introduced and its relation with the standard Fock particle states analized.}. While this is difficult or even incompatible with the standard techniques of QFT\\footnote{By standard QFT we mean the Hilbert-space approach of QFT, in which the central object is the vector space of states given by a complex Hilbert space, with observables represented by self-adjoint operators acting on it. We do not refer here to the algebraic approach to QFT.}, the GBF offers the appropriate framework for such description.\nMore serious problems emerge in the study of quantum fields in curved spacetimes, apart from the one just mentioned relative to anti-de Sitter space. The class of problems we are referring to originates form the absence (in general) of symmetries of the spacetime metric preventing the availability of a privileged criterion to select a specific vacuum state of the field, which leads consequently to many inequivalent quantum theories. Furthermore, in this case, evolution is understood between arbitrary spacelike Cauchy surfaces that provide a foliation of the spacetime and in general results to be non-unitary \\citep{BiDa:qfcs}.\nThe description of physical processes in a background-independent quantum context, as would be the case in quantum gravity, will entail even more radical problems requiring drastic departure from both technical and conceptual aspects of standard QFT\\footnote{Quantum gravity was indeed among the main motivations for the GBF. All the interest in a general boundary approach to quantum gravity resides in the possibility that the GBF can handle some of the major conceptual problems posed by any quantum theory not defined on a fixed background metric, such as the problem of time and the problem of locality \\citep{Oe:bqgrav}. By the problem of locality we mean the separation of the system of interest from the rest of the universe. While this is possible within quantum field theory, due to causality and the cluster decomposition principle, in the absence of background metric distant systems can not be separated and treated independently.}. \n\nThe GBF not only brings a new viewpoint on QFT but may also solve some of the problems mentioned above. Indeed within the GBF a consistent description is implemented for physical processes taking place in \\textsl{arbitrary} spacetime regions that are not consequently restricted to those involved in equation (\\ref{eq:standardTA}), namely regions defined by a time interval. The major novelty is then represented by considering regions with compact boundary, i.e. regions with a boundary hypersurface having both spacelike and timelike components. On the other hand, standard QFT can be recovered from the GBF by specializing the boundary hypersurface to the disjoint union of two Cauchy surfaces. In that sense the GBF represents an extension of the quantum theory. The basic ingridients of this new formulation are inspired by topological quantum field theory: state spaces are associated with the boundary of any spacetime region, and amplitudes are associated with the region and are given by maps from these state spaces to the complex numbers. These structures are required to satisfy a set of axioms assuring their consistency. Section \\ref{sec:qt} is devoted to the elucidation of these state spaces and amplitudes, which are introduced using the Schr\\\"odinger representation for the quantum states of the field, combined with the Feynman path integral quantization prescription\\footnote{It is important to mention that the GBF is a general framework to formulate quantum theories and is not bound to any particular theory. Nor to any specific quantization scheme consequently. However, a Schr\\\"odinger-Feynman quantization turns out to be useful and to work, at least at a formal level, in the specific situations considered so far. We mention the existence of new and more rigorous quantization scheme proposed by Oeckl in \\citep{Oe:newGBF}.}. Finally, a consistent physical interpretation can be given to such amplitudes and an appropriate notion of probability can be extracted from them \\citep{Oe:GBQFT,Oe:KGtl}. It is important to emphasize that the GBF implements a manifestly local description of the quantum theory: Indeed the computation of the amplitudes takes into account only the states defined on the boundary and the dynamics, compatible with the specified boundary states, taking place in the spacetime region of interest. \n\nIn a series of papers the GBF has been applied to study interacting scalar fields in Minkowski \\citep{CoOe:spsmatrix,CoOe:smatrixgbf} and Euclidean spacetime \\citep{CoOe:smatrix2d}. On the one hand the usual results of standard QFT have been recovered by considering time interval regions. On the other hand dynamics taking place in a new type of spacetime regions, namely regions bounded by a timelike and \\textsl{connected} hypersurface, were described and a consistent probability interpretation was implemented. In Minkowski spacetime the region considered was a timelike hypercylinder, i.e. a ball in space extended over all of time, first introduced in \\citep{Oe:KGtl}. In the 2-dimensional Euclidean space of \\citep{CoOe:smatrix2d} the region was given by a circle. Notice that the connectedness of the boundary prevents a natural decomposition of the state space associated with it into a state space containing \\textsl{in}-states and one containing \\textsl{out}-states. Thus, the traditional picture of dynamics, entails by expression (\\ref{eq:standardTA}), has to be extended to a more general one. The key point is that the GBF provides a precise mathematical description (at the level of rigor of the path integral) as well as a suitable physical interpretation for processes characterized by these geometries. Furthermore, it has been shown the existence of an isomorphism between the state spaces associated with the boundaries of the hypercylinder in Minkowski space with the state space associated with the time interval region. Due to this isomorphism the interacting asymptotic amplitude defined from the large radius limit of the hypercylinder results to be equivalent to the S-matrix in the standard setting, when both can be defined. Analogue results were obtained for field theory in Euclidean space.\n\nThe cases studied so far are restricted to flat-spacetime-based QFT. Here we take a further step by considering the quantum theory of a scalar field in a curved spacetime, specifically in de Sitter spacetime. The de Sitter spacetime presents many interesting features. First, because of its high degree of symmetry, the dynamics of quantized fields propagating in it is exactly solvable and many of the properties of the fields can be studied analytically \\citep{BiDa:qfcs}. Moreover, in the inflationary cosmology scenario de Sitter space models an exponentially expanding universe at the initial stage of inflation \\citep{Mukhanov:2005sc}. \nDe Sitter space has also attracted new interest in connection with the conjecture of the dS\/CFT correspondence proposed almost a decade ago by Strominger \\citep{Strominger:2001pn}.\n\nIn this article we will quantize a real massive minimally coupled scalar field in de Sitter spacetime within the GBF. The main result consists in the derivation of the expression of the S-matrix for the general interacting theory in two different contexts. The first one is represented by the standard settings of QFT, where the field evolves between spacelike Cauchy surfaces. The second one is inspired by the geometry of the hypercylinder in Minkowski space: evolution takes place in a non-compact spacetime region bounded by one timelike connected hypersurface on which the quantum states of the field are defined. In particular this hypersurface is at fixed distance from the temporal axis of de Sitter space, and will be called the hypercylinder in analogy to the Minkowski case. The S-matrix for the hypercylinder region is given by the asymptotic amplitude (at spatial infinity) for the specified quantum states. These two S-matrices will be shown to be equivalent by the existence of an isomorphism between the state spaces associated with the corresponding asymptotic hypersurfaces, hence extending to de Sitter space the results of \\citep{CoOe:spsmatrix,CoOe:smatrixgbf,CoOe:smatrix2d}. Part of the results presented here were announced in a previous paper \\citep{Co:dS}.\n\nThe outline of the paper is as follows. In Sec. \\ref{sec:classical} we present the two spacetime regions we will be interested in and solve the Klein-Gordon equation in the different coordinate systems chosen in the two regions. In Sec. \\ref{sec:qt} the main structures of the GBF are introduced. In Sec. \\ref{sec:free} the free scalar field is quantized in the region with spacelike boundary, namely the time interval region bounded by two equal-time hypersurfaces. We start by expressing the field propagator and then introduce the vacuum state and coherent states. Finally the asymptotic amplitude, interpreted as the $S$-matrix for the free theory is computed. We then treat the interacting theory in Sec. \\ref{sec:int} in two steps: First the asymptotic amplitude in the case of an interaction with a source field is derived and subsequently we use functional methods to work out the $S$-matrix for the general interacting theory. In Sec. \\ref{sec:hypfree} we quantize the free field in the hyercylinder region, following the treatment of Sec. \\ref{sec:free}: after introducing the main structures we obtain the expression of the asymptotic amplitude for coherent states defined on the hypercylinder. Sec. \\ref{sec:hypint} deals with the interacting theory and the asymptotic amplitude is derived following the same steps of Sec. \\ref{sec:int}. In Sec. \\ref{sec:iso} we show the existence of an isomorphism between the asymptotic Hilbert spaces associated with the boundaries of the two geometries in question and prove the equivalence of the asymptotic amplitudes under such isomorphism. Sec. \\ref{sec:con} contains our conclusions and an outlook. A few technical details are collected in appendix \\ref{sec:apA} and appendix \\ref{sec:apB}.\n\\section{Isomorphism}\n\\label{sec:iso}\n\nHaving in mind previous results obtained applying the GBF to the theory of quantum fields in Minkowski spacetime \\cite{CoOe:spsmatrix,CoOe:smatrixgbf} and Euclidean spacetime \\cite{CoOe:smatrix2d}, the similarity between the asymptotic amplitudes in the presence of a source field in the two geometries considered, namely, expressions (\\ref{eq:amplsrc1}) and (\\ref{eq:amplsrchyp4}), should not surprise us. In particular, based on the cited works and on the appearance of the same Feynman propagator in these amplitudes, we expect the two amplitudes to be equivalent under the action of an isomorphism between the corresponding spaces of states. To be more precise the isomorphism ${\\cal H}_{t_1} \\otimes {\\cal H}_{t_2}^* \\rightarrow {\\cal H}_{\\varrho}$, where ${\\cal H}_{\\varrho}$ denotes the Hilbert space associated with the hypercylinder, is determined by the identification of the complex solutions of the Klein-Gordon equation (\\ref{eq:hatxi}) and (\\ref{eq:hatxihyp}). Indeed these solutions establish a one-to-one correspondence between complex classical solutions in spacetime and coherent states in ${\\cal H}_{t_1} \\otimes {\\cal H}_{t_2}^*$ and ${\\cal H}_{\\varrho}$ respectively. Hence, for a given complex classical solution we implement the correspondence $\\psi_{t_1,\\xi_1} \\otimes \\overline{\\psi_{t_2,\\xi_2}} \\mapsto \\psi_{\\varrho,\\xi}$ that fixes the isomorphism. The equivalence of the amplitudes (\\ref{eq:amplsrc1}) and (\\ref{eq:amplsrchyp4}) is then realized if the free amplitudes (\\ref{eq:freeampl}) and (\\ref{eq:freeamplhyp}) coincide under the action of the isomorphism.\n\nIdentifying the complex classical solutions (\\ref{eq:hatxi}) and (\\ref{eq:hatxihyp}) leads to the following relations for the modes $\\xi_{l,m}(k)$ and $\\xi_{1,2}(\\underline{k})$,\n\\be\n\\xi_{l,m}(k) = e^{\\im \\nu \\pi \/2}\\frac{ \\im^l}{2 \\pi} \\int \\xd \\Omega_k \\, \\xi_1(\\underline{k}) \\, Y_l^m(\\Omega_k),\\qquad\n\\xi_{l,-m}(-k) = - e^{-\\im \\nu \\pi \/2} \\frac{(- \\im )^l}{2 \\pi} \\int \\xd \\Omega_k \\, \\overline{\\xi_2(\\underline{k})} \\, \\overline{Y_l^m}(\\Omega_k).\n\\label{eq:isom}\n\\ee\nwhere the angular coordinates $\\Omega_k$ determine the direction of the vector $\\underline{k}$. We substitute (\\ref{eq:isom}) in the expression of the free amplitude associated with the hypercylinder (\\ref{eq:freeamplhyp}),\n\\bea\n\\rho_{\\varrho,0}(\\psi_{\\varrho, \\xi} ) &=& \n \\exp \\left( - \\frac{1}{64 \\pi^2 R^2} \\int_{0}^{\\infty} \\xd k \\int \\xd \\Omega_k \\int \\xd \\Omega_k' \\sum_{l,m} k^2 \\, \\overline{Y_l^m(\\Omega_k)} \\, Y_l^m(\\Omega_k') \\right. \\nonumber\\\\\n&& \\times \\left( \\overline{\\xi_{1}(\\underline{k})} \\,\\xi_{1}(\\underline{k}') + \\overline{\\xi_{2}(\\underline{k})} \\, \\xi_{2}(\\underline{k}') - 2 \\overline{\\xi_{2}(\\underline{k})} \\, \\xi_{1}(\\underline{k}') \\right) \\bigg).\n\\eea\nThe sum over $l,m$ is performed using the completeness relation of spherical harmonics, see formula (B.88) of \\cite{Mes:qm1},\n\\be\n\\rho_{\\varrho,0}(\\psi_{\\varrho, \\xi} ) =\\exp \\left( - \\frac{\\pi}{8 R^2} \\int \\frac{\\xd^3 \\underline{k}}{(2 \\pi)^3} \\left( |\\xi_{1}(\\underline{k})|^2 + |\\xi_{2}(\\underline{k})|^2 - 2 \\overline{\\xi_{2}(\\underline{k})} \\, \\xi_{1}(\\underline{k}) \\right) \\right) = \\rho_{[t_1,t_2],0}(\\psi_{t_1,\\xi_1} \\otimes \\overline{\\psi_{t_2,\\xi_2}} ).\n\\ee\nThis result proves the equivalence of the free amplitudes (\\ref{eq:freeampl}) and (\\ref{eq:freeamplhyp}).\n\nThe map between coherent states $\\psi_{t_1,\\xi_1} \\otimes \\overline{\\psi_{t_2,\\xi_2}} \\mapsto \\psi_{\\varrho,\\xi}$ implemented by the isomorphism ${\\cal H}_{t_1} \\otimes {\\cal H}_{t_2}^* \\rightarrow {\\cal H}_{\\varrho}$ leads to a map between multiparticle states. We are interested here in the expression of a multiparticle state defined in one setting in terms of multiparticle states defined in the other setting. Consider a state with $n$ particles in ${\\cal H}_{t_1} \\otimes {\\cal H}_{t_2}^*$ with $q$ incoming particles with quantum numbers $\\underline{k}_1, \\dots, \\underline{k}_q$ and $n-q$ outgoing particles with quantum numbers $\\underline{k}_{q+1}, \\dots, \\underline{k}_n$, denoted as\n\\be\n\\psi_{\\underline{k}_1, \\cdots, \\underline{k}_q|\\underline{k}_{q+1}, \\cdots, \\underline{k}_n} = | \\psi_{\\underline{k}_1, \\cdots, \\underline{k}_q} \\rangle \\otimes \\langle \\psi_{\\underline{k}_{q+1}, \\cdots, \\underline{k}_n} |,\n\\label{eq:npart}\n\\ee\nwhere $| \\psi_{\\underline{k}_1, \\cdots, \\underline{k}_q} \\rangle$ is $q$-particle state introduced in (\\ref{eq:mulpar}). We want to evaluate the inner product between the state (\\ref{eq:npart}) and the $n$-particle state in ${\\cal H}_{\\varrho}$ with quantum numbers $(k_1,l_1,m_1), \\dots, (k_n,l_n,m_n)$ introduced in (\\ref{eq:mulparhyp}). Inserting the completeness relation of coherent states (\\ref{eq:complrelhyp2}), \n\\be\n\\langle \\psi_{\\underline{k}_1, \\cdots, \\underline{k}_q|\\underline{k}_{q+1}, \\cdots, \\underline{k}_n}| \\psi_{(k_1,l_1,m_1), \\dots, (k_n,l_n,m_n)} \\rangle = \\tilde{D}^{-1} \\int \\xd \\xi \\, \\xd \\overline{\\xi} \\langle \\psi_{\\underline{k}_1, \\cdots, \\underline{k}_q|\\underline{k}_{q+1}, \\cdots, \\underline{k}_n}| \\psi_{\\xi} \\rangle \\langle \\psi_{\\xi} | \\psi_{(k_1,l_1,m_1), \\dots, (k_n,l_n,m_n)} \\rangle.\n\\ee\nand using (\\ref{eq:mulpar}), (\\ref{eq:mulparhyp}), (\\ref{eq:Dtilde}) and the relations (\\ref{eq:isom}) we perform the integration in $\\xd \\xi \\, \\xd \\overline{\\xi}$ and obtain,\n\\begin{multline}\n\\langle \\psi_{\\underline{k}_1', \\cdots, \\underline{k}_q'|\\underline{k}_{q+1}', \\cdots, \\underline{k}_n'}| \\psi_{(k_1,l_1,m_1), \\dots, (k_n,l_n,m_n)} \\rangle = (-1)^{n-q} \\left( \\frac{2 \\pi^2}{R^2} \\right)^n \\, \\im^{l_{q+1} + \\cdots + l_n- l_1- \\cdots -l_q} \\\\\n\\times Y_{l_1}^{-m_1}(\\Omega_{k_1}) \\cdots Y_{l_n}^{-m_n}(\\Omega_{k_n})\n \\, \\delta(k_1'-k_1) \\cdots \\delta(k_q'-k_q) \\delta(k_{q+1}'+k_{q+1}) \\cdots \\delta(k_n'+k_n).\n\\end{multline}\nWe can now write an $n$-particle state in ${\\cal H}_{\\varrho}$ as a linear combination of $n$-particle states in ${\\cal H}_{t_1} \\otimes {\\cal H}_{t_2}^*$,\n\\begin{align}\n\\psi_{\\underline{k}_1, \\cdots, \\underline{k}_q|\\underline{k}_{q+1}, \\cdots, \\underline{k}_n} =& \\, \n(-1)^{n-q} \\, \\frac{(16 \\pi^3)^n}{k_1^2 \\cdots k_n^2} \\, \\sum_{l_1,m_1} \\cdots \\sum_{l_n,m_n} \\, \\im^{-l_{q+1} - \\cdots - l_n+ l_1+ \\cdots +l_q} \\, \nY_{l_1}^{m_1}(\\Omega_{k_1}) \\cdots Y_{l_n}^{m_n}(\\Omega_{k_n}) \\nonumber\\\\ \n& \\times \\psi_{(k_1,l_1,m_1), \\dots, (k_q,l_q,m_q),(-k_{q+1},l_{q+1},m_{q+1}), \\dots, (-k_n,l_n,m_n)}.\n\\end{align}\nReciprocally an $n$-particle state in ${\\cal H}_{t_1} \\otimes {\\cal H}_{t_2}^*$ is a linear combination of $n$-particle states in ${\\cal H}_{\\varrho}$,\n\\begin{align}\n\\psi_{(k_1,l_1,m_1), \\dots, (k_n,l_n,m_n)} =&\n\\, (-1)^{n-q} \\, (8 \\pi)^n \\int \\xd \\Omega_{k_1} \\cdots \\int \\xd \\Omega_{k_n} \\, \\im^{l_{q+1} + \\cdots + l_n- l_1- \\cdots -l_q}\n\\, Y_{l_1}^{m_1}(\\Omega_{k_1}) \\cdots Y_{l_n}^{m_n}(\\Omega_{k_n}) \\nonumber\\\\ \n& \\times \\psi_{\\underline{k}_1, \\cdots, \\underline{k}_q|\\underline{k}_{q+1}, \\cdots, \\underline{k}_n}.\n\\end{align}\nOn the hypercylinder, the incoming or outgoing character of particle states is encoded in the quantum numbers, in particular in the sign of $k$. By means of the isomorphism (\\ref{eq:isom}) between the Hilbert states associated with the different geometries considered, a particle in ${\\cal H}_{\\varrho}$ can be mapped, according to its quantum number, into an incoming or outgoing particle in ${\\cal H}_{t_1}$ or ${\\cal H}_{t_2}$ respectively.\n\n\\section{Quantum theory}\n\\label{sec:qt}\n\nAccording to the axioms of the GBF, a Hilbert space ${\\cal H}_{\\Sigma}$ of states is associated with each hypersurface $\\Sigma$. The quantum states in this Hilbert space are described in the Schr\\\"odinger representation, namely quantum states are wave functionals on the space of field configurations $K_{\\Sigma}$. The inner product in ${\\cal H}_{\\Sigma}$ is defined through an integral over field configurations,\n\\be\n\\langle \\psi_{\\Sigma} | \\psi_{\\Sigma}' \\rangle \\defeq \\int_{K_{\\Sigma}} \\xD \\varphi \\, \\overline{\\psi_{\\Sigma}(\\varphi)} \\, \\psi_{\\Sigma}(\\varphi).\n\\ee \nThe evolution of a quantum state $\\psi_{\\Sigma} \\in {\\cal H}_{\\Sigma}$ to a quantum state $\\psi_{\\hat{\\Sigma}} \\in {\\cal H}_{\\hat{\\Sigma}}$ is given in terms of the field propagator $Z_{[\\Sigma, \\hat{\\Sigma}]}$ associated with the spacetime region bounded by the two hypersurfaces $\\Sigma$ and $\\hat{\\Sigma}$,\n\\be\n\\psi_{\\hat{\\Sigma}} (\\hat{\\varphi}) = \\int_{K_{\\Sigma}} \\xD \\varphi \\, \\psi_{\\Sigma}(\\varphi) \\, Z_{[\\Sigma, \\hat{\\Sigma}]} (\\varphi, \\hat{\\varphi}).\n\\ee\nThe field propagator $Z_{[\\Sigma, \\hat{\\Sigma}]}$ encodes the evolution from the field configuration $\\varphi$ on the hypersurface $\\Sigma$ to the field configuration $\\hat{\\varphi}$ on the hypersurface $\\hat{\\Sigma}$. It is defined by the Feynman path integral as\n\\be\nZ_{[\\Sigma, \\hat{\\Sigma}]} (\\varphi, \\hat{\\varphi}) = \\int_{\\phi|_{\\Sigma} = \\varphi, \\, \\phi|_{\\hat{\\Sigma}} = \\hat{\\varphi}} \\xD \\phi \\, e^{\\im S_{[\\Sigma, \\hat{\\Sigma}]}(\\phi)},\n\\label{eq:prop}\n\\ee\nthe integral is extended over all field configurations $\\phi$ in the spacetime region bounded by the two hypersurfaces $\\Sigma$ and $\\hat{\\Sigma}$ that reduce to $\\varphi$ on $\\Sigma$ and to $\\hat{\\varphi}$ on $\\hat{\\Sigma}$, and $S_{[\\Sigma, \\hat{\\Sigma}]}$ is the action integral over this spacetime region. The path integral (\\ref{eq:prop}) can be formally evaluated by shifting the integration variable by a classical solution $\\phi_{cl}$ of the equation of motion derived from $S_{[\\Sigma, \\hat{\\Sigma}]}$, matching the boundary configurations $\\varphi$ and $\\hat{\\varphi}$ on the boundaries of the region. Explicitly,\n\\be\nZ_{[\\Sigma, \\hat{\\Sigma}]} (\\varphi, \\hat{\\varphi}) = \\int_{\\phi|_{\\Sigma} = \\varphi, \\, \\phi|_{\\hat{\\Sigma}} = \\hat{\\varphi}} \\xD \\phi \\, e^{\\im S_{[\\Sigma, \\hat{\\Sigma}]}(\\phi)}\n= \\int_{\\phi|_{\\Sigma} = \\phi|_{\\hat{\\Sigma}} = 0} \\xD \\phi \\, e^{\\im S_{[\\Sigma, \\hat{\\Sigma}]}(\\phi+ \\phi_{cl})}\n= N_{[\\Sigma, \\hat{\\Sigma}]} e^{\\im S_{[\\Sigma, \\hat{\\Sigma}]}( \\phi_{cl})},\n\\label{eq:propcl}\n\\ee\nwhere the normalization faction is formally given by\n\\be\nN_{[\\Sigma, \\hat{\\Sigma}]} = \\int_{\\phi|_{\\Sigma} = \\phi|_{\\hat{\\Sigma}} = 0} \\xD \\phi \\, e^{\\im S_{[\\Sigma, \\hat{\\Sigma}]}(\\phi)}.\n\\ee\nFinally, an amplitude $\\rho_{[\\Sigma, \\hat{\\Sigma}]}$ is associated with the spacetime region $[\\Sigma, \\hat{\\Sigma}]$ and a state $\\psi_{\\Sigma} \\otimes \\overline{\\psi_{\\hat{\\Sigma}}}$ in the Hilbert space associated with the boundary $\\partial [\\Sigma, \\hat{\\Sigma}]$, ${\\cal H}_{\\partial [\\Sigma, \\hat{\\Sigma}]} = {\\cal H}_{\\Sigma} \\otimes {\\cal H}_{\\hat{\\Sigma}}^*$. The amplitude $\\rho_{[\\Sigma, \\hat{\\Sigma}]} : {\\cal H}_{\\partial [\\Sigma, \\hat{\\Sigma}]} \\rightarrow {\\mathbb C}$ is defined as\n\\be\n\\rho_{[\\Sigma, \\hat{\\Sigma}]} (\\psi_{\\Sigma} \\otimes \\overline{\\psi_{\\hat{\\Sigma}}})= \\int \\xD \\varphi \\, \\xD \\hat{\\varphi} \\, \\psi_{\\Sigma}(\\varphi) \\, \\overline{\\psi_{\\hat{\\Sigma}}(\\hat{\\varphi})} \\, Z_{[\\Sigma, \\hat{\\Sigma}]} (\\varphi, \\hat{\\varphi}).\n\\label{eq:ampl}\n\\ee\nIn the following sections we will apply this formulation to describe the quantum dynamics of a scalar field in de Sitter space. In particular the expressions of the field propagator and amplitudes of quantum states will be explicitly worked out in three cases: we will start with the free theory defined by the free action (\\ref{eq:action}), then the interaction with an external source field will be considered, and finally we will use functional derivatives techniques to treat the case of a general interaction.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}