Abstract
The in-in formalism provides a way to systematically organize the calculation of primordial correlation functions. Although its theoretical foundations are now firmly settled, the treatment of total time derivative interactions, incorrectly trivialized as “boundary terms”, has been the subject of intense discussions and conceptual mistakes. In this work, we demystify the use of total time derivatives — as well as terms proportional to the linear equations of motion — and show that they can lead to artificially large contributions cancelling at different orders of the in-in operator formalism. We discuss the treatment of total time derivative interactions in the Lagrangian path integral formulation of the in-in perturbation theory, and we showcase the importance of interaction terms proportional to linear equations of motion. We then provide a new route to the calculation of primordial correlation functions, which avoids the generation of total time derivatives, by working directly at the level of the full Hamiltonian in terms of phase-space variables. Instead of integrating by parts, we perform canonical transformations to simplify interactions. We explain how to retrieve correlation functions of the initial phase-space variables from the knowledge of the ones after canonical transformations. As an important first application, we find the explicit sizes of Hamiltonian cubic interactions in single-field inflation with canonical kinetic terms and for any background evolution, straight in terms of the primordial curvature perturbation and its canonical conjugate momentum, as well as the corresponding ones in the tensor sector, and the ones mixing scalars and tensors. We also briefly comment on quartic interactions. Our results are important for performing complete calculations of exchange diagrams in inflation, such as the (scalar and tensor) exchange trispectrum and the one-loop power spectrum. Being already written in a form amenable to characterize quantum properties of primordial fluctuations, they also promise to shed light on the non-linear dynamics of quantum states during inflation.
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Acknowledgments
We are grateful to Xingang Chen, David Langlois, Jérôme Martin, Sébastien Renaux-Petel, David Seery and Denis Werth for useful discussions at various stages of this project. We also thank Xingang Chen, Guillem Domenech, Jacopo Fumagalli, Sebastian Garcia-Saenz, Sébastien Renaux-Petel and Denis Werth for comments on a draft of this paper. Some calculations in this paper were performed with the Mathematica software MathGR [71]. We would like to acknowledge CERN and the EuCAPT Consortium for the 2023 EuCAPT International Travel Award that helped foster this work. L.P. acknowledges funding support from the Initiative Physique des Infinis (IPI), a research training program of the Idex SUPER at Sorbonne Université. L.P. would also like to thank New York University for hospitality during the latest stages of this project.
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Braglia, M., Pinol, L. No time to derive: unraveling total time derivatives in in-in perturbation theory. J. High Energ. Phys. 2024, 68 (2024). https://doi.org/10.1007/JHEP08(2024)068
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DOI: https://doi.org/10.1007/JHEP08(2024)068