No time to derive: unraveling total time derivatives in in-in perturbation theory

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 [...]. 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.

69 pages + appendices, 1 figure, version published in JHEP with minor modifications (technical details moved to appendices)

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• The Open Effective Field Theory of Inflation
• Ultra-Slow-Roll Inflation on the Lattice I: Backreaction and Nonlinear Effects
• Comparing sharp and smooth transitions of the second slow-roll parameter in single-field inflation
• Absence of one-loop effects on large scales from small scales in non-slow-roll dynamics II: Quartic interactions and consistency relations
• Proving the absence of large one-loop corrections to the power spectrum of curvature perturbations in transient ultra-slow-roll inflation within the path-integral approach
• In-in formalism for the entropy of quantum fields in curved spacetimes
• Gravitational waves in ultra-slow-roll and their anisotropy at two loops
• Ultra-Slow-Roll Inflation on the Lattice II: Nonperturbative Curvature Perturbation
• Cosmological Bell Tests with Decoherence Effects
• One-loop corrections to the E-type $\alpha$-attractor models of inflation and primordial black hole production
• Generating large primordial fluctuations in single-field inflation for PBH formation
• Loops in inflation with strongly non-geodesic motion
• Bispectrum from five-dimensional inflation
• Stochastic Inflation with Interacting Noises
• New leading contributions to non-gaussianity in single field inflation
• Separate universe in multifield inflation: a phase-space approach