Abstract
We review the Schwinger-Keldysh, or in-in, formalism for studying quantum dynamics of systems out-of-equilibrium. The main motivation is to rephrase well known facts in the subject in a mathematically elegant setting, by exhibiting a set of BRST symmetries inherent in the construction. We show how these fundamental symmetries can be made manifest by working in a superspace formalism. We argue that this rephrasing is extremely efficacious in understanding low energy dynamics following the usual renormalization group approach, for the BRST symmetries are robust under integrating out degrees of freedom. In addition we discuss potential generalizations of the formalism that allow us to compute out-of-time-order correlation functions that have been the focus of recent attention in the context of chaos and scrambling. We also outline a set of problems ranging from stochastic dynamics, hydrodynamics, dynamics of entanglement in QFTs, and the physics of black holes and cosmology, where we believe this framework could play a crucial role in demystifying various confusions. Our companion paper [1] describes in greater detail the mathematical framework embodying the topological symmetries we uncover here.
Highlights
The study of quantum dynamics out of equilibrium and in open systems necessarily involves working with mixed states
We review the Schwinger-Keldysh, or in-in, formalism for studying quantum dynamics of systems out-of-equilibrium
We argue that this rephrasing is extremely efficacious in understanding low energy dynamics following the usual renormalization group approach, for the BRST symmetries are robust under integrating out degrees of freedom
Summary
The study of quantum dynamics out of equilibrium and in open systems necessarily involves working with mixed states. Our motivation for getting intrigued by the problem of constructing effective Schwinger-Keldysh theories was primarily to understand the general structure of such effective actions in the fluid dynamical regime and beyond [9, 12].2 These are qualitatively similar to the classic problem of the Brownian oscillator which motivated [2], or linear dissipative systems which inspired [4], albeit with a necessary upgrade to non-Gaussian interactions. Understanding the emergence of this gauge symmetry led us to revisiting the essentials of the Schwinger-Keldysh formalism, which as we outlined in [9] are best viewed by extracting the topological invariances inherent in the construction This suffices to reproduce the effective actions of [12, 21] for fluid dynamics as we recently explained in [22].4.
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