Schwinger-Keldysh formalism. Part I: BRST symmetries and superspace

• Published in: The journal of high energy physics Vol. 2017; no. 6; pp. 1 - 91
• Main Authors: Haehl, Felix M., Loganayagam, R., Rangamani, Mukund
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2017; Springer Nature B.V; SpringerOpen
• Subjects: BRST Quantization; Chaos theory; Classical and Quantum Gravitation; Computational fluid dynamics; Correlation; Cosmology; Dynamical systems; Elementary Particles; Entanglement; Fluid flow; Formalism; Functions (mathematics); Group dynamics; High energy physics; Holography and condensed matter physics (AdS/CMT); Hydrodynamics; Mathematical analysis; Physics; Physics and Astronomy; Probability theory; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Robustness (mathematics); Stochastic Processes; String Theory; Topological Field Theories; Topology; BRST Quantization; Topological Field Theories; Stochastic Processes; Holography and condensed matter physics (AdS/CMT)
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP06(2017)069

A bstract 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.