Schwinger-Keldysh formalism I: BRST symmetries and superspace

TL;DR

This work reframes the Schwinger-Keldysh (SK) in-in formalism as a BRST-invariant, superspace theory with a topological sector, enabling a manifestly symmetry-based approach to non-equilibrium QFT. By doubling the field content and introducing ghost partners, the authors derive a rich SK-KMS algebra that unifies field redefinitions, thermal translations, and their superspace representations, providing systematic constraints on correlators, causality, and fluctuation-dissipation relations. The framework is illustrated with explicit free-field examples (scalar, fermion, vector) and extended to thermal settings via KMS conditions, retarded-advanced bases, and thermofield-double interpretations. The paper further explores timefold (OTO) contours, predicting an enlarged N_T symmetry in OTO settings and outlining how BRST structures localize and constrain OTO observables. Potential applications span stochastic dynamics, hydrodynamics, entanglement, gravity, and amplitude cutting rules, suggesting broad utility for deepened understanding of dissipation, chaos, and information scrambling in QFT and holography.

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 arXiv:1610.01941 describes in greater detail the mathematical framework embodying the topological symmetries we uncover here.

Figures (5)

• Figure 1: Illustration of the generic Schwinger-Keldysh complex time contour. Every operator $\widehat{\mathbb{O}}$ in the original theory has two representations in the Schwinger-Keldysh path integral, viz., $\mathbb{O}_\text{\tiny R}$ and $\mathbb{O}_\text{\tiny L}$, which can be thought of as the distinction as to what part of the contour the operator is inserted on. Right operators are time-ordered, while left operators are anti-time ordered.
• Figure 2: SK time contour in the case where the initial state at time $t_0$ is known and the latest operator insertion happens at time $t$. The indicated operator insertions correspond to a real-time correlator $G_<(x,x')$.
• Figure 3: SK time contour in thermal physics, where the initial state is a thermal state with an entanglement pattern encoded in a Euclidean partition function. The starting and end points of the contour are identified. The associated Euclidean (imaginary time) periodicity is set by the inverse temperature $\beta_0$.
• Figure 4: The 2-OTO contour computing the correlation functions with operators inserted out of the conventional Schwinger-Keldysh time-ordering, cf. Eq. \ref{['eq:2oto']}. As usual the initial state is prepared at time $t_0$ and the latest operator insertion happens at time $t$. The indicated operator insertions correspond to the correlation function $\langle \mathbb{O}_{i\text{\tiny R}}(t_1) \mathbb{O}_{o\text{\tiny L}}(t_2) \mathbb{O}_{o\text{\tiny R}}(t_3) \mathbb{O}_{o\text{\tiny L}}(t_4) \rangle$.
• Figure 5: The k-OTO contour computing the out-of-time-ordered correlation functions encoded in the generating functional \ref{['eq:koto']}.