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arXiv:2604.26941

Schwinger-Keldysh Path Integral for Gauge theories

Greg Kaplanek, Maria Mylova, Andrew J. Tolley

hep-thquant-phSubmitted Apr 29, 2026• Announced Apr 30, 2026• Revised May 22, 2026 (v2)

v1 · Apr 29, 2026 v2 · May 22, 2026

Abstract

We develop the Schwinger-Keldysh path-integral formalism for open non-Abelian gauge theories that are gauge-fixed via the BRST method in covariant gauges. We focus on generic initial states, pure and mixed, specified at finite times suitable for non-equilibrium processes. We pay particular attention to the handling of the indefinite Hilbert space, the construction of BRST-invariant Schrodinger picture wavefunctionals, density matrices and inner product, the implementation of the Hata-Kugo prescription, and the role of boundary terms at both the initial and final times. We highlight the advantages of the Nakanishi-Lautrup field representation in dealing with initial/final conditions. The resulting Schwinger-Keldysh path integral is manifestly invariant under a diagonal (retarded) BRST symmetry for arbitrary physical initial states, whether pure or mixed. From this, we obtain the corresponding Ward-Takahashi-Slavnov-Taylor identities, valid perturbatively. Non-perturbatively the Gribov ambiguity is expected to break or modify the BRST symmetry. The naive advanced BRST symmetry is shown to be explicitly violated by the in-in boundary conditions. We show that the Feynman-Vernon influence functional derived by integrating out charged matter and/or hard gluon modes remains (perturbatively) BRST invariant. When the Open EFT action is expanded to second order in advanced fields it exhibits an exact symmetry under a contraction of the original BRST symmetry. This Keldysh BRST symmetry is equivalent to the BRST associated with the retarded gauge transformations together with a linearly realized BRST transformation of the advanced fields. These govern the structure of the leading terms in an Open EFT. We illustrate this with the explicit example of Hard Thermal Loop Effective Theory, and construct the general form of the Open EFT in a Higgs phase when all gauge symmetries are spontaneously broken.

Comments: 61 pages; (v2) references added & typos fixed

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We develop the Schwinger-Keldysh path-integral formalism for open non-Abelian gauge theories that are gauge-fixed via the BRST method in covariant gauges. We focus on generic initial states, pure and mixed, specified at finite times suitable for non-equilibrium processes. We pay particular attention to the handling of the indefinite Hilbert space, the construction of BRST-invariant Schrodinger picture wavefunctionals, density matrices and inner product, the implementation of the Hata-Kugo prescription, and the role of boundary terms at both the initial and final times. We highlight the advantages of the Nakanishi-Lautrup field representation in dealing with initial/final conditions. The resulting Schwinger-Keldysh path integral is manifestly invariant under a diagonal (retarded) BRST symmetry for arbitrary physical initial states, whether pure or mixed. From this, we obtain the corresponding Ward-Takahashi-Slavnov-Taylor identities, valid perturbatively. Non-perturbatively the Gribov ambiguity is expected to break or modify the BRST symmetry. The naive advanced BRST symmetry is shown to be explicitly violated by the in-in boundary conditions. We show that the Feynman-Vernon influence functional derived by integrating out charged matter and/or hard gluon modes remains (perturbatively) BRST invariant. When the Open EFT action is expanded to second order in advanced fields it exhibits an exact symmetry under a contraction of the original BRST symmetry. This Keldysh BRST symmetry is equivalent to the BRST associated with the retarded gauge transformations together with a linearly realized BRST transformation of the advanced fields. These govern the structure of the leading terms in an Open EFT. We illustrate this with the explicit example of Hard Thermal Loop Effective Theory, and construct the general form of the Open EFT in a Higgs phase when all gauge symmetries are spontaneously broken.

We present explicit top-down calculations of Open EFTs for gauged degrees of freedom with a focus on the effects of gauge fixing. Starting from the in-in contour with two copies of the action, we integrate out the charged matter in various

U(1)

gauge theories to obtain the Feynman-Vernon influence functional for the photon, or, in the case of symmetry breaking, for the photon and Stückelberg fields. The influence functional is defined through a quantum path integral, which -- as is always the case when quantizing gauge degrees of freedom -- contains redundancies that must be eliminated via a gauge-fixing procedure. We implement the BRST formalism in this setting. The in-in boundary conditions break the two copies of BRST symmetry down to a single diagonal copy. Nevertheless the single diagonal BRST is sufficient to ensure that the influence functional is itself gauge invariant under two copies of gauge symmetries, retarded and advanced, regardless of the choice of state or symmetry-breaking phase. We clarify how this is consistent with the decoupling limit where the global advanced symmetry is generically broken by the state. We illustrate our results with several examples: a gauge field theory analogue of the Caldeira-Leggett model, spinor QED with fermions integrated out, scalar QED in a thermal state, the Abelian Higgs-Kibble model in the spontaneously broken state with the Higgs integrated out, and Abelian Higgs-Kibble model coupled to a charged bath in a symmetry-broken phase. The latter serves as an example of an open system for Stückelberg/Goldstone fields.

We examine, in a quantum mechanical setting, the Hilbert space representation of the Becchi, Rouet, Stora, and Tyutin (BRST) symmetry associated with Schwinger-Keldysh path integrals. This structure had been postulated to encode important constraints on influence functionals in coarse-grained systems with dissipation, or in open quantum systems. Operationally, this entails uplifting the standard Schwinger-Keldysh two-copy formalism into superspace by appending BRST ghost degrees of freedom. These statements were previously argued at the level of the correlation functions. We provide herein a complementary perspective by working out the Hilbert space structure explicitly. Our analysis clarifies two crucial issues not evident in earlier works: first, certain background ghost insertions necessary to reproduce the correct Schwinger-Keldysh correlators arise naturally, and, second, the Schwinger-Keldysh difference operators are systematically dressed by the ghost bilinears, which turn out to be necessary to give rise to a consistent operator algebra. We also elaborate on the structure of the final state (which is BRST closed) and the future boundary condition of the ghost fields.

Abstract We develop the effective field theoretical (EFT) approach to time-translational symmetry breaking of nonequilibrium open systems based on the Schwinger-Keldysh formalism. In the Schwinger-Keldysh formalism, all the symmetries of the microscopic Lagrangian are doubled essentially because the dynamical fields are doubled to describe the time-evolution along the closed-time-path. The effective Lagrangian for open systems are then obtained by coarse-graining the microscopic Schwinger-Keldysh Lagrangian. As a consequence of coarse-graining procedure, there appear the noise and dissipation effects, which explicitly break the doubled time-translational symmetries into a diagonal one. We therefore need to incorporate this symmetry structure to construct the EFT for Nambu-Goldstone bosons in symmetry broken phases of open systems. Based on this observation together with the consistency of the Schwinger-Keldysh action, we construct and study the general EFT for time-translational symmetry breaking in particular, having in mind applications to synchronization, time crystal, and cosmic inflation.

We develop a bottom-up open effective field theory (EFT) for non-Abelian gauge theories within the Schwinger--Keldysh formalism. Instead of integrating out the environment completely and starting from a nonlocal influence functional, we retain the slow environmental response variables explicitly and construct a local system-environment EFT. The environmental sector is described by a dynamical color-frame variable, Stückelberg-like field, and an associated color-current sector, which gives the nontrivial interactions and dissipation between the system and the environment. The resulting construction provides a gauge-covariant Markov embedding of nonlocal and non-Markovian color response. After integrating out the retained environmental variables with retarded boundary conditions, the reduced system theory acquires nonlocal dissipative kernels and stochastic sources. We show that the hard thermal loop response arises naturally as a particular realization of the retained environmental response. Our framework provides a local open-EFT description of color transport, memory effects, and fluctuation-dissipation structure in non-Abelian plasmas, and offers a systematic starting point for dissipative Yang--Mills EFTs with dynamical environments.

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.

We present a pedagogical showcase of BRST quantization for restoring symmetries in nuclear many-body systems as a reformulation and potential alternative to conventional projection methods. The formalism is illustrated using translational invariance for a simple system of two interacting masses in one dimension, but with an eye toward generalizing to more particles, higher dimensions, and other symmetries. We explain the considerations underlying particular choices within the BRST construction, and develop both Hamiltonian and path integral formulations to provide guidance for the variety of many-body and effective field theory contexts where gauge fixing for symmetry restoration might be useful. For the demonstration system we show how to diagonalize within the extended BRST phase space, how variation after projection is recovered for product reference states, how the corresponding gauge-fixed functional integral is constructed, and how collective zero modes are isolated and controlled. Throughout, we keep in mind extensions of BRST symmetry to various approximation schemes as a guide for consistent symmetry restoration.