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arXiv:1712.04459 · Physical review. D/Physical review. D.

Schwinger-Keldysh superspace in quantum mechanics

Michael Geracie, Felix M. Haehl, R. Loganayagam, Prithvi Narayan, David M. Ramirez, Mukund Rangamani

Published May 29, 2018• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevd.97.105023

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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.
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.
In order to preserve the leading role of the action principle in formulating all field theories one needs quantum field theory, with the associated BRST symmetry, and Feynman–DeWitt–Faddeev–Popov ghost fields. Such fields result from the fiber-bundle structure of the space of histories, but the physics-oriented literature used them formally because a rigorous theory of measure and integration was lacking. Motivated by this framework, this paper exploits the previous work of Gill and Zachary, where the use of Banach spaces for the Feynman integral was proposed. The Henstock–Kurzweil integral is first introduced, because it makes it possible to integrate functions like [Formula: see text]. The Lebesgue measure on [Formula: see text] is then built and used to define the measure on every separable Hilbert space. The subsequent step is the construction of a new Hilbert space [Formula: see text], which contains [Formula: see text] as a continuous dense embedding, and contains both the test functions [Formula: see text] and their dual [Formula: see text], the Schwartz space of distributions, as continuous embeddings. This space allows us to construct the Feynman path integral in a manner that maintains its intuitive and computational advantages. We also extend this space to [Formula: see text], where [Formula: see text] is any separable Hilbert space. Last, the existence of a unique universal definition of time, [Formula: see text], that we call historical time, is proven. We use [Formula: see text] as the order parameter for our construction of Feynman’s time ordered operator calculus, which in turn is used to extend the path integral in order to include all time-dependent groups and semigroups with a reproducing kernel representation.
The connections between standard theoretical tools used to study open quantum systems can sometimes seem opaque. Whether it is a Lindblad master equation, the equation of motion for the Wigner function, or a dissipative Keldysh action, features evident in one formalism are often masked in another. Here, we reformulate the technique of third quantization in a way that explicitly connects all three methods. We first show that our formulation reveals a fundamental dissipative symmetry present in all quadratic bosonic or fermionic Lindbladians. This symmetry can then be used to easily diagonalize these models, and it provides an intuitive way to demonstrate the separation of dissipation and fluctuations in linear systems. For bosons, we then show that the Wigner function and the characteristic function can be thought of as ``wave functions'' of the density matrix in the eigenbasis of the third-quantized superoperators we introduce. The field-theory representation of the time-evolution operator in this basis is then the Keldysh path integral. To highlight the utility of our approach, we apply our version of third quantization to a dissipative nonlinear oscillator, and we use it to obtain new exact results.
We study the Caldeira-Leggett model where a quantum Brownian particle interacts with an environment or a bath consisting of a collection of harmonic oscillators in the path-integral formalism. Compared to the contours that the paths take in the conventional Schwinger-Keldysh formalism, the paths in our study are deformed in the complex time plane as suggested by the recent study by C. Aron, G. Biroli, and L. F. Cugliandolo [SciPost Phys. 4, 008 (2018)10.21468/SciPostPhys.4.1.008]. This is done to investigate the connection between the symmetry properties in the Schwinger-Keldysh action and the equilibrium or nonequilibrium nature of the dynamics in an open quantum system. We derive the influence functional explicitly in this setting, which captures the effect of the coupling to the bath. We show that in equilibrium the action and the influence functional are invariant under a set of transformations of path-integral variables. The fluctuation-dissipation relation is obtained as a consequence of this symmetry. When the system is driven by an external time-dependent protocol, the symmetry is broken. From the terms that break the symmetry, we derive a quantum Jarzynski-like equality for a quantum mechanical worklike quantity given as a function of fluctuating quantum trajectory. In the classical limit, the transformations becomes those used in the functional integral formalism of the classical stochastic thermodynamics to derive the classical fluctuation theorem.
This chapter examines the Schwinger-Keldysh technique, applying path integrals, and the functional method of 2PI effective actions—with the latter serving to construct powerful symmetry-preserving nonperturbative approximations for equilibrium as well as nonequilibrium descriptions of strongly interacting quantum many-body systems. It provides a rigorous mathematical basis for exploring the quantum mechanical basis of thermalization and decoherence. Mesoscopic systems with quantum processes, in particular, play a key role. The domain of nonequilibrium mesoscopic systems is flooded with open questions some of which are solved but still others have provisional or incomplete solutions. A correlation function encodes the correlation between the field operators at different times and is an inference to all thermodynamic quantities. The beauty of the Keldysh time path is the definition of the Green's function, which plays an analogous role in the nonequilibrium formalism similar to the role the causal Green's function plays in the equilibrium theory.
Relativistic quantum field theory (QFT) is commonly formulated in terms of operators, asymptotic states, and covariant amplitudes, a perspective that tends to obscure the real-time origin of field dynamics and correlations. Here we formulate quantum fields in a real-time Schrödinger-picture framework, in which fields evolve as probability ensembles on the space of field configurations. Within this formulation, the wavefunctional Ψ[φ,t]Ψ[φ,t]Ψ[φ,t] encodes a first-order, causal ensemble dynamics on configuration space. Interactions appear as couplings between configuration-space directions, while propagators arise as derived correlation structures rather than as fundamental postulates. Entanglement, scattering amplitudes, and conformal field theory correlators emerge as distinct projections of the same underlying ensemble evolution, corresponding to equal-time, asymptotic, and symmetry-organized observables. Standard operator, diagrammatic, and path-integral formulations are recovered as computational representations of this single real-time dynamics. This organization makes explicit the distinction between fundamental dynamical structure and representational tools in QFT, and clarifies the scope within which ensemble-averaged correlators account for quantum fluctuations, while also delineating the level at which questions associated with individual realizations and randomness would arise beyond the correlator-based field-theoretic description.