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arXiv:hep-ph/9605372

Boundary Conditions and Correlations in Path Integrals for Quantum Field Theory of Thermal and Non-Equilibrium States

Peter A. Henning, R. Fauser

Published May 22, 1996• 1 citation
arXivHTMLPDF

Abstract

For thermal equilibrium systems it is shown, how the Kubo-Martin-Schwinger boundary condition may be used to factorize the generating functional of Green functions at least on the level of the full two-point function. Genuine non-equilibrium system exhibit correlations that one may also incorporate in the path integral. One one hand this provides a natural tool for a perturbative expansion including these correlations. On the other hand it allows to prove that in general non-equilibrium systems the generating functional does not factorize.

DOI
10.48550/arxiv.hep-ph/9605372

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For thermal equilibrium systems it is shown, how the Kubo-Martin-Schwinger boundary condition may be used to factorize the generating functional of Green functions at least on the level of the full two-point function. Genuine non-equilibrium system exhibit correlations that one may also incorporate in the path integral. One one hand this provides a natural tool for a perturbative expansion including these correlations. On the other hand it allows to prove that in general non-equilibrium systems the generating functional does not factorize.
Thermalization of a system when interacting with a thermal bath is an interesting problem. If a system eventually reaches a thermal state in the long time limit, it's expected that its density matrix would resemble the mean-force Gibbs state. Moreover, the correlation function must satisfy the Kubo-Martin-Schwinger (KMS) condition, or equivalently, the fluctuation-dissipation relation (FDR). In this paper, we derive a formal expression for the non-Markovian two-point function within the context of the weak coupling limit. Using this expression, we explicitly compute the two-point function for specific models, demonstrating their adherence to the KMS. In addition, we formulate a nonperturbative approach in the form of a self-consistent approximation that includes a partial resummation of perturbation theory. This approach can capture strong coupling phenomena while still relying on simple equations. Notably, we verify that the two-point function obtained through this method also satisfies the KMS condition.
Thermalization of a system when interacting with a thermal bath is an interesting problem. If a system eventually reaches a thermal state in the long time limit, it's expected that its density matrix would resemble the mean-force Gibbs state. Moreover, the correlation function must satisfy the Kubo-Martin-Schwinger (KMS) condition or equivalently the Fluctuation-Dissipation Relation (FDR). In this paper, we derive a formal expression for the non-Markovian two-point function within the context of the weak coupling limit. Using this expression, we explicitly compute the two-point function for specific models, demonstrating their adherence to the KMS. In addition, we have formulated a non-perturbative approach in the form of a self-consistent approximation that includes a partial resummation of perturbation theory. This approach can capture strong coupling phenomena while still relying on simple equations. Notably, we verify that the two-point function obtained through this method also satisfies the KMS condition.
We derive the generating functional of thermal Green's functions for an interacting electron-phonon system through the use of functional integrals in holomorphic representations (c numbers for phonons and Grassmann numbers for electrons). These integrals are over paths obeying one-sided boundary conditions at either 0 or \ensuremath{\beta}=1/kT, and do not require normalization factors unlike previous derivations. We recover the Matsubara perturbation formalism without using any results from the standard operator method.
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.
Abstract We discuss the regularized boundary state e−τ0H∣B⟩a {e}^{-{\tau}_0H}\Big|{\left.B\right\rangle}_a e−τ0​H​B⟩a​ e − τ 0 H | B a on two aspects in both 2D CFT and higher dimensional free field theory. One is its entanglement and correlation properties, which exhibit exponential decay in 2D CFT, the parameter 1/τ 0 works as a mass scale. The other concerns with its time evolution, i.e., e−itHe−τ0H∣B⟩a {e}^{-itH}{e}^{-{\tau}_0H}\Big|{\left.B\right\rangle}_a e−itHe−τ0​H​B⟩a​ e − i t H e − τ 0 H | B a . We investigate the Kubo-Martin-Schwinger (KMS) condition on correlation function of local operators to detect the thermal properties. Interestingly we find the correlation functions in the initial state e−τ0H∣B⟩a {e}^{-{\tau}_0H}\Big|{\left.B\right\rangle}_a e−τ0​H​B⟩a​ e − τ 0 H | B a also partially satisfy the KMS condition. In the limit t → ∞, the correlators will exactly satisfy the KMS condition. We generally analyse quantum quench by a pure state and obtain some constraints on the possible form of 2-point correlation function in the initial state if assuming they satisfies KMS condition in the final state. As a byproduct we find in an large τ 0 limit the thermal property of 2-point function in e−τ0H∣B⟩a {e}^{-{\tau}_0H}\Big|{\left.B\right\rangle}_a e−τ0​H​B⟩a​ e − τ 0 H | B a also appears.