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arXiv:hep-th/0607130

Thermal boundary conditions as constraints

César Daniel Fosco, Adolfo P. C. Malbouisson, I. Roditi

Published Jul 19, 2006• 0 citations
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Abstract

We introduce the boundary conditions corresponding to the imaginary-time (Matsubara) formalism for the finite-temperature partition function in $d+1$ dimensions as {\em constraints} in the path integral for the vacuum amplitude (the zero-temperature partition function). We implement those constraints by using Lagrange multipliers, which are static fields, two of them associated to each physical degree of freedom. After integrating out the original, physical fields, we obtain an effective representation for the partition function, depending only on the Lagrange multipliers. The resulting functional integral has the appealing property of involving only $d$-dimensional, {\em time independent} fields, looking like a non local version of the classical partition function. We analyze the main properties of this novel representation for the partition function, developing the formalism within the context of two concrete examples: the real scalar and Dirac fields.

DOI
10.48550/arxiv.hep-th/0607130

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We introduce the boundary conditions corresponding to the imaginary-time (Matsubara) formalism for the finite-temperature partition function in d+1d+1d+1 dimensions as {\em constraints} in the path integral for the vacuum amplitude (the zero-temperature partition function). We implement those constraints by using Lagrange multipliers, which are static fields, two of them associated to each physical degree of freedom. After integrating out the original, physical fields, we obtain an effective representation for the partition function, depending only on the Lagrange multipliers. The resulting functional integral has the appealing property of involving only ddd-dimensional, {\em time independent} fields, looking like a non local version of the classical partition function. We analyze the main properties of this novel representation for the partition function, developing the formalism within the context of two concrete examples: the real scalar and Dirac fields.
The boundary conditions corresponding to the Matsubara formalism for the T>0T > 0T>0 partition function may be introduced as {\em constraints} in the path integral for the vacuum amplitude. We implement those constraints with time-independent Lagrange multipliers and, by integrating out the original fields, we obtain an alternative representation for the partition function, in terms of the Lagrange multipliers as dynamical fields. The resulting functional integral has the appealing property of involving only ddd-dimensional, {\em time independent} fields, and looks like a nonlocal version of the classical partition function. We develop this formalism within the context of the scalar and Dirac fields.
We impose the periodicity conditions corresponding to the Matsubara formalism for thermal field theory as constraints in the imaginary-time path integral. These constraints are introduced by means of time-independent auxiliary fields which, by integration of the original variables, become dynamical fields in the resulting ``dual'' representation for the theory. This alternative representation has the appealing property of involving fields that live in one dimension less than the original ones, with a quantum partition function whose integration measure is identical to the one of its classical counterpart, albeit with a different (spatially nonlocal) action.
We present a unified formulation for quantum statistical physics based on the representation of the density matrix as a functional integral. For quantum statistical (thermal) field theory, the stochastic variable of the statistical theory is a boundary field configuration. We explore the properties of an effective theory for such boundary configurations and apply it to the computation of the partition function of an interacting one-dimensional quantum-mechanical system at finite temperature. Plots of free energy and specific heat show excellent agreement with more involved semiclassical results. The method of calculation provides an alternative to the usual sum over periodic trajectories: it sums over paths with coincident end points and includes nonvanishing boundary terms. An appropriately modified expansion into modified Matsubara modes is presented.
Representations for the generating functionals of static correlators of zzz-components of spins in the XY and XXXXXX Heisenberg spin chains are obtained in the form of sums of the fermionic functional integrals. The peculiarity of the functional integrals in question is because of the fact that the integration variables depend on the imaginary time ``automorphically''. In other words, the integration variables are multiplied with a certain complex number when the imaginary time is shifted by a period. Therefore, the corresponding boundary conditions at the ends of the imaginary time segment are not of the form corresponding to fermionic, or bosonic, variables taken in the Matsubara representation at nonzero temperature. In fact, one part of sites of the models corresponds to the integration variables which are subjected to the unusual boundary conditions, while the variables on the other sites depend on the imaginary time conventionally, i.e., as fermions (or bosons). Thus a situation, when an ``automorphic'' boundary condition is the same for all sites of a chain spin model, is generalized. The results of the functional integration are obtained in the form of determinants of the matrix operators which are regularized by means of the generalized zeta-function approach. The partition functions of the models and certain correlation functions at nonzero temperature are obtained explicitly thus demonstrating correctness of the functional integral representations proposed.
Abstract The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.
Using the eigenfunctional method, we derive general functional expressions of the partition function and electron Green's function of the DDD-dimensional Hubbard model for zero and finite temperatures, in which the sign problem of fermions is avoided. These expressions can be calculated by solving the equation of a phase field and taking the functional average over a Lagrange multiplier field, while for some special cases they can be solved exactly. Moreover, we compare our results with the exact solution of the dynamical mean-field theory in the infinite-dimension limit.