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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Statistical mechanics of a (1 + 1)-dimensional quantum field theory at finite density and temperature

H. B. Thacker

Published Oct 15, 1977• 26 citations
Publisher

Abstract

The quantum field theory in one space + one time dimension described by the Lagrangian $\mathfrak{L} = (\frac{i}{2}){\ensuremath{\varphi}}^{*}{\stackrel{}{\ensuremath{\partial}}}_{0}\ensuremath{\varphi}\ensuremath{-}({\ensuremath{\partial}}_{1}{\ensuremath{\varphi}}^{*})({\ensuremath{\partial}}_{1}\ensuremath{\varphi})\ensuremath{-}c{|\ensuremath{\varphi}|}^{4}$ is studied for systems with finite temperature and particle density. Using momentum-space techniques previously developed, a graphical procedure is obtained for calculating inner products of many-particle scattering state wave functions. The unitarity of the wave operator $U(0,\ensuremath{-}\ensuremath{\infty})$ is demonstrated as a pattern of graphical cancellations. An operator formulation of statistical mechanics is derived in which partition functions are given in terms of matrix elements having the form of diagonal (forward) inner products. The importance of noncommutativity of the forward limit and the $i\ensuremath{\epsilon}\ensuremath{\rightarrow}0$ limit is noted and traced to the presence of forward singular graphs in the inner product. Combining this observation with wave-operator unitarity, we obtain a graphical recipe for calculating $N$-body partition functions. The thermodynamics first described by Yang and Yang is obtained by summation of the fugacity series for the grand partition function.

DOI
10.1103/physrevd.16.2515

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The quantum field theory in one space + one time dimension described by the Lagrangian \mathfrak{L} = (\frac{i}{2}){\ensuremath{\varphi}}^{*}{\stackrel{}{\ensuremath{\partial}}}_{0}\ensuremath{\varphi}\ensuremath{-}({\ensuremath{\partial}}_{1}{\ensuremath{\varphi}}^{*})({\ensuremath{\partial}}_{1}\ensuremath{\varphi})\ensuremath{-}c{|\ensuremath{\varphi}|}^{4} is studied for systems with finite temperature and particle density. Using momentum-space techniques previously developed, a graphical procedure is obtained for calculating inner products of many-particle scattering state wave functions. The unitarity of the wave operator U(0,\ensuremath−\ensuremath∞)U(0,\ensuremath{-}\ensuremath{\infty})U(0,\ensuremath−\ensuremath∞) is demonstrated as a pattern of graphical cancellations. An operator formulation of statistical mechanics is derived in which partition functions are given in terms of matrix elements having the form of diagonal (forward) inner products. The importance of noncommutativity of the forward limit and the i\ensuremathϵ\ensuremath→0i\ensuremath{\epsilon}\ensuremath{\rightarrow}0i\ensuremathϵ\ensuremath→0 limit is noted and traced to the presence of forward singular graphs in the inner product. Combining this observation with wave-operator unitarity, we obtain a graphical recipe for calculating NNN-body partition functions. The thermodynamics first described by Yang and Yang is obtained by summation of the fugacity series for the grand partition function.
The Statistical Mechanics of the N-particle system is derived from Quantum Random Field Theory. An asymptotically measure preserving, flow operator has been obtained from the QFT evolution operator. A basic assumption is that the field Lagrangian is a generalized, infinitely divisible random field, and the interaction Hamiltonian varies in time in a step-wise manner at the transition lattice space-time points. Macroscopic boundary conditions are imposed on the wave functions. The density matrix, the partition function, the Bose-Einstein, the Fermi-Dirac distributions and the thermodynamic temperature follow. The equilibrium or non-equilibrium temperature have been obtained as functionals of the interaction Hamiltonian. The chemical potential has also been derived from the theory.
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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.
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.
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