Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Further Multiparticle Topicst

Stephen L. Adler

Published Apr 27, 1995• 0 citations
Publisher

Abstract

Abstract In the third section, we examine the implementation of statistical mechanical methods in quaternionic quantum mechanics. We define thermal averages and construct thermally averaged retarded and “temperature” Green’s functions, which obey the usual temporal boundary conditions, but which are no longer related to one another by analytic continuation in Fourier space.

DOI
10.1093/oso/9780195066432.003.0010

Related papers

Thermodynamic Green’s Functions and Spectral Structure

Norman J. M. Horing

Abstract Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.

•Jul 27, 2017•0 citations

Variational Formulation for Quaternionic Quantum Mechanics

C. A. M. de Melo, B. M. Pimentel

arXiv:0809.4280•Advances in Applied Clifford Algebras•May 8, 2010•6 citations

Quantum Mechanical Ensemble Averages and Statistical Thermodynamics

Norman J. M. Horing

Abstract Chapter 6 introduces quantum-mechanical ensemble theory by proving the asymptotic equivalence of the quantum-mechanical, microcanonical ensemble average with the quantum grand canonical ensemble average for many-particle systems, based on the method of Darwin and Fowler. The procedures involved identify the grand partition function, entropy and other statistical thermodynamic variables, including the grand potential, Helmholtz free energy, thermodynamic potential, Gibbs free energy, Enthalpy and their relations in accordance with the fundamental laws of thermodynamics. Accompanying saddle-point integrations define temperature (inverse thermal energy) and chemical potential (Fermi energy). The concomitant emergence of quantum statistical mechanics and Bose–Einstein and Fermi–Dirac distribution functions are discussed in detail (including Bose condensation). The magnetic moment is derived from the Helmholtz free energy and is expressed in terms of a one-particle retarded Green’s function with an imaginary time argument related to inverse thermal energy. This is employed in a discussion of diamagnetism and the de Haas-van Alphen effect.

•Jul 27, 2017•0 citations

Quantum Ensembles

Robert H. Swendsen

This chapter begins Part 4 of the text, which is concerned with quantum statistical mechanics. A brief review of quantum mechanics is followed by an extension of statistical mechanics to include averages over microscopic quantum states.

Oxford University Press eBooks•Mar 1, 2012•0 citations

Ionic conductivity in a quantum lattice gas model with three-particle interactions

Jeremiah H. Barry, K. A. Muttalib, T. Tanaka

A system of mesoscopic ions with dominant three-particle interactions is modeled by a quantum lattice liquid on the planar kagome lattice. The two-parameter Hamiltonian contains localized attractive triplet interactions as potential energy and nearest neighbor hopping-type terms as kinetic energy. The dynamic ionic conductivity σ(ω) is theoretically investigated for 'weak hopping' via a quantum many-body perturbation expansion of the thermal (Matsubara) Green function (current-current correlation). A simple analytic continuation and mapping of the thermal Green function provide the temporal Fourier transform of the physical retarded Green function in the Kubo formula. Substituting pertinent exact solutions for static multi-particle correlations known from previous work, Arrhenius relations are revealed in zeroth-order approximation for the dc ionic conductivity σdc along special trajectories in density–temperature space. The Arrhenius plots directly yield static activation energies along the latter loci. Experimental possibilities relating to σdc are discussed in the presence of equilibrium aggregation.This article is part of 'Lattice models and integrability', a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.

Journal of Physics A Mathematical and Theoretical•Nov 27, 2012•2 citations

Non-invariant ground states, thermal average, and generalized fermionic statistics

R. Mignani, Roberto Scipioni

arXiv:cond-mat/9910151•Physics Letters A•Dec 1, 1999•0 citations
Abstract In the third section, we examine the implementation of statistical mechanical methods in quaternionic quantum mechanics. We define thermal averages and construct thermally averaged retarded and “temperature” Green’s functions, which obey the usual temporal boundary conditions, but which are no longer related to one another by analytic continuation in Fourier space.
Abstract Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.
Abstract Chapter 6 introduces quantum-mechanical ensemble theory by proving the asymptotic equivalence of the quantum-mechanical, microcanonical ensemble average with the quantum grand canonical ensemble average for many-particle systems, based on the method of Darwin and Fowler. The procedures involved identify the grand partition function, entropy and other statistical thermodynamic variables, including the grand potential, Helmholtz free energy, thermodynamic potential, Gibbs free energy, Enthalpy and their relations in accordance with the fundamental laws of thermodynamics. Accompanying saddle-point integrations define temperature (inverse thermal energy) and chemical potential (Fermi energy). The concomitant emergence of quantum statistical mechanics and Bose–Einstein and Fermi–Dirac distribution functions are discussed in detail (including Bose condensation). The magnetic moment is derived from the Helmholtz free energy and is expressed in terms of a one-particle retarded Green’s function with an imaginary time argument related to inverse thermal energy. This is employed in a discussion of diamagnetism and the de Haas-van Alphen effect.
A system of mesoscopic ions with dominant three-particle interactions is modeled by a quantum lattice liquid on the planar kagome lattice. The two-parameter Hamiltonian contains localized attractive triplet interactions as potential energy and nearest neighbor hopping-type terms as kinetic energy. The dynamic ionic conductivity σ(ω) is theoretically investigated for 'weak hopping' via a quantum many-body perturbation expansion of the thermal (Matsubara) Green function (current-current correlation). A simple analytic continuation and mapping of the thermal Green function provide the temporal Fourier transform of the physical retarded Green function in the Kubo formula. Substituting pertinent exact solutions for static multi-particle correlations known from previous work, Arrhenius relations are revealed in zeroth-order approximation for the dc ionic conductivity σdc along special trajectories in density–temperature space. The Arrhenius plots directly yield static activation energies along the latter loci. Experimental possibilities relating to σdc are discussed in the presence of equilibrium aggregation.This article is part of 'Lattice models and integrability', a special issue of Journal of Physics A: Mathematical and Theoretical in honour of F Y Wu's 80th birthday.