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arXiv:hep-th/9606019

Algebra of Conserved Generators and Statistical Ensembles in Generalized Quantum Dynamics

Stephen L. Adler, Lawrence Paul Horwitz

Published Jun 4, 1996• 0 citations
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Abstract

We study here the algebraic structure of the conserved generators from which the microcanonical and canonical ensembles are constructed on an underlying generalized quantum dynamics, and the flows they induce on the phase space. We also discuss briefly the structure of the microcanonical and canonical ensembles.
DOI
10.48550/arxiv.hep-th/9606019

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The phase space Γ of quantum mechanics can be viewed as the complex projective space CPn endowed with a Kählerian structure given by the Fubini-Study metric and an associated symplectic form. We can then interpret the Schrödinger equation as generating a Hamiltonian dynamics on Γ. Based upon the geometric structure of the quantum phase space we introduce the corresponding natural microcanonical and canonical ensembles. The resulting density matrix for the canonical Γ-ensemble differs from the density matrix of the conventional approach. As an illustration, the results are applied to the case of a spin one-half particle in a heat bath with an applied magnetic field.
The phase space of quantum mechanics can be viewed as the complex projective space CP^n endowed with a Kaehlerian structure given by the Fubini-Study metric and an associated symplectic form. Therefore, we can interpret the Schrodinger equation as generating a natural Hamiltonian dynamics on CP^n. Based upon the geometric structure of the quantum phase space we introduce the corresponding natural microcanonical and canonical ensembles. The resulting distribution implied by the quantum canonical ensemble differs from distribution of the conventional density matrix approach, which in some respects can be viewed as semiclassical. The results are applied in detail to the case of a spin one-half particle in a heat bath with an applied magnetic field.
We develop a regularization of the quantum microcanonical ensemble, called a Gaussian ensemble, which can be used for derivation of the canonical ensemble from microcanonical principles. The derivation differs from the usual methods by giving an explanation for the, at the first sight unreasonable, effectiveness of the canonical ensemble when applied to certain small, isolated, systems. This method also allows a direct identification between the parameters of the microcanonical and the canonical ensemble and it yields simple indicators and rigorous bounds for the effectiveness of the approximation. Finally, we derive an asymptotic expansion of the microcanonical corrections to the canonical ensemble for those systems, which are near, but not quite, at the thermodynamical limit and show how and why the canonical ensemble can be applied also for systems with an exponentially increasing density of states. The aim throughout the paper is to keep mathematical rigour intact while attempting to produce results which are both physically and practically interesting.