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

arXiv:hep-th/9606023 · Journal of Mathematical Physics

Microcanonical ensemble and algebra of conserved generators for generalized quantum dynamics

Stephen L. Adler, Lawrence Paul Horwitz

Published Nov 1, 1996• 13 citations
arXivHTMLPDF

Abstract

It has recently been shown, by application of statistical mechanical methods to determine the canonical ensemble governing the equilibrium distribution of operator initial values, that complex quantum field theory can emerge as a statistical approximation to an underlying generalized quantum dynamics. This result was obtained by an argument based on a Ward identity analogous to the equipartition theorem of classical statistical mechanics. We construct here a microcanonical ensemble which forms the basis of this canonical ensemble. This construction enables us to define the microcanonical entropy and free energy of the field configuration of the equilibrium distribution and to study the stability of the canonical ensemble. We also study the algebraic structure of the conserved generators from which the microcanonical and canonical ensembles are constructed, and the flows they induce on the phase space.

DOI
10.1063/1.531714

Related papers

Algebra of Conserved Generators and Statistical Ensembles in Generalized Quantum Dynamics

Stephen L. Adler, Lawrence Paul Horwitz

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.

arXiv:hep-th/9606019•Jun 4, 1996•0 citations

A generalized quantum microcanonical ensemble

Jan Naudts, Erik Van der Straeten

We discuss a generalized quantum microcanonical ensemble. It describes isolated systems that are not necessarily in an eigenstate of the Hamilton operator. Statistical averages are obtained by combining a time average and a maximum entropy argument to resolve the lack of knowledge about initial conditions. As a result, statistical averages of linear observables coincide with values obtained in the canonical ensemble. Non-canonical averages can be obtained by taking into account conserved quantities which are non-linear functions of the microstate.

arXiv:quant-ph/0602039•Journal of Statistical Mechanics Theory and Experiment•Jun 23, 2006•26 citations

The Constrained Origin of Canonical and Microcanonical Ensembles in Quantum Theory

Loris Di Cairano

In quantum theory, equilibrium statistical mechanics is usually formulated through the canonical ensemble, whose privileged status is tied to the Euclidean continuation of time evolution. The microcanonical ensemble, by contrast, is commonly introduced as a separate spectral construction. In this work we show that this asymmetry is representational rather than structural. We formulate the system in an extended Hilbert space in which time is promoted to an auxiliary clock degree of freedom and physical states are selected by a reparametrization-invariant constraint operator $\hat C = \hat P_T + \hat H$. The corresponding projector $δ(\hat C)$ provides a single unified object from which both canonical and microcanonical ensembles emerge as complementary projections in the clock sector. In the clock-time representation, a purely imaginary clock separation yields the Euclidean kernel and the canonical partition function. In the conjugate clock-energy representation, the same projector reduces to the spectral operator $δ(\hat H-E)$ and hence to the microcanonical density of states. The main consequence is structural: canonical and microcanonical statistics need not be introduced as independent constructions, since both are already encoded in the same constrained quantum dynamics.

quant-pharXiv:2603.12148•Mar 13, 2026•0 citations

Generalized quantum dynamics as pre-quantum mechanics

Stephen L. Adler, Millard, A C

We address the issue of when generalized quantum dyanmics, based on a total trace Hamiltonian {\\bf H}, reduces to Heisenberg picture complex quantum mechanics. We begin by showing that when {\\bf H}={\\rm{\\bf Tr}} H_{OP}, with H_{OP} a Weyl ordered operator Hamiltonian, then the generalized quantum dynamics operator equations of motion agree with those obtained from H_{OP} in the Heisenberg picture by using canonical commutation relations. Hence generalized quantum dynamics conicides with complex quantum mechanics in this case, provided the initial values of the operators obey the canonical algebra. The remainder of the paper is devoted to a study of when this initial value condition is satisfied. We first show that for any total trace Hamiltonian which involves no noncommutative constants, there is a conserved operator \\tilde C which is closely related to the canonical algebra. We demonstrate that for a system in equilbrium, the equilbrium ensemble of initial values is governed by the distribution exp(-|\\tilde C|/\\kappa). An argument analogous to that for equipartition of energy, together with the assumption that ensemble averages of products can be replaced by products of ensemble averages, then implies that the ensemble averaged operator initial values obey the standard canonical commutation relations of complex quantum mechanics, with the imaginary unit I identified as -\\tilde C / |\\tilde C|, and with the constant \\kappa playing the role of the reduced Planck constant \\hbar. Thus there is a well--defined sense in which complex quantum mechanics and quantum field theory emerge as a statistical approximation to generalized quantum dynamics.

•Aug 16, 1995•37 citations

The Quantum Canonical Ensemble

Brody, D C, Lane P. Hughston

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.

•Jan 1, 1998•34 citations

An alternative definition of the quantum microcanonical ensemble

Jan Naudts, Erik Van der Straeten

We discuss an alternative definition of the quantum microcanonical ensemble, in which statistical averages are obtained by a combination of a time average and a maximum entropy argument to resolve the lack of knowledge about initial conditions. As a result, statistical averages of linear observables coincide with values obtained in the canonical ensemble. Non-canonical averages can be obtained by taking into account conserved quantities which are non-linear functions of the microstate.

•Feb 3, 2006•0 citations
It has recently been shown, by application of statistical mechanical methods to determine the canonical ensemble governing the equilibrium distribution of operator initial values, that complex quantum field theory can emerge as a statistical approximation to an underlying generalized quantum dynamics. This result was obtained by an argument based on a Ward identity analogous to the equipartition theorem of classical statistical mechanics. We construct here a microcanonical ensemble which forms the basis of this canonical ensemble. This construction enables us to define the microcanonical entropy and free energy of the field configuration of the equilibrium distribution and to study the stability of the canonical ensemble. We also study the algebraic structure of the conserved generators from which the microcanonical and canonical ensembles are constructed, and the flows they induce on the phase space.
We discuss a generalized quantum microcanonical ensemble. It describes isolated systems that are not necessarily in an eigenstate of the Hamilton operator. Statistical averages are obtained by combining a time average and a maximum entropy argument to resolve the lack of knowledge about initial conditions. As a result, statistical averages of linear observables coincide with values obtained in the canonical ensemble. Non-canonical averages can be obtained by taking into account conserved quantities which are non-linear functions of the microstate.
In quantum theory, equilibrium statistical mechanics is usually formulated through the canonical ensemble, whose privileged status is tied to the Euclidean continuation of time evolution. The microcanonical ensemble, by contrast, is commonly introduced as a separate spectral construction. In this work we show that this asymmetry is representational rather than structural. We formulate the system in an extended Hilbert space in which time is promoted to an auxiliary clock degree of freedom and physical states are selected by a reparametrization-invariant constraint operator C^=P^T+H^\hat C = \hat P_T + \hat HC^=P^T​+H^. The corresponding projector δ(C^)δ(\hat C)δ(C^) provides a single unified object from which both canonical and microcanonical ensembles emerge as complementary projections in the clock sector. In the clock-time representation, a purely imaginary clock separation yields the Euclidean kernel and the canonical partition function. In the conjugate clock-energy representation, the same projector reduces to the spectral operator δ(H^−E)δ(\hat H-E)δ(H^−E) and hence to the microcanonical density of states. The main consequence is structural: canonical and microcanonical statistics need not be introduced as independent constructions, since both are already encoded in the same constrained quantum dynamics.
We address the issue of when generalized quantum dyanmics, based on a total trace Hamiltonian {\\bf H}, reduces to Heisenberg picture complex quantum mechanics. We begin by showing that when {\\bf H}={\\rm{\\bf Tr}} H_{OP}, with H_{OP} a Weyl ordered operator Hamiltonian, then the generalized quantum dynamics operator equations of motion agree with those obtained from H_{OP} in the Heisenberg picture by using canonical commutation relations. Hence generalized quantum dynamics conicides with complex quantum mechanics in this case, provided the initial values of the operators obey the canonical algebra. The remainder of the paper is devoted to a study of when this initial value condition is satisfied. We first show that for any total trace Hamiltonian which involves no noncommutative constants, there is a conserved operator \\tilde C which is closely related to the canonical algebra. We demonstrate that for a system in equilbrium, the equilbrium ensemble of initial values is governed by the distribution exp(-|\\tilde C|/\\kappa). An argument analogous to that for equipartition of energy, together with the assumption that ensemble averages of products can be replaced by products of ensemble averages, then implies that the ensemble averaged operator initial values obey the standard canonical commutation relations of complex quantum mechanics, with the imaginary unit I identified as -\\tilde C / |\\tilde C|, and with the constant \\kappa playing the role of the reduced Planck constant \\hbar. Thus there is a well--defined sense in which complex quantum mechanics and quantum field theory emerge as a statistical approximation to generalized quantum dynamics.
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.