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

Journal of Physics A Mathematical and General

Entropy of ideal quantum gases as a Lie function

B Bruhn

Published Dec 21, 1993• 1 citation
Publisher

Abstract

We investigate the Lie series representation of the canonical transformations in a complex phase space. It is shown that any canonical mapping in the complex domain can be labelled by two different functions. One of these functions corresponds to an observable in the sense of classical mechanics. The second one has special analytic properties and can be used to form quantities which are important in quantum statistical mechanics. In particular we show that the entropy of ideal quantum gases generates a special canonical transformation and, moreover, the entropy itself can be represented as a Lie function formed by a special characteristic function.

DOI
10.1088/0305-4470/26/24/003

Related papers

Factorization of complex canonical transformations

B. Bruhn

We investigate the Lie series representation of the canonical transformations in a finite dimensional complex phase space. It is shown that any transformation of this type can be factorized into a product of three factors associated with a pure imaginary generating function, a holomorphic function, and an element of the cyclic group C4. The imaginary function can be considered as an observable in the sense of classical mechanics. Some hints are given which suggest that the holomorphic function can be connected with the notion of the state of a physical system. Moreover, a special kind of mappings is studied which provides a link between entropy, action, and state functions. The occurrence of these important physical quantities shows that the mathematical structure goes beyond a formal analogy to quantum physics at least in the finite dimensional case.

Journal of Mathematical Physics•Jul 1, 1997•0 citations

The Group-Algebraic Formalism of Quantum Probability and Its Applications in Quantum Statistical Mechanics

Yan Gu, Jiao Wang

We show that the theory of quantum statistical mechanics is a special model in the framework of the quantum probability theory developed by mathematicians, by extending the characteristic function in the classical probability theory to the quantum probability theory. As dynamical variables of a quantum system must respect certain commutation relations, we take the group generated by a Lie algebra constructed with these commutation relations as the bridge, so that the classical characteristic function defined in a Euclidean space is transformed to a normalized, non-negative definite function defined in this group. Indeed, on the quantum side, this group-theoretical characteristic function is equivalent to the density matrix; hence, it can be adopted to represent the state of a quantum ensemble. It is also found that this new representation may have significant advantages in applications. As two examples, we show its effectiveness and convenience in solving the quantum-optical master equation for a harmonic oscillator coupled with its thermal environment, and in simulating the quantum cat map, a paradigmatic model for quantum chaos. Other related issues are reviewed and discussed as well.

Entropy•Jan 10, 2025•0 citations

Canonical Transformations and Squeezing in Quantum Mechanics

José M. Cerveró, Rodriguez-Gonzalez, A.

In this Paper we present an approach to Quantum Mechanical Canonical Transformations. Our main result is that Time Dependent Quantum Canonical Transformations can always be cast in the form of Squeezing Operators. We revise the main properties of these operators in regard to its Lie group properties, how two of them can be combined to yield another operator of the same class and how can also be decomposed and fragmented. In the second part of the paper we show how this procedure works extremely well for the Time Dependent Quantum Harmonic Oscillator. The issue of the systematic construction of Quantum Canonical Transformations is also discussed along the lines of Dirac, Wigner and Schwinger ideas and to the more recent work by Lee. The main conclusion is that the Classical Phase Space Transformation can be maintained in the operator formalism but the construction of the Quantum Canonical Transformation is not clearly related to the Classical Generating Function of a Classical Canonical Transformation. We propose the road of Squeezing Operators rather than the old one attached to Quantum Operators constructed under the guideline of the exponential of the Classical Generating Function.

arXiv:quant-ph/0106157•Jun 28, 2001•0 citations

Point transformations in quantum mechanics

Kurt Bernardo Wolf

First a review is presented of the definition and properties of classical and quantum canonical transformations, from the point of view of a current program on the role of canonical transformations in quantum mechanics. The groups and the corresponding infinitesimal algebras are explored. The subgroup of point transformations (i.e. canonical transformations between pairs of conjugate observables, where one of a pair is function only of one of the other pair) is of special interest since it is a group of transformations for which classical and quantum mechanics follow each other. Only the group of inhomogeneous symplectic transformations has similar characteristics. Point transformations are treated in detail and a principal series of unitary representations is constructed. (auth)

•Jan 1, 1973•3 citations

Geometric Transformations in Classical Phase Space and their Representation in Quantum Mechanics

Peter Krämer, G. John, D. Schenzle

The geometry of classical phase space is determined by the Poisson bracket of functions on phase space. Canonical transformations of the classical coordinates and momenta are defined by the property that the fundamental Poisson brackets be preserved. In this section we shall study in detail the linear canonical transformations in phase space and their representations in quantum mechanics. The reason for developing these concepts is twofold. First of all there are a number of geometric transformations common to classical and quantum mechanics which appear naturally in the analysis of many-body systems. Examples are rotations of the three space coordinates, permutations of single-particle coordinates and transformations to various types of relative vectors. To these we shall add translations of coordinates and momenta and construct the operators which represent these geometric transformations in the Hilbert space of quantum mechanics. The second reason for dealing with these transformations is the fact that certain operators encountered in quantum mechanics may be interpreted as representatives of underlying geometric transformations in classical phase space. This applies in particular to dilatation operators and to Gaussian interactions. Recognition of the underlying geometric transformation allows us to reduce operator multiplication to combination on the geometric level like matrix multiplication.

Vieweg+Teubner eBooks•Jan 1, 1981•0 citations

Quantum statistical mechanics as entropy collapse in classical phase space

Phil Attard

The seventh chapter is concerned with the formulation of quantum statistical mechanics from first principles. It is shown that interactions with the reservoir or environment cause the subsystem wave function to collapse into a mixture of pure entropy states, which obey the axioms of classical probability theory. A dual representation in position and momentum states allows a permutation loop expansion for particle symmetrisation, and a second expansion accounting for the non-commutativity of the position and momentum operators. The leading term gives classical statistical mechanics, and higher order terms gives successive quantum corrections that become increasingly important for low temperatures, high densities, and small masses. From the permutation loop expansion, explicit results for the quantum ideal gas and the quantum harmonic oscillator are derived. The role of entropy in the classical manifestation of the universe given the underlying quantum mechanical wave equation is discussed.

Institute of Physics eBooks•Apr 1, 2023•0 citations
We investigate the Lie series representation of the canonical transformations in a complex phase space. It is shown that any canonical mapping in the complex domain can be labelled by two different functions. One of these functions corresponds to an observable in the sense of classical mechanics. The second one has special analytic properties and can be used to form quantities which are important in quantum statistical mechanics. In particular we show that the entropy of ideal quantum gases generates a special canonical transformation and, moreover, the entropy itself can be represented as a Lie function formed by a special characteristic function.
We investigate the Lie series representation of the canonical transformations in a finite dimensional complex phase space. It is shown that any transformation of this type can be factorized into a product of three factors associated with a pure imaginary generating function, a holomorphic function, and an element of the cyclic group C4. The imaginary function can be considered as an observable in the sense of classical mechanics. Some hints are given which suggest that the holomorphic function can be connected with the notion of the state of a physical system. Moreover, a special kind of mappings is studied which provides a link between entropy, action, and state functions. The occurrence of these important physical quantities shows that the mathematical structure goes beyond a formal analogy to quantum physics at least in the finite dimensional case.
We show that the theory of quantum statistical mechanics is a special model in the framework of the quantum probability theory developed by mathematicians, by extending the characteristic function in the classical probability theory to the quantum probability theory. As dynamical variables of a quantum system must respect certain commutation relations, we take the group generated by a Lie algebra constructed with these commutation relations as the bridge, so that the classical characteristic function defined in a Euclidean space is transformed to a normalized, non-negative definite function defined in this group. Indeed, on the quantum side, this group-theoretical characteristic function is equivalent to the density matrix; hence, it can be adopted to represent the state of a quantum ensemble. It is also found that this new representation may have significant advantages in applications. As two examples, we show its effectiveness and convenience in solving the quantum-optical master equation for a harmonic oscillator coupled with its thermal environment, and in simulating the quantum cat map, a paradigmatic model for quantum chaos. Other related issues are reviewed and discussed as well.
In this Paper we present an approach to Quantum Mechanical Canonical Transformations. Our main result is that Time Dependent Quantum Canonical Transformations can always be cast in the form of Squeezing Operators. We revise the main properties of these operators in regard to its Lie group properties, how two of them can be combined to yield another operator of the same class and how can also be decomposed and fragmented. In the second part of the paper we show how this procedure works extremely well for the Time Dependent Quantum Harmonic Oscillator. The issue of the systematic construction of Quantum Canonical Transformations is also discussed along the lines of Dirac, Wigner and Schwinger ideas and to the more recent work by Lee. The main conclusion is that the Classical Phase Space Transformation can be maintained in the operator formalism but the construction of the Quantum Canonical Transformation is not clearly related to the Classical Generating Function of a Classical Canonical Transformation. We propose the road of Squeezing Operators rather than the old one attached to Quantum Operators constructed under the guideline of the exponential of the Classical Generating Function.
First a review is presented of the definition and properties of classical and quantum canonical transformations, from the point of view of a current program on the role of canonical transformations in quantum mechanics. The groups and the corresponding infinitesimal algebras are explored. The subgroup of point transformations (i.e. canonical transformations between pairs of conjugate observables, where one of a pair is function only of one of the other pair) is of special interest since it is a group of transformations for which classical and quantum mechanics follow each other. Only the group of inhomogeneous symplectic transformations has similar characteristics. Point transformations are treated in detail and a principal series of unitary representations is constructed. (auth)
The geometry of classical phase space is determined by the Poisson bracket of functions on phase space. Canonical transformations of the classical coordinates and momenta are defined by the property that the fundamental Poisson brackets be preserved. In this section we shall study in detail the linear canonical transformations in phase space and their representations in quantum mechanics. The reason for developing these concepts is twofold. First of all there are a number of geometric transformations common to classical and quantum mechanics which appear naturally in the analysis of many-body systems. Examples are rotations of the three space coordinates, permutations of single-particle coordinates and transformations to various types of relative vectors. To these we shall add translations of coordinates and momenta and construct the operators which represent these geometric transformations in the Hilbert space of quantum mechanics. The second reason for dealing with these transformations is the fact that certain operators encountered in quantum mechanics may be interpreted as representatives of underlying geometric transformations in classical phase space. This applies in particular to dilatation operators and to Gaussian interactions. Recognition of the underlying geometric transformation allows us to reduce operator multiplication to combination on the geometric level like matrix multiplication.
The seventh chapter is concerned with the formulation of quantum statistical mechanics from first principles. It is shown that interactions with the reservoir or environment cause the subsystem wave function to collapse into a mixture of pure entropy states, which obey the axioms of classical probability theory. A dual representation in position and momentum states allows a permutation loop expansion for particle symmetrisation, and a second expansion accounting for the non-commutativity of the position and momentum operators. The leading term gives classical statistical mechanics, and higher order terms gives successive quantum corrections that become increasingly important for low temperatures, high densities, and small masses. From the permutation loop expansion, explicit results for the quantum ideal gas and the quantum harmonic oscillator are derived. The role of entropy in the classical manifestation of the universe given the underlying quantum mechanical wave equation is discussed.