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Quantum Boltzmann Statistics in Interacting Systems

Luigi Accardi, Sergei Vladimirovich Kozyrev

Published Jan 1, 2003• 0 citations

Abstract

Collective operators for generic quantum system with discrete spectrum are investigated. These operators, considered as operators in the entangled Fock space (space generated by action of collective creations on the vacuum) satisfy a particular kind of Quantum Boltzmann (or free) commutational relations.

DOI: 10.1007/978-3-0348-8018-3_1

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Quantum Boltzmann statistics in interacting systems

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Collective operators that describe interaction of generic quantum system with discrete spectrum with a quantum field are investigated. These operators, considered as operators in the entangled Fock space (space generated by action of collective creations on the vacuum) in the stochastic limit satisfy a particular kind of Quantum Boltzmann (or free) commutational relations. This clarifies a general phenomenon of arising of Quantum Boltzmann relations in interacting systems.

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Birkhäuser Basel eBooks

Quantum Boltzmann Statistics in Interacting Systems

Luigi Accardi, Sergei Vladimirovich Kozyrev

Published Jan 1, 2003• 0 citations

Publisher

Abstract

DOI: 10.1007/978-3-0348-8018-3_1

Related papers

Quantum Boltzmann statistics in interacting systems

Luigi Accardi, Sergei Vladimirovich Kozyrev

arXiv:quant-ph/0105009May 3, 20010 citations

QUANTUM BOLTZMANN STATISTICS IN INTERACTING SYSTEMS

Luigi Accardi, Sergei Vladimirovich Kozyrev

Quantum Probability and Infinite Dimensional AnalysisJan 1, 20030 citations

Particles, Generalized Statistics and Categories

Władysław Marcinek

Interacting systems of particles with generalized statistics are considered on both classical and quantum level. It is shown that all possible quantum states and corresponding processes can be represented in terms of certain specific categories. The corresponding Fock space representation is discussed. The problem of existence of well--defined scalar product is considered. It is shown that commutation relations corresponding to a system with generalized statistics can be constructed from relations corresponding to Bolzman statistics.Interacting systems of particles with generalized statistics are considered on both classical and quantum level. It is shown that all possible quantum states and corresponding processes can be represented in terms of certain specific categories. The corresponding Fock space representation is discussed. The problem of existence of well--defined scalar product is considered. It is shown that commutation relations corresponding to a system with generalized statistics can be constructed from relations corresponding to Bolzman statistics.

arXiv:math/0104141Apr 12, 20010 citations

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Alexei Daletskii, A. A. Kalyuzhnyi, Eugene Lytvynov +1 more

Let $H$ be a separable Hilbert space and $T$ be a self-adjoint bounded linear operator on $H^{\otimes 2}$ with norm $\le1$ , satisfying the Yang--Baxter equation. Bo\.zejko and Speicher (1994) proved that the operator $T$ determines a $T$ -deformed Fock space $\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)$ . We start with reviewing and extending the known results about the structure of the $n$ -particle spaces $\mathcal F_n(H)$ and the commutation relations satisfied by the corresponding creation and annihilation operators acting on $\mathcal F(H)$ . We then choose $H=L^2(X\to V)$ , the $L^2$ -space of $V$ -valued functions on $X$ . Here $X:=\mathbb R^d$ and $V:=\mathbb C^m$ with $m\ge2$ . Furthermore, we assume that the operator $T$ acting on $H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})$ is given by $(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)$ . Here, for a.a.\ $(x,y)\in X^2$ , $C_{x,y}$ is a linear operator on $V^{\otimes 2}$ with norm $\le1$ that satisfies $C_{x,y}^*=C_{y,x}$ and the spectral quantum Yang--Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function $C_{xy}$ in the case $d=2$ determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its $T$ -deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.Let $H$ be a separable Hilbert space and $T$ be a self-adjoint bounded linear operator on $H^{\otimes 2}$ with norm $\le1$, satisfying the Yang--Baxter equation. Bo\.zejko and Speicher (1994) proved that the operator $T$ determines a $T$-deformed Fock space $\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)$. We start with reviewing and extending the known results about the structure of the $n$-particle spaces $\mathcal F_n(H)$ and the commutation relations satisfied by the corresponding creation and annihilation operators acting on $\mathcal F(H)$. We then choose $H=L^2(X\to V)$, the $L^2$-space of $V$-valued functions on $X$. Here $X:=\mathbb R^d$ and $V:=\mathbb C^m$ with $m\ge2$. Furthermore, we assume that the operator $T$ acting on $H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})$ is given by $(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)$. Here, for a.a.\ $(x,y)\in X^2$, $C_{x,y}$ is a linear operator on $V^{\otimes 2}$ with norm $\le1$ that satisfies $C_{x,y}^*=C_{y,x}$ and the spectral quantum Yang--Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function $C_{xy}$ in the case $d=2$ determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its $T$-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.

Nov 7, 201910 citations

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Zhi-Tao Yan

A new kind of quantum statistics which interpolates between Bose and Fermi statistics is proposed beginning with the assumption that the quantum state of a many-particle system is a functional on the internal space of the particles. The quantum commutation relations for such particle creation and annihilation operators are derived, and statistical partition function and thermodynamical properties of an ideal gas of the particles are investigated. The application of this quantum statistics for the ensemble of extremal black holes are discussed.A new kind of quantum statistics which interpolates between Bose and Fermi statistics is proposed beginning with the assumption that the quantum state of a many-particle system is a functional on the internal space of the particles. The quantum commutation relations for such particle creation and annihilation operators are derived, and statistical partition function and thermodynamical properties of an ideal gas of the particles are investigated. The application of this quantum statistics for the ensemble of extremal black holes are discussed.

arXiv:quant-ph/9711038Physical Review LettersNov 19, 19970 citations

Let

H

be a separable Hilbert space and

T

be a self-adjoint bounded linear operator on

H^{\otimes 2}

with norm

\le1

, satisfying the Yang--Baxter equation. Bo\.zejko and Speicher (1994) proved that the operator

T

determines a

T

-deformed Fock space

\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)

. We start with reviewing and extending the known results about the structure of the

n

-particle spaces

\mathcal F_n(H)

and the commutation relations satisfied by the corresponding creation and annihilation operators acting on

\mathcal F(H)

. We then choose

H=L^2(X\to V)

, the

L^2

-space of

V

-valued functions on

X

. Here

X:=\mathbb R^d

and

V:=\mathbb C^m

with

m\ge2

. Furthermore, we assume that the operator

T

acting on

H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})

is given by

(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)

. Here, for a.a.\

(x,y)\in X^2

C_{x,y}

is a linear operator on

V^{\otimes 2}

with norm

\le1

that satisfies

C_{x,y}^*=C_{y,x}

and the spectral quantum Yang--Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function

C_{xy}

in the case

d=2

determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its

T

-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.