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arXiv:quant-ph/0105009

Quantum Boltzmann statistics in interacting systems

Luigi Accardi, Sergei Vladimirovich Kozyrev

Published May 3, 2001• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0105009

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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.
Abstract Emerging of free (or quantum Boltzmann) statistics for a model of quantum particle interacting with quantum field is described in the stochastic limit without dipole approximation. The quantum field is considered in a Gaussian (for example temperature) state. Entangled operators which describe interaction of the field and the particle satisfy the qqq -deformed relations which in the stochastic limit generate free 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.
We survey some results obtained in the last 25 years to illustrate the following probabilistic quantization statement: quantum probability is not a generalization of classical probability but a deeper level of it. Generalizing the classical theory of orthogonal polynomials, one can show that any random field XXX admitting all the moments can be represented as the sum of three operators, which are natural extensions of the creation, annihilation, and preservation operators in the usual boson Fock quantum theory. These operators generate a noncommutative ∗*∗-algebra on which the quantum extension of the expected value relative to the probability distribution of the field XXX induces a quantum state. So, any classical algebraic probability space generates a quantum space with its own commutation relations. The Heisenberg commutation relation characterizes the classical fields, while the new type of commutation relations (of type II) appears in non-Gaussian cases. The same machinery, but applied to Bernoulli fields, leads to Fermi--Dirac anticommutation relations (see the introduction).
We survey some results obtained in the last 25 years to illustrate the following probabilistic quantization statement: quantum probability is not a generalization of classical probability but a deeper level of it. Generalizing the classical theory of orthogonal polynomials, one can show that any random field XXX admitting all the moments can be represented as the sum of three operators, which are natural extensions of the creation, annihilation, and preservation operators in the usual boson Fock quantum theory. These operators generate a noncommutative ∗*∗-algebra on which the quantum extension of the expected value relative to the probability distribution of the field XXX induces a quantum state. So, any classical algebraic probability space generates a quantum space with its own commutation relations. The Heisenberg commutation relation characterizes the classical fields, while the new type of commutation relations (of type II) appears in non-Gaussian cases. The same machinery, but applied to Bernoulli fields, leads to Fermi-Dirac anticommutation relations (see the introduction).