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Chinese Physics

Multi-mode q-oscillator algebras with q2(k + 1) = 1 and related thermo field dynamics

Ya-Jun Gao

Published May 14, 2002• 1 citation
Publisher

Abstract

A type of multi-mode q-oscillator algebra with q2(k + 1) = 1 is set up and the associated qk-thermo field dynamics is constructed for all k = 1,2,,∞ in a unified form. It is demonstrated that these qk-thermo field dynamics can all be nicely fitted into the algebraic formulation of statistical mechanics (axiomatized form for statistical physics). This means that we obtain infinitely many realizations of the algebraic scheme, which extend the consideration of Ojima [1981 Ann. Phys. 137 1] and contain the usual thermo field dynamics for the fermionic (k = 1) and bosonic (k = ∞) systems as special cases. As simple applications, the qk-statistical average of some operators are given.

DOI
10.1088/1009-1963/11/6/307

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A type of multi-mode q-oscillator algebra with q2(k + 1) = 1 is set up and the associated qk-thermo field dynamics is constructed for all k = 1,2,,∞ in a unified form. It is demonstrated that these qk-thermo field dynamics can all be nicely fitted into the algebraic formulation of statistical mechanics (axiomatized form for statistical physics). This means that we obtain infinitely many realizations of the algebraic scheme, which extend the consideration of Ojima [1981 Ann. Phys. 137 1] and contain the usual thermo field dynamics for the fermionic (k = 1) and bosonic (k = ∞) systems as special cases. As simple applications, the qk-statistical average of some operators are given.
A type of multi-mode q-oscillator algebra with q^2(k+1)=1 is set up and the associated qk-thermo field dynamics is constructed for all k=1,2,…，∞ in a unified form.It is demonstrated that these qk-thermo field dynamics can all be nicely fitted into the algebraic formulation of statistical mechanics(axiomatized form for statistical physics).This means that we obtain infinitely many realizations of the algebraic scheme,which extend the consideration of Ojima [1981 Ann.Phys.137 1] and contain the usual thermo field dynamics for the fermionic (k=1) and bosonic (k=∞) systems as special cases.As simple applications,the qk-statistical average of some operators are given.
The algebraic structure of Thermo Field Dynamics lies in the qqq-deformation of the algebra of creation and annihilation operators. Doubling of the degrees of freedom, tilde-conjugation rules, and Bogoliubov transformation for bosons and fermions are recognized as algebraic properties of hq(1)h_{q}(1)hq​(1) and of hq(1∣1)h_{q}(1|1)hq​(1∣1), respectively.
In this paper, we present two new realizations of infinite statistics and construct their respective number operators in a simple manner. One of the realizations assumes the individual oscillators to be bosonic but the mutually commutation relationship (CR) to be of the quonic form. Interestingly, the system of mutually commuting oscillators satisfying individual relations of the type [Formula: see text] (no sum over i) is shown to describe a collection of bosons. This demonstrates that for these examples the mutual CRs between the oscillators decide the statistics.
Problem Non Ideal Bose Gas, Superfluidity and Fundamental Aspects of Quasiaverages Application Quasiaverages in the Theory of Superconductivity Correlations Weakening and Theorems on Singularities of the Type 1/Q2 Some Remarks on the Ergodic Theory (and Real Example From Statistical Mechanics Where the Usual Ergodic Conditions Not Satisfed) Presented Revised (More Simplified Version) Method of Second Quantized Representation for Wave Functions of Boson and Fermion Particles and Dynamical Operators.
We presents a unified and concise exposition of key topics in the mathematical theory of open quantum systems, developed within the framework of operator algebras. The manuscript consolidates and extends a series of invited articles originally prepared for the Modern Encyclopedia of Mathematical Physics, combining foundational material with modern perspectives on non-equilibrium quantum statistical mechanics. After introducing the C*- and W*-algebraic formulation of quantum mechanics, the paper reviews quantum dynamical systems, KMS states, and Tomita-Takesaki modular theory, as well as CCR and CAR algebras for bosonic and fermionic systems. Particular emphasis is placed on infinite systems, non-equilibrium steady states, entropy production, and linear response theory. The later sections develop a systematic treatment of small systems coupled to reservoirs, open lattice quantum spin systems, culminating in a detailed discussion of competing notions of quantum entropy production. The presentation highlights structural insights, conceptual clarity, and connections between equilibrium and non-equilibrium phenomena, providing a self-contained reference for researchers and graduate students in mathematical physics.
The description of the statistical dynamics of quantum oscillators is formulated in terms of the Wigner distribution, analogous to the more commonly used PPP distribution, with explicit formulas being obtained for its time evolution and for average values. This formulation is desirable because, e.g., the Wigner distribution always exists whereas the PPP distribution does not. The formalism is applied to the process of parametric amplification in a single mode, which may be considered as the degenerate form of the well-known two-mode case. This degeneracy gives rise to significantly different properties; for example, the PPP distribution for the single mode of interest evolves from a circularly symmetric two-dimensional Gaussian into an elliptically symmetric form and ceases to exist after a finite time, even for amplification in the presence of losses. This is contrary to the two-mode case. The corresponding Wigner distribution is found to exist as a well-behaved function for all time as expected, regardless of the amount of losses, and is used to calculate average values of various quantities of interest. It is found, e.g., that in the lossless case the average number of photons in the signal mode always becomes infinite as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. This is in contrast to the corresponding classical result for the lossless case which allows the signal to decay rather than to grow with time, depending on the relative phase between the signal and the pump. Field fluctuations are discussed and found to have some unusual properties. The combination of frequency up-conversion with single-mode amplification is also described briefly. The effect of the quantization of the pump oscillator is considered in an Appendix.