The thermal Green functions in nonextensive quantum statistical mechanics

Abstract

The thermal Green functions of the quantum-mechanical harmonic oscillator are constructed within the framework of nonextensive statistical mechanics with normalized q-expectation values. For the Tsallis index q greater than unity, these functions are found to be expressed analytically in terms of the Hurwitz zeta function. It is found that influence of the nonextensivity on the time-ordered thermal propagator is relevant only at the "on-shell" states. In particular, the finite-temperature contribution to the thermal propagator becomes enhanced for the strong nonextensivity. PACS. 05.30.-d Quantum statistical mechanicsThermal field theory is attracting continuous interest in various research areas including solid state physics, particle physics, and cosmology. Among others, quark-gluon plasma (i.e., quantum chromodynamics at finite temperature) and quantum processes in the early Universe may be thought of as typical examples. There is a common important point behind them: the systems treated there include long-range interactions in themselves. In the former example, there is no natural mechanism which shields the chromomagnetic force, in contrast to the chromoelectric component [1]. In the latter one, quantum fields are put under influence of gravitational interaction [2]. Both of them are nonextensive systems.Nonextensivity is also relevant when a system has long-time memory, (multi)fractal configuration, or quantum group structure. A feature of a nonextensive system is that the total internal energy does not become proportional to the number of microscopic elements of the system in the thermodynamic limit. To treat such a system thermodynamically, suitable generalization of Boltzmann-Gibbs statistical mechanics seems to be essential [3]. Nowadays a possible approach to this problem is known. It was initiated and developed by Tsallis. This formalism [4,5], referred to as nonextensive statistical mechanics (NSM), and related mathematical frameworks have been successfully applied to, e.g., the Ising model with long-range interactions [6], astrophysical and cosmological problems [7], Lévy-type random walks [8], the traveling salesman problem [9], studies of biomolecules [10], and quantum groups [11]. (A comprehensive list of references is currently available at URL [12].)Unfortunately, nonextensive generalization of thermal field theory is still far out of reach, mainly due to mathematical difficulty of treating infinite degrees of freedom. One should however recall that within the framework of Boltzmann-Gibbs statistical mechanics some important physical properties of thermal field theory can still be found in dynamics of a single quantum-mechanical particle in the heat bath: e.g., the spectral property and the imaginary time periodicity of the thermal Green functions.In this paper, we study the quantum-mechanical thermal Green functions based on NSM, in hope of giving a step toward nonextensive thermal field theory. In particular, we use NSM with normalized q-expectation values [13], which has recently been propos...

III. Tsallis Theory, the Maximum Entropy Principle, and Evolution Equations

“…Many relevant mathematical properties of the standard thermostatistics are preserved by Tsallis' formalism or admit suitable generalizations [12][13][14][15][16][17][18][19][20]. Tsallis' proposal was shown to be consistent both with Jaynes' information theory formulation of statistical mechanics [21], and with the dynamical thermostatting approach to statistical ensembles [22].…”

Two-time Green’s functions and spectral density method in nonextensive quantum statistical mechanics

“…[25,26] in the context of the Kadanoff-Baym formalism [4]. This representation allows to express the relevant q quantities in terms of appropriate parametric integrals involving the corresponding q = 1-ones.…”

“…Some years ago the GF method in the Kadanoff-Baym framework [4] was generalized to the Tsallis quantum statistical mechanics adopting the second quantized representation for many-particle systems [25,26]. In these works, the q GFs for a nonextensive many-body system were formally expressed in terms of parametric integrals over the corresponding extensive (q = 1) quantities, q denoting the so-called Tsallis parameter which measures the nonextensivity degree.…”

Two-Time Green's Functions and the Spectral Density Method in Nonextensive Classical Statistical Mechanics

“…Non-normalized [8] and normalized [9] q-mean values have been used, where q is the Tsallis nonextensivity parameter. The q-GF technique provides new and effective methods for dealing with nontrivial physical problems in the Tsallis quantum thermostatistics.…”

“…Before introducing the method for a direct calculation of L ͑q͒ AB ͑v͒ we mention two procedures for calculating the relevant quantities. The first one consists of expressing the classical q-quantities by means of parametric integrals over corresponding extensive (q 1) integrals, as in the quantum counterpart [8,9]. This can be achieved simply by employing the contour integral representation G 21 ͑z͒ ib 12z R C du 2p exp͑2ub͒ ͑2u͒ 2z with b .…”