Quantum statistical mechanics as a construction of an embedding scheme

The aim of the present paper is to show that the formalism of equilibrium quantum statistical mechanics can fully be incorporated into Ludwig's embedding scheme for classical theories in many-body quantum mechanics. A construction procedure based on a recently developed reconstruction procedure for the so-called macro-observable is presented which leads to the explicit determination of the set of classical ensembles compatible with the embedding scheme.     

Field operators and their spectral properties in finite-dimensional quantum field theory

In Ref. 1 we have considered the finite-dimensional quantum mechanics. There the quantum mechanical space of states wasV=C ^r. It is known that the second quantization of this space is the space of square-summable functions of finite number of variables(L ²(R^r,dx)) (Segal isomorphism). Creation and annihilation operators were introduced in Ref. 1, and the former coincided with the usual position and momentum operators in the conventional quantum mechanics. In this paper we shall investigate the spectral properties of field operators. We shall show that the isomorphism between the exponential ofV andL ²(R^r,dx) can be understood as the decomposition by generalized eigenvectors of field operators (Fourier transform).     

Statistical mechanics and the ontological interpretation

To complete our ontological interpretation of quantum theory we have to conclude a treatment of quantum statistical mechanics. The basic concepts in the ontological approach are the particle and the wave function. The density matrix cannot play a fundamental role here. Therefore quantum statistical mechanics will require a further statistical distribution over wave functions in addition to the distribution of particles that have a specified wave function. Ultimately the wave function of the universe will he required, but we show that if the universe in not in thermodynamic equilibrium then it can he treated in terms of weakly interacting large scale constituents that are very nearly independent of each other. In this way we obtain the same results as those of the usual approach within the framework of the ontological interpretation.Professor D. Bohm died on 28 October 1992, shortly after this paper was completed.     

Hydrogen-like atom with nonnegative quantum distribution function

Among numerous approaches to probabilistic interpretation of conventional quantum mechanics (CQM), the closest to N. Bohr’s idea of the correspondence principle is the Blokhintzev-Terletsky approach of the quantum distribution function (QDF) on the coordinate-momentum (q, p) phase space. The detailed investigation of this approach has led to the correspondence rule of V.V. Kuryshkin parametrically dependent on a set of auxiliary functions. According to investigations of numerous authors, the existence and the explicit form of QDF depends on the correspondence rule between classical functions A(q, p) and quantum operator A. At the same time, the QDF corresponding to all known quantization rules turns out to be alternating in sign or overly complex valued. Finally nonexistence of nonnegative QDF in CQM was proved. On the other hand, from this follows the possibility to construct quantum mechanics where a nonnegative QDF exists. We consider a certain set of auxiliary functions to construct explicit expressions for operators O(H) for the hydrogen atom. Naturally, these operators differ from the related operator Ĥ in CQM, so that spherical coordinates are no longer separable for a hydrogen-like atom in quantum mechanics with nonnegative QDF. The text was submitted by the authors in English.     

Generalization of quantum statistics in statistical mechanics

We propose a generalization of quantum statistics in the framework of statistical mechanics. We derive a general formula which involves a wide class of equilibrium quantum statistical distributions, including the Bose and Fermi distributions. We suggest a way of evaluating the statistical distributions with the help of many-particle partition functions and apply it to studying some interesting distributions. A question on the statistical distribution for anyons is discussed, and the term following the Boltzmann one in the expansion of this distribution in powers of the Boltzmann factor, exp[(-_i)], is estimated. An ansatz is proposed for evaluating the statistical distribution forquons (particles whose creation and annihilation operators satisfy theq-commutation relations). We also treat non-equilibrium statistical mechanics, obtaining unified expressions for the entropy of a nonequilibrium quantum gas and for a collision integral which are valid for a wide class of statistics.     

Nonperturbative weak-coupling analysis of the quantum Liouville field theory

Two independent weak-coupling expansions are developed for the Liouville quantum field theory on a circle. In the first, the coupling of the nonzero modes is treated as a perturbation on the exact solution to the zero-mode problem (quantum mechanics with an exponential potential). The second approach is a weak-coupling approximation to an explicit operator solution which expresses various Liouville operators as functions of a free massless field using a Bäcklund transformation. It is shown that the free state space associated with the latter solution must be restricted to the sector which is odd with respect to a type of “parity.” Various matrix elements are computed to order g¹⁰ using both approaches, yielding identical results.     

C*-algebras and Mackey's axioms

A non-commutative version of probability theory is outlined, based on the concept of a*-algebra of operators (sequentially weakly closedC*-algebra of operators). Using the theory of*-algebras, we relate theC*-algebra approach to quantum mechanics as developed byKadison with the probabilistic approach to quantum mechanics as axiomatized byMackey. The*-algebra approach to quantum mechanics includes the case of classical statistical mechanics; this important case is excluded by theW*-algebra approach. By considering the*-algebra, rather than the von Neumann algebra, generated by the givenC*-algebraA in its reduced atomic representation, we show that a difficulty encountered byGuenin concerning the domain of a state can be resolved.     

Non-Equilibrium Steady States of Finite¶Quantum Systems Coupled to Thermal Reservoirs

We study the non-equilibrium statistical mechanics of a 2-level quantum system, ?, coupled to two independent free Fermi reservoirs ?₁, ?₂, which are in thermal equilibrium at inverse temperatures β₁≠β₂. We prove that, at small coupling, the combined quantum system ?+?₁+?₂ has a unique non-equilibrium steady state (NESS) and that the approach to this NESS is exponentially fast. We show that the entropy production of the coupled system is strictly positive and relate this entropy production to the heat fluxes through the system. A part of our argument is general and deals with spectral theory of NESS. In the abstract setting of algebraic quantum statistical mechanics we introduce the new concept of the C-Liouvillean, L, and relate the NESS to zero resonance eigenfunctions of L ^*. In the specific model ?+?₁+?₂ we study the resonances of L ^* using the complex deformation technique developed previously by the authors in [JP1]. Dedicated to Jean Michel Combes on the occasion of his sixtieth birthday Received: 12 July 2001 / Accepted: 11 October 2001     

Entanglement and relaxation in exactly solvable models

A system put in contact with a large heat bath normally thermalizes. This means that the state of the system ρ^ℐ(t) approaches an equilibrium state ρ_eq^ℐ, the latter depending only on macroscopic characteristics of the bath (e.g. temperature), but not on the initial state of the system. The above statement is the cornerstone of the equilibrium statistical mechanics; its validity and its domain of applicability are central questions in the studies of the foundations of statistical mechanics. In the present contribution we discuss the recently proven general theorems about thermalization and demonstrate how they work in exactly solvable models. In particular, we review a necessary condition for the system initial state independence (ISI) of ρ_eq^ℐ, which was proven in our previous work, and apply it for two exactly solvable models, the XX spin chain and a central spin model with a special interaction with the environment. In the latter case we are able to prove the absence of the system ISI. Also the Eigenstate Thermalization Hypothesis is discussed. It is pointed out that although it is supposed to be generically true in essentially not-integrable (chaotic) quantum systems, it is how-ever also valid in the integrable XX model.     

Nonrelativistic scale anomaly, and composite operators with complex scaling dimensions

It is demonstrated that a nonrelativistic quantum scale anomaly manifests itself in the appearance of composite operators with complex scaling dimensions. In particular, we study nonrelativistic quantum mechanics with an inverse square potential and consider a composite s-wave operator O=ψψ. We analytically compute the scaling dimension of this operator and determine the propagator 〈0|TOO^†|0〉. The operator O represents an infinite tower of bound states with a geometric energy spectrum. Operators with higher angular momenta are briefly discussed.     

Star-polarization: A natural link between phase space representation and operator representation of quantum mechanics

The method of *-polarization connects phase space mechanics to the usual operator formulation of quantum theory. A *-polarization is a linear submanifold of the space of C functions on phase space. Elements of a *-polarization are in direct correspondence with the Schroedinger wave functions and this correspondence induces the Weyl correspondence between classical observables and operators. All generalized Moyal algebras admit *-polarizations and a general method is thus available for translating *-quantization into operator language.     

Ursell operators in statistical physics of dense systems: the role of high order operators and of exchange cycles

The purpose of this article is to discuss cluster expansions in dense quantum systems, as well as their interconnection with exchange cycles. We show in general how the Ursell operators of order l≥ 3 contribute to an exponential which corresponds to a mean-field energy involving the second operator U₂, instead of the potential itself as usual - in other words, the mean-field correction is expressed in terms of a modification of a local Boltzmann equilibrium. In a first part, we consider classical statistical mechanics and recall the relation between the reducible part of the classical cluster integrals and the mean-field; we introduce an alternative method to obtain the linear density contribution to the mean-field, which is based on the notion of tree-diagrams and provides a preview of the subsequent quantum calculations. We then proceed to study quantum particles with Boltzmann statistics (distinguishable particles) and show that each Ursell operator U_n with n≥ 3 contains a “tree-reducible part”, which groups naturally with U₂ through a linear chain of binary interactions; this part contributes to the associated mean-field experienced by particles in the fluid. The irreducible part, on the other hand, corresponds to the effects associated with three (or more) particles interacting all together at the same time. We then show that the same algebra holds in the case of Fermi or Bose particles, and discuss physically the role of the exchange cycles, combined with interactions. Bose condensed systems are not considered at this stage. The similarities and differences between Boltzmann and quantum statistics are illustrated by this approach, in contrast with field theoretical or Green's functions methods, which do not allow a separate study of the role of quantum statistics and dynamics. Received 18 October 2001     

Stochastic phase spaces and master Liouville spaces in statistical mechanics

The concept of probability space is generalized to that of stochastic probability space. This enables the introduction of representations of quantum mechanics on stochastic phase spaces. The resulting formulation of quantum statistical mechanics in terms of -distribution functions bears a remarkable resemblance to its classical counterpart. Furthermore, both classical and quantum statistical mechanics can be formulated in one and the same master Liouville space overL ²(). A joint derivation of a classical and quantum Boltzman equation provides an illustration of the practical uses of these formalisms.Supported in part by an NRC grant.     

Thermostatistics of the g-on gas

In this study, the particles of the quantum gases, namely bosons and fermions are called g-ons by using the parameter of the fractional exclusion statistics g, where . With this point of departure, the distribution function of the g-on gas is obtained by the variational, steepest descent and statistical methods. The distribution functions which are found by means of these three methods are compared. It is shown that the thermostatistical formulations of quantum gases can be unified. By suitable choices of g, standard relations of statistical mechanics of the Bose and Fermi systems are recovered.Received: 26 March 2003, Published online: 22 September 2003PACS: 05.20.-y Classical statistical mechanics - 03.65.-w Quantum mechanics     

Diagram Expansions in Nonequilibrium Statistical Mechanics Classical Limit Considerations

A diagram expansion is proposed for calculating traces of the kind Tr{Ae^?itLB} which are of interest for calculating time correlation functions and expectation values in nonequilibrium statistical mechanics. Arbitrary initial conditions are considered. In the classical limit the diagram expansion of FUJITA is obtained. A systematic method for obtaining quantum corrections including exchange and symmetry effects is proposed.     

Quantum Logic and Non-Commutative Geometry

We propose a general scheme for the “logic” of elementary propositions of physical systems, encompassing both classical and quantum cases, in the framework given by Non-Commutative Geometry. It involves Baire^*-algebras, the non-commutative version of measurable functions, arising as envelope of the C ^*-algebras identifying the topology of the (non-commutative) phase space. We outline some consequences of this proposal in different physical systems. This approach in particular avoids some problematic features appearing in the definition of physical states in the standard (W ^*-)algebraic approach to classical mechanics.     

Foundations of the quantum statistics of systems in equilibrium: A measuretheoretic approach

The fundamental equations of equilibrium quantum statistical mechanics are derived in the context of a measure-theoretic approach to the quantum mechanical ergodic problem. The method employed is an extension, to quantum mechanical systems, of the techniques developed by R. M. Lewis for establishing the foundations of classical statistical mechanics. The existence of a complete set of commuting observables is assumed, but no reference is made a priori to probability or statistical ensembles. Expressions for infinite-time averages in the microcanonical, canonical, and grand canonical ensembles are developed which reduce to conventional quantum statistical mechanics for systems in equilibrium when the total energy is the only conserved quantity. No attempt is made to extend the formalism at this time to deal with the difficult problem of the approach to equilibrium.     

Positive Commutators in Non-Equilibrium Quantum Statistical Mechanics

The method of positive commutators, developed for zero temperature problems over the last twenty years, has been an essential tool in the spectral analysis of Hamiltonians in quantum mechanics. We extend this method to positive temperatures, i.e. to non-equilibrium quantum statistical mechanics. We use the positive commutator technique to give an alternative proof of a fundamental property of a certain class of large quantum systems, called Return to Equilibrium. This property says that equilibrium states are (asymptotically) stable: if a system is slightly perturbed from its equilibrium state, then it converges back to that equilibrium state as time goes to infinity. Received: 27 December 2000 / Accepted: 21 June 2001