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Equilibrium states in statistical mechanics C*-algebraic approach

Nicolaas Marinus Hugenholtz

Published Jan 1, 1973• 0 citations
OpenAlex

Abstract

Infinite systems, particularly infinite quantum systems are considered. It is pointed out that algebraic methods can be applied not only to statistical mechanics but also to quantum field theory. Inequivalent representations of canonical anticommutation relations, states and observables, observable algebras for certain simple cases, the ground state, the equilibrium state at T and O, the Heisenberg model with long-range interaction are covered. (JFP)

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Infinite systems, particularly infinite quantum systems are considered. It is pointed out that algebraic methods can be applied not only to statistical mechanics but also to quantum field theory. Inequivalent representations of canonical anticommutation relations, states and observables, observable algebras for certain simple cases, the ground state, the equilibrium state at T and O, the Heisenberg model with long-range interaction are covered. (JFP)
Quantum systems with infinite degrees of freedom exhibit properties, like relaxation to equilibrium, phase-transitions and the existence of inequivalent representations of the CAR and CCR , that can satisfactorily be dealt with by means of the methods and techniques of algebraic quantum statistical mechanics [80, 81, 130]. The point of departure from standard quantum mechanics, is that in an infinite dimensional context one is usually provided with the algebraic properties of the relevant observables, but, in general, not with an a priori given representation on a Hilbert space; the latter rather depends on to the physical properties of the systems under consideration [324, 342, 343].
Quantum field theory is the application of quantum mechanics to systems with infinitely many degrees of freedom. This 2007 textbook presents quantum field theoretical applications to systems out of equilibrium. It introduces the real-time approach to non-equilibrium statistical mechanics and the quantum field theory of non-equilibrium states in general. It offers two ways of learning how to study non-equilibrium states of many-body systems: the mathematical canonical way and an easy intuitive way using Feynman diagrams. The latter provides an easy introduction to the powerful functional methods of field theory, and the use of Feynman diagrams to study classical stochastic dynamics is considered in detail. The developed real-time technique is applied to study numerous phenomena in many-body systems. Complete with numerous exercises to aid self-study, this textbook is suitable for graduate students in statistical mechanics and condensed matter physics.
We construct a C*-algebraic formulation of the dynamics of a pair of mutually interacting quantum-mechanical systems S̃ and Ŝ, the former being finite and the latter infinite. Our basic assumptions are that: (i) Ŝ, when isolated, satisfies the Dubin-Sewell dynamical axioms; (ii) the coupling between S̃ and Ŝ is energetically bounded and spatially localized; and (iii) the initial states of S̃ and Ŝ are mutually uncorrelated, with S̃ in an arbitrary normal state and Ŝ in a Gibbs state. Our formulation leads to a rigorous theory of (a) the dynamics of a finite open system, i.e., of a finite system (S̃) coupled to an ``infinite reservoir'' (Ŝ), and of (b) the dynamics of an infinite system (Ŝ), driven from equilibrium by a ``signal generator'' (S̃). As regards (a), we show that the state of S̃ always remains normal, and we derive a generalized master equation (in an appropriate Banach space) governing its temporal evolution. As regards (b), we show that the state of Ŝ always corresponds to a (time-dependent) density matrix in the representation space of the algebra of observables for Ŝ, induced by the initial Gibbs state. By formulating the linear part (appropriately defined) of the response of Ŝ to S̃, we generalize the fluctuation-dissipation theorem to infinite systems. Further, we show that the total effect of S̃ on Ŝ reduces to that of a ``classical'' time-dependent external force in cases where the initial state of S̃ possesses certain coherence properties similar to those of the Glauber type.
Classical statistical mechanics tries to derive the macroscopic properties of matter starting from the mechanical laws that rule the behaviour of single particles. To describe equilibrium states, only observables accounting for correlations among states are considered, and only systems consisting of a large number of particles are taken into account. The observables are described by continuous functions on phase space, and the states are represented by linear assignments of a number to each observable. Within the framework of the canonical ensemble, one deals with mechanical systems in thermal equilibrium with a thermal reservoir, and the equilibrium state is reached as a result of the interaction with the external world. The external world may be really external, or equally well, the unrecognized internal degrees of freedom, like in the calculation of viscosity or thermal conductivity. If really only the external world is the cause we may expect some surface dependence while an internal unrecognized degree of freedom would have a volume effect, unless the interactions are long range. The microcanonical ensemble is instead introduced to study isolated mechanical systems, and the equilibrium is viewed as a temporal average, rather than as a limit. Attention is then focused on partition functions, the theorem of equipartition of energy and an elementary theory of specific heats. In the second part, the Planck derivation of the law of black-body radiation is analysed, presenting in chronological order the Kirchhoff theorem, the Stefan law, the Wien displacement law, the Rayleigh–Jeans formula and the Planck hypothesis. Further topics discussed are the Einstein and Debye quantum models for specific heats of solids. These topics prepare the ground for the introduction of quantum statistical mechanics. The third part is, in fact, devoted to the analysis of identical particles in quantum mechanics.
This is a non-standard exposition of the main notions of quantum mechanics and quantum field theory, including recent results. It is based on the algebraic approach in which the starting point is a star-algebra and on the geometric approach in which the starting point is a convex set of states. Standard formulas for quantum probabilities are derived from decoherence. This derivation allows us to go beyond quantum theory in the geometric approach. Particles are defined as elementary excitations of the ground state (and quasiparticles as elementary excitations of any translation invariant state). The conventional scattering matrix does not work for quasiparticles (or even for particles if the theory does not have particle interpretation). The analysis of scattering in these cases is based on the notion of an inclusive scattering matrix, which is closely related to inclusive cross-sections. It is proven that the conventional scattering matrix can be expressed in terms of Green functions (LSZ formula) and the inclusive scattering matrix can be expressed in terms of generalized Green functions that appear in the Keldysh formalism of non-equilibrium statistical physics. The derivation of the expression of the evolution operator and other physical quantities in terms of functional integrals is based on the notion of the symbol of an operator; these arguments can be applied in the geometric approach as well. In particular, this result can be used to provide a simple derivation of the diagram technique for generalized Green functions. The notion of an inclusive scattering matrix makes sense in the geometric approach, although it seems that a definition of the conventional scattering matrix cannot be provided in this situation. The geometric approach is used to show that quantum mechanics and its generalizations can be considered as classical theories where our devices are able to measure only a part of the observables.
A possible thermodynamic evolution to equilibrium of a quantum statistical system is discussed. The description is entirely algebraic and it is done by means of complex coherent states built in the Universal Enveloping Algebra of the complex algebra https://www.w3.org/1998/Math/MathML" display="inline"> A 1 . Finite dimensional representations of https://www.w3.org/1998/Math/MathML" display="inline"> A 1 give fermion distributions while infinite dimensional ones describe bosons. The scheme is thermodynamic in the sense that fluctuations are not considered and this allows us to resort to an evolution operator that realizes, for t → ∞, a projector on the state of equilibrium. The shrinking of the Hilbert space is related to non unitary evolution that change the normalization of the state vectors. The theory is invariant under time reversal: vectors at t → ±∞ differ only in phases and both correspond to the equilibrium probability distribution. Equilibrium and non-equilibrium thermodynamic quantities can be calculated at microscopic level in a framework that is in between thermodynamic and statistical mechanics. Our approach is a physically founded example of non-linear quantum theory.