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Physica

Complete sets of diagonal operators in quantum-statistical mechanics

W.J. Caspers, H. P. van de Braak, P.W. Verbeek, J.C. Verstelle

Published May 1, 1971• 6 citations
Publisher
DOI
10.1016/0031-8914(71)90071-1

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The statistical operator is a weighted sum, over eigenvalues of a complete commuting set of observables, of pure states each evolving unitarily from a microstate of the set. For every complete commuting set q̂ there is a latent ensemble, the diagonal projection of the statistical operator in q̂ representation, which determines the probabilities of observables in set q̂. When constraints imposed by a measurement situation makes these observables measurable, the latent q̂ ensemble is realized as the initial state of a new ensemble, which then evolves unitarily. Realization of a latent ensemble is an irreversible process with non-negative entropy increase. Reduced ensembles are realized when the constraints of the measurement situation permit only a subset of q̂ observables to be measurable. Successive realizations in response to changing measurement situations drive the system toward a state of maximum entropy. This mechanism explains the approach to equilibrium, and the change of an equilibrium system to an equilibrium state of increased entropy as constraints are relaxed.
Quantum physics consists of a formal core that is universally agreed upon (basically being a piece of mathematics with suggestive names for concepts matching related concepts of experimental culture) and an interpretational halo (trying to make this relation more precise) that remains highly disputed even after more than 90 years of modern quantum physics. The latter is the subject of the interpretation of quantum mechanics (see Part II), where many interpretations coexist and compete for the attention of those interested in what quantum physics really means. In this chapter, an axiomatic introduction to the undisputed formal core of quantum physics is given. The axioms described present nonrelativistic quantum statistical mechanics in the Schrödinger picture. The relativistic case is outside the scope of these axioms, as it must be treated by quantum field theory; see Chapter 7. As in any axiomatic setting (necessary for a formal discipline), there are a number of different but equivalent sets of axioms or postulates that can be used to define formal quantum physics. Since they are equivalent, their choice is a matter of convenience. The choice presented here is a formulation featuring three aspects: It emphasises the similarity of quantum mechanics and classical mechanics, gives most direct access to statistical mechanics, and is free from allusions to measurement. The reason for the first is that this similarity serves as a guide to meaning. From the axiomatic setting for quantum mechanics presented below, classical mechanics (in the Koopman setting discussed in Section 7.10) is obtained by simply restricting the operators to be diagonal, so that all operations happen pointwise on the diagonal elements. Thus, multiplication is commutative, and one can identify operators and functions. In particular, the density operator degenerates into a probability density. Therefore, the quantum case appears as a generalization of the classical case allowing for noncommutativity, so that both q-observables and the density operator are (usually infinite-dimensional) operators. The reason for the second is that statistical mechanics is the main tool for applications of quantum physics to the macroscopic systems we are familiar with. The reason for the third is that realistic measurements constitute a complex process involving macroscopic detectors, hence should be explained by quantum statistical mechanics rather than be part of the axiomatic foundations themselves. (This is in marked contrast to other foundations, and distinguishes the present system of axioms.)
One of us (C.G) has developed a transformation theory in nonequilibrium statistical mechanics that generalizes the usual transformation theory of quantum mechanics in the framework of the resolvent formalism of the Liouville-von Neumann operator extensively used by the Brussels school. Prigogine, George, and Henin have shown that the evolution of a dissipative system can be decomposed into separate subdynamics in orthogonal subspaces. The π subspace, containing all thermodynamics, can be seen as an extension of the subspace of the invariants in the absence of dissipation. In this subspace a suitable contracted description of the system can be given through a dressing operator χ, which has the property that HR = χ−1H0 can be considered as the renormalized energy. A differential equation that determines χ has been proposed by Mandel. It is useful to see, even for a nondissipative system, what relation the transformation theory in the Liouville-von Neumann formalism has to the standard transformations used in quantum mechanics. In this paper we consider a system that can be completely diagonalized by the Bogoliubov transformation. It is shown that in this case the Mandel equation can be solved and that χ−1H0 = HD, where HD is the diagonalized Hamiltonian, obtained by the Bogoliubov transformation.