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Physical Review A

Ehrenfest theorem and information theory

D. Otero, Angel Ricardo Plastino, Araceli N. Proto, G. Zannoli

Published Sep 1, 1982• 55 citations
Publisher

Abstract

An explicit connection via the information-theory formalism is made between Ehrenfest's theorem and entropy conservation that yields a useful prescription for describing the time evolution of observables.
DOI
10.1103/physreva.26.1209

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The Ehrenfest theorem is a well-known theoretical result of quantum mechanics. It relates the dynamical evolution of the expectation value of a quantum operator to the expectation value of its corresponding commutator with the Hermitian Hamiltonian operator. However, the proof of validity of the Ehrenfest theorem for quantum gravity field theory has remained elusive, while its validation poses challenging conceptual questions. In fact, this presupposes a number of minimum requirements, which include the prescription of quantum Hamiltonian operator, the definition of scalar product, and the identification of dynamical evolution parameter. In this paper, it is proven that the target can be established in the framework of the manifestly covariant quantum gravity theory (CQG theory). This follows as a consequence of its peculiar canonical Hamiltonian structure and the commutator-bracket algebra that characterizes its representation and probabilistic interpretation. The theoretical proof of the theorem for CQG theory permits to elucidate the connection existing between quantum operator variables of gravitational field and the corresponding expectation values to be interpreted as dynamical physical observables set in the background metric space-time.
We revisit the concept of phase time, which has been previously proposed as a solution to the problem of time in quantum gravity. Concretely, we show how the geometry of configuration space together with the phase of the wave function of the Universe can, under certain conditions, lead to the definition of a positive-definite squared norm from which probabilities can be defined. The norm is conserved under time evolution, and we obtain a version of Ehrenfest's theorem that is analogous to the one in ordinary quantum mechanics. We address how the present approach avoids some difficulties that were encountered in previous attempts at defining an Ehrenfest theorem relative to phase time in the context of relativistic quantum mechanics and canonical quantum gravity, and how it is connected to a notion of gauge fixing the inner product. We conclude with a brief outlook.