Space-Time Translation, Quantum Dynamics and Various Representations in Quantum Mechanics

Mustafa Amin

Classical mechanics, in the Koopman-von Neumann formulation, is described in Hilbert space. It is shown here that classical canonical transformations are generated by Hermitian operators that are in general noncommutative. This naturally brings about uncertainty relations inherent in classical mechanics, for example between position and the generator of space translations, between momentum and the generator of momentum translations, and between dynamical time and the Liouvillian, to name a few. Further, it is shown that the Wigner representation produces a quasi-probability distribution that can take on negative values. Thus, two of the hallmark features of quantum mechanics are reproduced, and become apparent, in a Hilbert-space formulation of classical mechanics.

Axel Pelster, A. Wunderlin, Karl Zeile

Hsiang Shun Chou

Abstract We investigate the time evolution for quantum systems with a dynamical Hilbert space within the framework of the pseudo-Hermitian representation of quantum mechanics. Each representation of quantum mechanics is characterized by a Hilbert space, a Hamiltonian, and a set of observables. The representation-transformation law of the time-evolution operator is derived from its formal solution in the dynamical Hilbert space. It ensures the unitarity of the dynamics and the representation independence of the transition amplitudes. In addition, we show that the equation of time evolution in the position space is independent of the representation, regardless of whether the Hilbert space is stationary or dynamical. Furthermore, we demonstrate the representation independence of the position wave function itself. As a concrete example of the representation independence of the quantum mechanics, we derive the reciprocity theorem in the dynamical Hilbert space. The material in the present paper makes a topic which can be covered in a graduate course on quantum mechanics.

James B. Hartle

Y. Strauss

In nonrelativistic quantum mechanics time enters as a parameter in the Schrödinger equation. However, there are various situations where the need arises to view time as a dynamical variable. In this paper we consider the dynamical role of time through the construction of a Lyapunov variable—i.e., a self-adjoint quantum observable whose expectation value varies monotonically as time increases. It is shown, in a constructive way, that a certain class of models admits a Lyapunov variable and that the existence of a Lyapunov variable implies the existence of a transformation mapping the original quantum mechanical problem to an equivalent irreversible representation. In addition, it is proven that in the irreversible representation there exists a natural time ordering observable splitting the Hilbert space at each t>0 into past and future subspaces.

Xinchuan Zeng