States and Dynamical Variables

Pedro Pereyra Padilla

Peter Leifer

State-dependent local dynamical variables (LDVs) sharply differ from the ordinary operators of quantum mechanics. The N -level model system shows the physical importance of such operators in the complex projective Hilbert state space CP(N −1). The process of quantum measurement in terms of LDVs is described.

P. A. M. Dirac

Abstract The general representation theory of quantum mechanics requires one to represent the states of a dynamical system by functions of a set of real variables q'r, each of which has a domain consisting of either discrete points or a continuous range of points, or possibly both together. A dynamical variable is represented by a function of two such sets of real variables q'r and q"r forming a generalized “matrix". In this paper we shall show that in certain cases it is advantageous to consider some of our variables q'r as complex variables and to suppose the representatives of states and dynamical variables to depend on them in accordance with the theory of functions of a complex variable. In the usual theory the domains of the s are the eigenvalues of certain observables q"r. This significance of the q"s of course gets lost when we consider them as complex variables, but we have, however, some beautiful mathematical features appearing instead, and we gain a considerable amount of mathematical power for the working out of particular examples.

Baidyanath Misra, Ilya Prigogine, M. Courbage

We discuss the question of the dynamical meaning of the second law of thermodynamics in the framework of quantum mechanics. Previous discussion of the problem in the framework of classical dynamics has shown that the second law can be given a dynamical meaning in terms of the existence of so-called Lyapounov variables-i.e., dynamical variables varying monotonically in time without becoming contradictory. It has been found that such variables can exist in an extended framework of classical dynamics, provided that the dynamical motion is suitably unstable. In this paper we begin to extend these results to quantum mechanics. It is found that no dynamical variable with the characteristic properties of nonequilibrium entropy can be defined in the standard formulation of quantum mechanics. However, if the Hamiltonian has certain well-defined spectral properties, such variables can be defined but only as a nonfactorizable superoperator. Necessary nonfactorizability of such entropy operators M has the consequence that they cannot preserve the class of pure states. Physically, this means that the distinguishability between pure states and corresponding mixtures must be lost in the case of a quantal system for which the algebra of observables can be extended to include a new dynamical variable representing nonequilibrium entropy. We discuss how this result leads to a solution of the quantum measurement problem. It is also found that the question of existence of entropy of superoperators M is closely linked to the problem of defining an operator of time in quantum mechanics.

Ray LaPierre