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Journal of Mathematical Physics

Relations among elements of the density matrix. II. Exotic conservation laws

James L Park, William Band

Published Jun 1, 1973• 2 citations
Publisher

Abstract

Ordinary quantal conservation laws are associated with null operators for time rate of change and are valid for causal evolution of the system through either pure or mixed states. There is, however, a larger class of quantal constants of the motion, some of which are conserved only for pure state evolution. After an analysis of the theoretical origins of these exotic conserved quantities, several illustrations are presented with empirical interpretations based on the quorum theory methods used in Paper I.

DOI
10.1063/1.1666378

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Ordinary quantal conservation laws are associated with null operators for time rate of change and are valid for causal evolution of the system through either pure or mixed states. There is, however, a larger class of quantal constants of the motion, some of which are conserved only for pure state evolution. After an analysis of the theoretical origins of these exotic conserved quantities, several illustrations are presented with empirical interpretations based on the quorum theory methods used in Paper I.
We propose an extended quantum mechanical formalism that is based on a wave operator $\stackrel{^}{\ensuremath{\varrho}}$, which is related to the ordinary density matrix via $\ensuremath{\rho}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\stackrel{^}{\ensuremath{\varrho}}{\stackrel{^}{\ensuremath{\varrho}}}^{\ifmmode\dagger\else\textdagger\fi{}}$. This formalism allows a (generalized) unitary evolution between pure and mixed states. It also preserves much of the connection between symmetries and conservation laws. The new formalism is illustrated for the case of a two-level system.
In the formulation of Banks, Peskin and Susskind, we show that one can construct evolution equations for the quantum mechanical density matrix ρρρ with operators which do not commute with hamiltonian which evolve pure states into mixed states, preserve the normalization and positivity of ρρρ and conserve energy. Furthermore, it seems to be different from a quantum mechanical system with random sources.
Abstract For present purposes, the first thing to recall about quantum mechanics is that all dynamical quantities, instead of being functions of the location of a system in phase space as in classical mechanics, are now represented by operators in a Hilbert space of quantum states. In particular, the quantum analog of the classical phase space distribution function is the quantum mechanical density matrix. Since this is seldom discussed in introductory texts on either quantum mechanics or statistical mechanics, it may be helpful to summarize some of its basic properties.
It has been argued that any evolution law taking pure states to mixed states in quantum field theory necessarily gives rise to violations of either causality or energy-momentum conservation in such a way as to have unacceptable consequences for ordinary laboratory physics. We show here that this is not the case by giving a simple class of examples of Markovian evolution laws where rapid evolution from pure states to mixed states occurs for a wide class of states with appropriate properties at the ``Planck scale,'' suitable locality and causality properties hold for all states, and the deviations from ordinary dynamics (and, in particular, violations of energy-momentum conservation) are unobservably small for all states which one could expect to produce in a laboratory. In addition, we argue (via consideration of other, non-Markovian models) that conservation of energy and momentum for all states is not fundamentally incompatible with causality in dynamical models in which pure states evolve to mixed states.