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Journal of Mathematical Physics

Background Dependence of Local States

A. L. Licht

Published Sep 1, 1968• 3 citations
Publisher

Abstract

We consider states prepared by a series of actions performed in a finite space-time region in the presence of a background state described by a density matrix. It is shown that there will always be some observable whose expectation value in such a state will depend to some extent on the background. There will, however, be a certain set of observables whose expectation values will be independent of the background. We obtain a characterization of such sets of observables.

DOI
10.1063/1.1664740

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We consider states prepared by a series of actions performed in a finite space-time region in the presence of a background state described by a density matrix. It is shown that there will always be some observable whose expectation value in such a state will depend to some extent on the background. There will, however, be a certain set of observables whose expectation values will be independent of the background. We obtain a characterization of such sets of observables.
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We propose a method to simulate the real time evolution of one-dimensional quantum many-body systems at finite temperature by expressing both the density matrices and the observables as matrix product states. This allows the calculation of expectation values and correlation functions as scalar products in operator space. The simulations of density matrices in inverse temperature and the local operators in the Heisenberg picture are independent and result in a grid of expectation values for all intermediate temperatures and times. Simulations can be performed using real arithmetics with only polynomial growth of computational resources in inverse temperature and time for integrable systems. The method is illustrated for the XXZ model and the single impurity Anderson model.
The conventional statement of statistical determinism is that “the expectation values of all (Heisenberg)observables are determined by the expectation values of the observables at one time.” This requires that a full algebra of self-adjoint operators be in one-to-one correspondence with measurement procedures performed at one time. For instance, it requires that if two noncommuting observables pand q are defined at t = 0, there should exist a measurement procedure at t = 0 corresponding to p+q. No such procedure is known. The contrast between the positive assertion of the existence of certain laboratory procedures and the inability to describe them constitutes perhaps the weakest point of quantum mechanics. However, the conventional statement of statistical causality is shown to be untenable in a relativistic theory. This paper proposes a weaker form of causality which (1) uses measurements made within a truncated light cone rather than at one time for predictive purposes, and (2)which involves only strictly localized states, i.e., states which are vacuumlike outside a finite volume. Failure of the conventional causality statement implies that the set of quasilocal observables is not necessarily linear, i.e., if A and B are in a set, A+B is not necessarily in it. This remark may open the way to a systematic inquiry into the problems of associating laboratory procedures to self-adjoint operators.