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arXiv:quant-ph/0406220

On superselection rules for macroscopic objects

Прохоров Лев Васильевич

Published Jun 29, 2004• 0 citations
arXivHTMLPDF

Abstract

It is shown that for "ideal" macroscopic objects there are superselection rules forbidding superpositions of macroscopically distinguishable states of the objects. For real macroscopic bodies the notion of "weak" superselection rules is introduced. Some other aspects of the measurement problem are discussed.
DOI
10.48550/arxiv.quant-ph/0406220

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In quantum theory, physically measurable quantities of a microscopic system are represented by self-adjoint operators. However, not all of the self-adjoint operators correspond to measurable quantities. The superselection rule is a criterion to distinguish measurable quantities. Any measurable quantity must obey the superselection rules. By contraposition, any quantity which does not obey the superselection rules cannot be measured. Although some of superselection rules were proved, the raizon d'être of the superselection rules has been still obscure. In this paper we deduce the superselection rules from an assumption on symmetry property of measurement process. We introduce the notion of covariant indicator, which is a macroscopic observable whose value indicates the value of a microscopic object observable. We prove that if an object system has a quantity that is conserved during the measurement process, other quantities that do not commute with the conserved quantity are non-measurable by the covariant indicator. Our derivation of superselection rules is compared with the uncertainty relation under the restriction by a conservation law. An implication of the color superselection rule for the color confinement is discussed. It is also argued that spontaneous symmetry breaking enables a measurement that the superselection rule prohibits.
A superselection rule advanced in the course of a quantum-mechanical treatment of some phenomenon is an assertion to the effect that the superposition principle of quantum mechanics is to be restricted in the application at hand. Superselection accounts of measurement all have in common a decision to represent the indicator states of detectors by eigenspaces of superselection operators named in a superselection rule, on the grounds that the states in question are states of a so-called classical quantity and therefore not subject to quantum interference effects. By this strategy superselectionists of measurement expect to dispense with use of projection postulates in treatments of measurement. I shall argue that superselection accounts of measurement are self-contradictory, and that treatments of infinite systems, if they can avoid the contradiction, are not true superselection accounts.
The superposition principle is a very basic ingredient of quantum theory. What may come as a surprise to many students, and even to many practitioners of the quantum craft, is tha superposition has limitations imposed by certain requirements of the theory. The discussion of such limitations arising from the so-called superselection rules is the main purpose of this paper. Some of their principal consequences are also discussed. The univalence, mass and particle number superselection rules of non-relativistic quantum mechanics are also derived using rather simple methods.
For orthodox quantum systems, pure states correspond one-to-one to all one-dimensional subspaces and observables correspond one-to-one to all selfadjoint operators in the state space. A relaxation of this one-to-one correspondence will bring about superselection rules. A general formulation of superselection rules is technically complicated. 1 We shall adopt an intuitive approach here and limit our discussion to some special cases to see how superselection rules can be incorporated in a Hilbert space structure of quantum theory.