A remark on the definition of superselection rules in terms of unbounded operators

Abstract

The mathematical definition of superselection rules in the case when observables are described by unbounded operators in a fixed Hilbert space (for instance, in the frame of Wightman's axioms) is examined. The additional condition PHqD c D (where D is the common domain of definition of the operators, Hq is the qth sector, and PHq is the projection on Hq) is found to be sufficient in order to preserve-as in the case of bounded observables-the one-to-one correspondence between reducing subspaces Hq and projections PHq from the commutant ~' of the algebra ~ of observables. This additional condition is equivalent to the physical requirement that every physical vector state can be uniquely represented as a linear combination of physical states, each belonging to some sector. In the following Note, observations are made on some elementary facts from reduction theory of sets of unbounded operators in Hilbert space H. These facts are used to define mathematically superselection rules for the case when the observables of the quantum system are described directly in terms of unbounded operators but are not substituted by bounded operators. The description is similar to the one given by Streater & Wightman (1964). To achieve a complete analogy with the case of bounded operators, the introduction of an additional condition in the definition of superselection sectors is found to be necessary. (It is automatically fulfilled in the case of bounded observables.) If D(D 4= H) is the common domain of definition of the unbounded operators used as observables and Hq is the subspace of H representing the qth sector, the condition PHqD C D should be fulfilled with PHq the projection on Hq. This turns out to be equivalent to the physical requirement that every physical vector state from D can be uniquely represented as a linear combination of physical states from D, each belonging to some sector Hq. In the course of the discussion an interesting result of © 1976 Plenum Publishing Corporation. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission of the publisher.