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Compatibility of representations of quantum systems
Robert A. Van Wesep
abstract: There is a natural equivalence relation on representations of the states of a
given quantum system in a Hilbert space, two representations being equivalent
iff they are related by a unitary transformation. There are two equivalence
classes, with members of opposite classes being related by a conjugate-unitary
(anti-unitary) transformation. These two conjugacy classes are related in much
the same way as are the two imaginary units of a complex field, and there is a
priori no basis on which to prefer one over the other in any individual case.
This is potentially problematic in that the choice of conjugacy class of a
representation determines the sign of energy and other quantities defined as
generators of continuous symmetries of the system in question, so that it would
appear that principles like conservation of energy for a compound system may
hold or fail depending on relative choices of conjugacy class of
representations of its subsystems. We show that for any finite set of quantum
systems there are exactly two ways of choosing conjugacy classes of
representations consistent with the usual tensor-product construction for
representing the compound system composed of these. Each is obtained from the
other by reversing the conjugacy of all the representations at once. The
relation of unitary equivalence for representations of a single system is
therefore uniquely extendible to representations of all systems that can
interact with it.
- oai_identifier:
- oai:arXiv.org:quant-ph/0612096
- categories:
- quant-ph
- comments:
- The main theorem of this article is very basic, but I have been
unable to find any discussion of it or the issue it addresses in the existing
literature. As I do not wish to present this as new if it is not, I would
greatly appreciate any pertinent reference--even if only a published
statement that it is "well known" or "folklore"
- arxiv_id:
- quant-ph/0612096
- created:
- 2006-12-12
Full article ▸
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