dorsal/arxiv
View SchemaCompatibility of representations of quantum systems
| Authors | Robert A. Van Wesep |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0612096 |
| URL | https://arxiv.org/abs/quant-ph/0612096 |
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.
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"abstract": "There is a natural equivalence relation on representations of the states of a\ngiven quantum system in a Hilbert space, two representations being equivalent\niff they are related by a unitary transformation. There are two equivalence\nclasses, with members of opposite classes being related by a conjugate-unitary\n(anti-unitary) transformation. These two conjugacy classes are related in much\nthe same way as are the two imaginary units of a complex field, and there is a\npriori no basis on which to prefer one over the other in any individual case.\nThis is potentially problematic in that the choice of conjugacy class of a\nrepresentation determines the sign of energy and other quantities defined as\ngenerators of continuous symmetries of the system in question, so that it would\nappear that principles like conservation of energy for a compound system may\nhold or fail depending on relative choices of conjugacy class of\nrepresentations of its subsystems. We show that for any finite set of quantum\nsystems there are exactly two ways of choosing conjugacy classes of\nrepresentations consistent with the usual tensor-product construction for\nrepresenting the compound system composed of these. Each is obtained from the\nother by reversing the conjugacy of all the representations at once. The\nrelation of unitary equivalence for representations of a single system is\ntherefore uniquely extendible to representations of all systems that can\ninteract with it.",
"arxiv_id": "quant-ph/0612096",
"authors": [
"Robert A. Van Wesep"
],
"categories": [
"quant-ph"
],
"title": "Compatibility of representations of quantum systems",
"url": "https://arxiv.org/abs/quant-ph/0612096"
},
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