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Progress of Theoretical Physics

A Phase Space Approach to the Projective Representation of the Canonical Commutation Relations

Loucas C. Papaloucas

Published Jan 1, 1996• 0 citations
Publisher

Abstract

The projective nature of the representations of canonical commutation relations on the space of states is probed via cohomological considerations. Some interesting geometrical pictures arise, which associate themselves with the phase descriptions of quantum systems.
DOI
10.1143/ptp.95.37

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The basis-independent approach to canonical commutation relations (CCR's), which allows arbitrary test-function spaces for smearing the field operators, is a generalization of the basis-dependent approach, in which the fields are smeared with an orthonormal system (or finite linear combinations thereof) to obtain an infinite set of qk and pk. Using recent results on continuity properties of representations of the CCR's, we show that every representation of the basis-independent type in a separable Hilbert space can be obtained by continuous extension of a suitable representation of the basis-dependent type where properties like irreducibility, cyclicity, etc., remain unaffected. In this sense, both approaches are equivalent, and the classification problem for CCR's is reduced from the simultaneous consideration of all representations for all possible test-function spaces to those for a single one (up to isomorphism).
The dissipative models in string theory can have more broad range of application: 1) Noncritical strings are dissipative systems in the "coupling constant" phase space. 2) Bosonic string in the affine-metric curved space is dissipative system. But the quantum descriptions of the dissipative systems have well known ambiguities. In order to solve the problems of the quantum description of dissipative systems we suggest to introduce an operator W in addition to usual (associative) operators. The suggested operator algebra does not violate Heisenberg algebra because we extend the canonical commutation relations by introducing an operator W of the nonholonomic quantities in addition to the usual (associative) operators of usual (holonomic) coordinate -momentum functions. To satisfy the generalized commutation relations the operator W must be nonassociative nonLieble (does not satisfied the Jacobi identity) operator. As the result of these properties the total time derivative of the multiplication and commutator of the operators does not satisfies the Leibnitz rule. This lead to compatibility of quantum equations of motion for dissipative systems and canonical commutation relations. The suggested generalization of the von Neumann equation is connected with classical Liouville equation for dissipative systems.