Locality in GNS Representations of Deformation Quantization

Details Locality in GNS Representations of Deformation Quantization Locality in GNS Representations of Deformation Quantization

1999-03-09

arxiv.org/abs/math/9903053v1

Mathematics

Quantum Algebra

LaTeX2e, 29 pages

Scientific paper

10.1007/s002200050788

In the framework of deformation quantization we apply the formal GNS construction to find representations of the deformed algebras in pre-Hilbert spaces over $\mathbb C[[\lambda]]$ and establish the notion of local operators in these pre-Hilbert spaces. The commutant within the local operators is used to distinguish `thermal' from `pure' representations. The computation of the local commutant is exemplified in various situations leading to the physically reasonable distinction between thermal representations and pure ones. Moreover, an analogue of von Neumann's double commutant theorem is proved in the particular situation of a GNS representation with respect to a KMS functional and for the Schr\"odinger representation on cotangent bundles. Finally we prove a formal version of the Tomita-Takesaki theorem.

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