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arXiv · hep-th/0306221

Quantum Deformed Canonical Transformations, W_{\infty}-algebras and Unitary Transformations

E. Gozzi·M. Reuter
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Abstract

We investigate the algebraic properties of the quantum counterpart of the classical canonical transformations using the symbol-calculus approach to quantum mechanics. In this framework we construct a set of pseudo-differential operators which act on the symbols of operators, i.e., on functions defined over phase-space. They act as operatorial left- and right- multiplication and form a $W_{\infty}\times W_{\infty}$- algebra which contracts to its diagonal subalgebra in the classical limit. We also describe the Gel'fand-Naimark-Segal (GNS) construction in this language and show that the GNS representation-space (a doubled Hilbert space) is closely related to the algebra of functions over phase-space equipped with the star-product of the symbol-calculus.

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E. Gozzi, M. Reuter. 2003-06-23. Quantum Deformed Canonical Transformations, W_{\infty}-algebras and Unitary Transformations. https://doi.org/10.1142/s0217751x94002405

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