, Volume 304, Issue 1, pp 95-123,
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Date: 09 Oct 2010

Warped Convolutions, Rieffel Deformations and the Construction of Quantum Field Theories

Abstract

Warped convolutions of operators were recently introduced in the algebraic framework of quantum physics as a new constructive tool. It is shown here that these convolutions provide isometric representations of Rieffel’s strict deformations of C *–dynamical systems with automorphic actions of \({\mathbb R^n}\) , whenever the latter are presented in a covariant representation. Moreover, the device can be used for the deformation of relativistic quantum field theories by adjusting the convolutions to the geometry of Minkowski space. The resulting deformed theories still comply with pertinent physical principles and their Tomita–Takesaki modular data coincide with those of the undeformed theory; but they are in general inequivalent to the undeformed theory and exhibit different physical interpretations.

Communicated by Y. Kawahigashi
Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen.
Research supported by the NSF Grant DMS-0901370.
Dedicated to Sergio Doplicher on the occasion of his seventieth birthday