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Quantum mechanics and field theory with momentum defined on an anti-de Sitter space

Abstract

Relativistic dynamics with energy and momentum restricted to an anti-de Sitter space is presented. Coordinate operators conjugate to such momenta are introduced. Definition of functions of these operators, their differentiation and integration, all necessary for the development of dynamics is presented. The resulting algebra differs from the standard Heisenberg one, notably in that the space-time coordinates do not commute among each other. The resulting time variable is discrete and the limit to continuous time presents difficulties. A parallel approach, in which an overlap function, between position and momentum states, is obtained from solutions of wave equations on this curved space are also investigated. This approach, likewise, has problems in the that high energy behavior of these overlap functions precludes a space-time definition of action functionals. © 2010 World Scientific Publishing Company.

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