Geometry, stochastic calculus and quantum fields in a non-commutative space-time

Abstract

The algebras of non-relativistic and of classical mechanics are unstable algebraic structures. Their deformation towards stable structures leads, respectively, to relativity and to quantum mechanics. Likewise, the combined relativistic quantum mechanics algebra is also unstable. Its stabilization requires the non-commutativity of the space-time coordinates and the existence of a fundamental length constant. The new relativistic quantum mechanics algebra has important consequences on the geometry of space-time, on quantum stochastic calculus and on the construction of quantum fields. Some of these effects are studied in this paper.

36 pages Latex, 1 eps figure

References in corpus (3)

• The quantum structure of spacetime at the Planck scale and quantum fields
• Minimal Length Uncertainty Relation and Ultraviolet Regularisation
• Time and the Higgs (with apologies to J. B. Priestley)

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• A freely falling frame at the interface of gravitational and quantum realms
• Some consequences of a noncommutative space-time structure
• Non-commutative space-time and the uncertainty principle
• Entropy In The Present And Early Universe: New Small Parameters And Dark Energy Problem
• Minimal spatio-temporal extent of events, neutrinos, and the cosmological constant problem
• The deformation-stability fundamental length and deviations from c
• Space-time: Commutative or noncommutative ?
• Entropy bound for the photon gas in noncommutative spacetime
• Exotic Statistics for Ordinary Particles in Quantum Gravity
• The geometry of noncommutative space-time
• Local Currents for a Deformed Algebra of Quantum Mechanics with a Fundamental Length Scale
• An extended Dirac equation in noncommutative space-time
• Nonabelian lattice theories: Consistent measures and strata
• Soft singularity and the fundamental length
• Non-commutative probability and non-commutative processes