Symmetries of fundamental interactions in quantum phase space

Abstract

Generators of spacetime translations and Lorentz group transformations form the Lie algebra of the Poincar\'e group and give rise to the Casimir invariants for a specification of elementary particle characteristics. Moreover quantum operators of coordinate and momentum components of a particle in Minkowski spacetime together with Lorentz group generators belong to the known noncommutative algebra. This algebra can be generalized under some constraints, in particular, the Lorentz invariance condition. The generalized algebra depends on the new fundamental constants with dimensions of length (L), mass (M) and action (H). Quantum fields, which can be constructed with the help of representations of this algebra, are referred to as HLM generalized quantum fields and the associated particles as HLM quantum particles. Relations between spacetime observables of a HLM quantum particle depend on the new constants and lead to complication of the Poincar\'e invariant equations for canonical fields. The modification of a quantum measurement procedure is needed in order to take into account inherent features of HLM quantum particles.