Representations of the canonical group, (the semidirect product of the unitary and Weyl$ndash$Heisenberg groups), acting as a dynamical group on noncommutative extended phase space

Abstract

The unitary irreducible representations of the covering group of the Poincare group define the framework for much of particle physics on the physical Minkowski space = /, where is the Lorentz group. While extraordinarily successful, it does not provide a large enough group of symmetries to encompass observed particles with a (3) classification. Born proposed the reciprocity principle that states physics must be invariant under the reciprocity transform that is heuristically {t, e, qi, pi} → {t, e, pi, −qi} where {t, e, qi, pi} are the time, energy, position and momentum degrees of freedom. This implies that there is reciprocally conjugate relativity principle such that the rates of change of momentum must be bounded by b, where b is a universal constant. The appropriate group of dynamical symmetries that embodies this is the canonical group (1, 3) = (1, 3) ⊗s (1, 3) = (1, 3) ⊗s (1, 3) and in this theory the non-commuting space = (1, 3)/(1, 3) is the physical quantum space endowed with a metric that is the second Casimir invariant of the canonical group, T2 + E2/c2b2 − Q2/c2 − P2/b2 + 2I/bc (Y/bc − 2) where {T, E, Qi, Pi, I, Y} are the generators of the algebra of (1, 3) = (1) ⊗s (1, 3). The idea is to study the representations of the canonical dynamical group using Mackey's theory to determine whether the representations can encompass the spectrum of particle states. The unitary irreducible representations of the canonical group contain a direct product term that is a representation of (1, 3) that Kalman has studied as a dynamical group for hadrons. The (1, 3) representations contain discrete series that may be decomposed into infinite ladders where the rungs are representations of (3) (finite dimensional) or (2) (with degenerate (1) ⊗ (2) finite-dimensional representations) corresponding to the rest or null frames.