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 Poincaré group P define the framework for much of particle physics on the physical Minkowski space M = P êê ê ê L êêê , where L is the Lorentz group. While extraordinarily successful, it does not provide a large enough group of symmetries to encompass observed particles with a SU(3) classification. Born proposed the reciprocity principle that states physics must be invariant under the reciprocity transform that is heuristically8t, e, q i , p i < Ø 8t, e, p i , -q i < where 8t, e, q i , p i < 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 CH1, 3L = UH1, 3L ≈ s HH1, 3L = SUH1, 3L ≈ s OsH1, 3L and in this theory the non-commuting space Q = CH1, 3L ê SUH1, 3L is the physical quantum space endowed with a metric that is the second Casimir invariant of the Canonical group,-2 L where 8T, E, Q i , P i , I, Y < are the generators of the algebra of OsH1, 3L = UH1L ≈ s HH1, 3L. 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 UH1, 3L that Kalman has studied as a dynamical group for hadrons. The UH1, 3L representations contain discrete series that may be decomposed into infinite ladders where the rungs are representations of UH3L (finite dimensional) or CH2L (with degenerate UH1L ≈ SUH2L finite dimensional representations) corresponding to the rest or null frames.

The Extended Relativity Theory in Clifford Phase Spaces and Modifications of Gravity at the Planck/Hubble Scales

“…The U (1, 3) = SU (1, 3)⊗ U (1) Group transformations which leave invariant the phase-space intervals under rotations, velocity and acceleration boosts, were found by Low [10] and can be simplified drastically when the velocity/acceleration boosts are taken to lie in the z-direction, leaving the transverse directions x, y, p x , p y intact ; i.e., the U (1, 1) = SU (1, 1) ⊗ U (1) subgroup transformations leave invariant the phasespace interval given by (in units ofh = c = 1)…”

“…In Born's relativity we have the invariance group which is the intersection of SO(4 + 4) and the ordinary symplectic group Sp (8). The intersection contains the unitary group U (4) which allowed Low [10] to write down the symmetry transformations under velocity and acceleration boosts, etc .. One may ask the question is : what is the intersection of the group SO(12) with the ternary group ( SO(4n) and n-ary group in general ) which leaves invariant the triplewedge product (4.2) and the interval (4.1) ? A careful study reveals that this is the wrong question.…”

“…An upper limit on the maximal acceleration of particles was proposed long ago by Cainello [9]. This idea is a direct consequence of a suggestion made years earlier by Max Born on a Reciprocal Relativity principle operating in Phase Spaces [8], [10] where there is an upper bound on the four-force (maximal string tension or tidal forces in strings ) acting on a particle as well as an upper bound in the particle's velocity given by the speed of light.…”

“…The quantity g(τ ) is the proper four-acceleration of a particle of mass m in the z-direction which we take to be defined by the X coordinate. The interval (dω) 2 described by Low [10] is U (1, 3)-invariant for the most general transformations in the 8D phase-space. These transformations are rather elaborate, so we refer to the references [10] for details.…”

“…The transformations laws of the coordinates in that leave invariant the interval (1.1) were given by [10]:…”