Reciprocal relativity of noninertial frames: quantum mechanics

Noninertial transformations on time-position-momentum-energy space {t,q,p,e} with invariant Born-Green metric ds^2=-dt^2+dq^2/c^2+(1/b^2)(dp^2-de^2/c^2) and the symplectic metric -de/\dt+dp/\dq are studied. This U(1,3) group of transformations contains the Lorentz group as the inertial special case. In the limit of small forces and velocities, it reduces to the expected Hamilton transformations leaving invariant the symplectic metric and the nonrelativistic line element ds^2=dt^2. The U(1,3) transformations bound relative velocities by c and relative forces by b. Spacetime is no longer an invariant subspace but is relative to noninertial observer frames. Born was lead to the metric by a concept of reciprocity between position and momentum degrees of freedom and for this reason we call this reciprocal relativity. For large b, such effects will almost certainly only manifest in a quantum regime. Wigner showed that special relativistic quantum mechanics follows from the projective representations of the inhomogeneous Lorentz group. Projective representations of a Lie group are equivalent to the unitary reprentations of its central extension. The same method of projective representations of the inhomogeneous U(1,3) group is used to define the quantum theory in the noninertial case. The central extension of the inhomogeneous U(1,3) group is the cover of the quaplectic group Q(1,3)=U(1,3)*s H(4). H(4) is the Weyl-Heisenberg group. A set of second order wave equations results from the representations of the Casimir operators

Representations of the Canonical group, (the semi-direct product of the Unitary and Weyl-Heisenberg groups), acting as a dynamical group on noncommuting extended phase space

The unitary irreducible representations of the covering group of the Poincare group P define the framework for much of particle physics on the physical Minkowski space P/L, 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 heuristically {t,e,q,p}->{t,e,p,-q} where {t,e,q,p} 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 C(1,3) = U(1,3) *s H(1,3) and in this theory the non-commuting space Q= C(1,3)/ SU(1,3) is the physical quantum space endowed with a metric that is the second Casimir invariant of the Canonical group, T^2 + E^2 - Q^2/c^2-P^2/b^2 +(2h I/bc)(Y/bc -2) where {T,E,Q,P,I,Y} are the generators of the algebra of Os(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 U(1,3) that Kalman has studied as a dynamical group for hadrons. The U(1,3) representations contain discrete series that may be decomposed into infinite ladders where the rungs are representations of U(3) (finite dimensional) or C(2) (with degenerate U(1)* SU(2) finite dimensional representations) corresponding to the rest or null frames.Comment: 25 pages; V2.3, PDF (Mathematica 4.1 source removed due to technical problems); Submitted to J.Phys.