The quantum relativistic two-body bound state. II. The induced representation of SL(2,C)

Abstract

It was shown in I [J. Math. Phys. 30, 66 (1989)] that the eigenfunctions for the reduced motion of the quantum relativistic bound state with O(3,1) symmetric potential have support in an O(2,1) invariant subregion of the full spacelike region. They form irreducible representations of SU(1,1) [in the double covering of O(2,1)] parametrized by the unit spacelike vector nμ, taken in I as the direction of the z axis (the spectrum is independent of this choice), for which this O(2,1) is the stabilizer. Lorentz transformations move these representations on an orbit whose range is the single-sheeted hyperboloid covered by this spacelike vector, providing a set of induced representations of SL(2,C). From linear combinations of functions from the irreducible representations of SU(1,1), the representations of the SU(2) subgroup of SL(2,C) on the orbit are extracted and the differential equations that are the eigenvalue equations for the Casimir operators of SL(2,C) are solved. It is found that these SU(2) representations form a basis for the principal series in the canonical representations of Gel’fand. There is a natural scalar product, obtained from group integration on SL(2,C), for which the canonical basis forms an orthogonal set, and the representation is unitary. Since the scalar product [over the O(2,1) invariant measure space] of SU(1,1) irreducible representations is invariant under the action of the little group, the remaining group measure [on the coset space SL(2,C)/SU(1,1)] is the volume on the hyperboloidal spacelike hypersurface dμn=d4n δ(n2−1). The family of Hilbert spaces (ℋn) that carries the representations of O(3,1) is therefore embedded in a larger Hilbert space ℋ with measure d4y dμn, where the { y} are the space-time coordinates of the restricted region associated with nμ. The representations with nonrelativistic limit coinciding with the known Schrödinger solutions for corresponding spherically symmetric potential problems are in the double covering (half-integer values for the lowest L level) of O(3,1).

The Particle as a Statistical Ensemble of Events in Stueckelberg–Horwitz–Piron Electrodynamics

“…Perhaps most significantly, while defining the system in an unconstrained 8D phase space relaxes the a priori mass shell relationẋ 2 = c 2 and thus permits classical trajectories that reverse the direction of their time evolution, it also eliminates reparameterization invariance. This is because the mass shell constraint and reparameterization invariance are related features of a Lagrangian that is homogeneous of first degree in the velocities, which is not the case for (8). Moreover, in Stueckelberg-Horwitz-Piron (SHP) electrodynamics, the evolution parameter τ cannot be identified as the proper time of the motion, but is a dynamical quantity proportional to it through…”

“…in the argument of nonrelativistic scalar potentials, Horwitz et al found solutions for relativistic generalizations of the standard central force problems, including quantum mechanical potential scattering and bound states [4][5][6][7][8]. Examination of radiative transitions [9][10][11] associated with these bound states suggests that the scalar interaction V is required along with the four-vector potential A µ in order to account for known phenomenology.…”

Discrete Symmetries of Off-Shell Electromagnetism

“…The connection with Maxwell theory enlarges on Stueckelberg's observation in (13). Under the conditions j 5 → 0 and f 5µ → 0, pointwise in x as τ → ±∞, integration of (24) over τ , called concatenation of events into a worldlines [9], recovers the relations…”

“…Within this framework, solutions have been found for the generalizations of the standard central force problem, including potential scattering [11] and bound states [12,13]. Examination of radiative transitions [14], in particular the Zeeman [15] and Stark effects [16], indicate that all five components of the gauge potential are necessary for an adequate explanation of observed phenomenology.…”

A Perturbative Approach to the Relativistic Harmonic Oscillator and Unitarity

“…The Relativistic Harmonic Oscillator is probably the simplest relativistic system containing bound states, yet it exhibits the typical problems of Relativistic Quantum Mechanics. Many papers have been devoted to the solution of this relativistc system [1,2,3,4,5] although the first question is probably to define what we understand by a Relativistic Harmonic Oscillator.…”