Dirac Representation of Dynamically-Generated Elementary-Particle Mass
  Special-relativistic dynamically-generated elementary-particle mass is represented by a self-adjoint energy operator acting on a rigged Hilbert space (RHS) of functions over the 6-dimensional Euclidean-group manifold. The energy operator is not the generator of infinitesimal wave-function evolution in classical time. Ray evolution is generated by action-carrying Feynman paths. Extending quantum-theoretic formalism which Dirac invented and applied non-relativistically, unitary Poincar\'e -group representation is provided by the wave functions of a spacelike entity that we call 'preon'. Although the term 'preon observable' is misleading, six continuous Feynman-path-contacting preon coordinates specify spatial location (3 coordinates), lightlike-velocity-direction (2 coordinates) and transverse polarization (1 coordinate). Velocity and spatial location combine to define a preon time operator conjugate to the energy operator. In RHS bases alternative to functions over the group manifold, the wave function depends on a preon 'velocity-helicity' integer or half-odd integer) and a positive-lightlike velocity 4-vector that is Lorentz-orthogonal to a canonically-conjugate pair of spacelike 4-vectors. One 4-vector prescribes location in preon spacetime while its conjugate prescribes preon energy-momentum. Emulating the Schr\"odinger-dubbed 'zitterbewegung' of Dirac's 'relativistic' electron, mass for any spinning (positive-timelike) elementary particle accompanies a reflection-symmetric fluctuation of preon lightlike velocity and velocity-helicity. But (departing from Dirac), a tiny elementary-particle 'longitudinal' spatial extension accompanies a huge fluctuation of preon longitudinal momentum dictated by motionless-particle reflection symmetry about the plane perpendicular to spin direction. 
Geoffrey F. Chew