Parity Doubling , Zitterbewegung , and Rest Mass for Spin 1 ∗

A 6-component “wave function” (not field, but S-matrix interpretable) for a massive spin-1 particle parallels the Dirac “chiralitydoubled” 4-component wave function for a spin-1/2 particle, by pairing two wave functions for same spin but opposite “handedness”. The correlated “opposite-parity” pair of complex 3-vectors defines a fluctuating spin-correlated lightlike “internal velocity” as well as an independent “external rapidity”. Extension from fermions to vector bosons of the velocity-fluctuation (“zitterbewegung”) interpretation of rest mass weakens theoretical motivation for elementary scalar bosons. ∗This work is supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, Division of High Energy Physics, of the U.S. Department of Energy under Contract DE-AC03-76SF00098 Introduction The standard model accommodates massive spin-1 particles, with zero as well as ±1 helicity, by enlarging the fermion, vector-boson family of elementary particles to include scalar bosons. The present paper employs a 6-dimensional representation of spin 1—imitating the wave-function doubling that characterizes Dirac 4-spinors (a doubling that allows representation not only of parity but of charge conjugation)—to define a Hilbert space that accommodates (charged or neutral) spin 1 with nonvanishing rest mass even though based on elementary “lightlike” (zero-rest-mass) vectors (no scalars). Rest mass, for vector bosons as well as for fermions, is attributed to “zitterbewegung”. Dirac’s 4-component chirally-doubled-spinor wave function for a spin-1/2 particle recognizes a momentum-independent “internal” significance for velocity that parallels or antiparallels spin (not momentum) in its direction. The unique internal-velocity magnitude is that of light (Dirac’s “velocity operator” is γ5~σ). (2) The Dirac wave function for nonzero rest mass manifests a fluctuation, at fixed spin (and momentum), of a 2-valued internal-velocity direction; only with zero rest mass does internal velocity fail to fluctuate. (The expectation value of Dirac’s internal-velocity operator is equal to the ratio of momentum to energy—i.e., to “external velocity”.) Dirac’s association of rest mass with lightlike internal-velocity fluctuation—“zitterbewegung”— subsequently became obscured by continuous-field theory but accords with a dynamics, based on lightlike discrete-step Feynman paths, that is under investigation by one of the authors. We here call attention to a spin-correlated directional fluctuation of lightlike internal velocity within the wave function for a spin-1 particle with nonzero rest mass. A correlated pair of “opposite-parity” complex 3-vector wave functions—a 6-component wave function—depicts the “source” of rest mass in a sense similar to that of Dirac. Although our meaning for internal velocity is unaccompanied by any vector-boson Hamiltonian, wavefunction zitterbewegung propagation is prescribable, even in absence of either