(Received July 29, 1968)
The propagators for different arguments are explicitly given for any spin in terms of SO(N) projection operators (N = 2, 3, 4).
For massive particles of integer spin J, from among GL(4) representations, symmetry and tracelessness of the rank-J tensor fields take D(J/2, J/2) of the Lorentz group, out of which divergence-free conditions choose D(J) of SO(3), the little group of the Poincaré group. The propagators pass to to this little-group projection operators, which are then boosted to Minkowski space. Trace-free and divergence-free conditions are lifted off the mass shell, where the propagators turn to projection operators of a particular GL(4) representation, letting loose a set of superfluous spins (J-1, J-2, …, 0) initially excluded by the on-shell subsidiary conditions; these low spins do not propagate but give rise to transitions and cooperate in getting rid of singularity in the scattering amplitudes. This necessarily follows from the present tensor formalism, whence it may be regarded as a dynamical assumption for off-shell spin. For half-integer spin J+1/2, we make use of J+1/2 projection operators acting on the tensor (rank J)-spinor fields.
A transition to massless particles is, except for J = 0, 1/2, prohibited due to M-2 singularity, which is to be removed by gauge transformations peculiar to massless particles. In the radiation gauge of transversal states, the propagators are constructed by projection operators of SO(2), a compact subgroup of the little group E2. In the Feynman gauge involving unphysical polarizations, the propagators are given by SO(3, 1) projection operators; hence a series of redundant spins (J-2, J-4, …) propagates. The complication of gauges arises from the use of the D(J/2, J/2) fields; the present formalism is not so simple for massless particles as for massive particles.
Comparison with propagators derived from Lagrangians is made for several spins: Differences like in the terms multiplied by the Klein-Gordon operator, hence vanish on the mass shell. A choice of parameters left undetermined by the Lagrangian method fixes, off the mass shell, a particular composition of spin contents (J-1, J-2, …, 0), which can be further changed in view of non-uniqueness of the propagators for identical arguments.
URL : http://ptp.ipap.jp/link?PTP/41/214/
DOI : 10.1143/PTP.41.214