Twistor Construction of Massive Particles, Old and New

George Sparling's 1981 Theory of Massive Particles 1 constructs, by localizing the conformal algebra at nonzero mass, an internal SU(3) generated by spinor-transfer operators and a vector operator that commutes with four-momentum. Its finite multiplets contain several Wigner spins at correlated charges. The construction is algebraic, however: it does not provide an underlying incidence model or spacetime field equations. This talk supplies such a realization. The fully prepared Heisenberg-incidence Schrödinger equation encodes the massless helicity towers. We weaken that system by one contraction, obtaining an operator G. At fixed non-null Fourier momentum, G becomes a constant vector field along one of the residual spinor directions. Its quotient retains the two components of a spinor, together with a scalar invariant u. The complete polynomial kernel is therefore the polynomial algebra in these three variables. At fixed total degree, its decomposition into powers of u gives finite spin sectors together with their Heisenberg descendants. We show how these these sectors can be identified with massive Wigner bundles and their associated spacetime fields. Allowing momentum to vary over the future timelike cone assembles them into a direct integral over all positive masses. Thus the positive-timelike part of the kernel of G is a natural all-mass carrier on which mass becomes a spectral variable. We also obtain explicit field realizations of Sparling's operators: his r-operators transfer excitations between the scalar and spinor modes, while his R-operator preserves four-momentum and mixes the different Wigner spins within a fixed massive multiplet