Massive Particles from Massless Spinors

Spinors are lightlike. How do they combine to make massive particles? We visit the zoo of Lagrangian singularities, or caustics, in spacetime projections from spin spacethe phase space of lightlike, 8spinor flows. We find that the species living there are the elementary particles. Codimension J = (1, 2, 3, 4) phase singularities vortex lines, sheets, tubes, and knots, are classified by the Coxeter groups generated by multiplicity s reflections : ”mass scatterings” off the vacuum spinors, that keep chiral pairs of matter envelopes confined to a timelike world tube, endowing a bispinor particle with mass. Using the volume in spin space as the action, the particle masses emerge in terms of the multiplicities, s: the number of null zigzags needed to close a cycle of mass scatterings. These mass values (calculated to lowest order in the vacuum intensity) are within a few percent of the observed masses for the leptons (J = 1) and hadrons (J = 3) ; but are up to 25 percent off for the mesons (J = 2). 1 From Spin Space to spacetime Spinors live in the root space of vector and tensor fields, much as complex numbers live in the root space of real polynomials. Left (l ∈ L) or Right (r ∈ R) chirality spinors have L and R-handed su (2) twists with spatial translation: they are Clifford (C)-analytic and conjugate C-analytic, respectively [BdS]. Complexified spinors [M.C.1] have an additional counter-clockwise or clockwise u (1) phase advance with cosmic time, T : the log of the distance back to the big bang, in units of the compactification radius, a# : T = a# ln γ; γ (t) = alt) a# . Here a(t) is the scale factor of our spatial hypersurface, S3[a(t)],as a function of Minkowsky time, or arctime, t : |△x| = ± △ T = △t, the distance traveled by a photon.. Arctime and cosmic time combine to make complex time, z ≡ t + iT. A spinor is either analytic (+) or conjugate analytic (−) in complex time; the sign, Sgn[∂T θ ] is its charge [MC.1][M.C.2]. Complex conjugation, ( z )∗ ≡ t− iT , gives charge reversal, C.