Geometro-stochastically quantized fields with internal spin variables

Abstract

The use of internal variables for the description of relativistic particles with arbitrary mass and spin in terms of scalar functions is reviewed and applied to the stochastic phase space formulation of quantum mechanics. Following Bacry and Kihlberg a four-dimensional internal spin spaceS is chosen possessing an invariant measure and being able to represent integer as well as half integer spins.S is a homogeneous space of the group SL(2, C) parametrized in terms of spinors α ∈ C 2 and their complex conjugatesᾱ. The generalized scalar quantum mechanical wave functions may be reduced to yield irreducible components of definite physical mass and spin [m, s], with m ≥ 0 and s = 0, 1 2 , 1, 3 2 . . . , with spin described in terms of the usual (2s + 1)-component fields. Viewed from the internal space description of spin this reduction amounts to a restriction of the variable α to a compact subspace ofS, i.e. a "spin shell" S 2 r=2s of radius r = 2s in C 2 . This formulation of single particles or single antiparticles of type [m, s] is then used to study the geometro-stochastic (i.e. quantum) propagation of amplitudes for arbitrary spin on a curved background space-time possessing a metric and axial vector torsion treated as external fields. A Poincaré gauge covariant path integral-like representation for the probability amplitude (generalized wave function) of a particle with arbitrary spin is derived satisfying a second order wave equation on the Hilbert bundle constructed over curved space-time. The implications for the stochastic nature of polarization effects in the presence of gravitation are pointed out and the extension to Fock bundles of bosonic and fermionic type is briefly mentioned.

Maxwell Field on the Poincaré Group

“…In the Ref. 43 Drechsler considered the two-sphere as a "spin shell" S 2 r=2s of radius r = 2s, where s = 0, It is easy to see that the first equation is equivalent to the second, and third equation…”

“…For that reason M 6 is a minimal homogeneous space of the Poincaré group (the spacetime translations act trivially on S 2 ). Field models on the configuration space M 6 have been considered in recent works [41,42,43]. In the Ref.…”

CPT Groups of Higher Spin Fields

“…All the possible double coverings C a,b,c,d,e,f,g are given in the and respectively non-Cliffordian group when C a,b,c,d,e,f,g is Abelian. It is easy to see that in the case of the algebra Cℓ p,q (or subalgebra Cℓ p,q ⊂ C n ) with the real division ring K ≃ R, p − q ≡ 0, 2 (mod 8), CP T -structures, defined by the groups (17), are reduced to the eight Shirokov-Dabrowski P T -structures [47,48,16].…”

“…In 1955, Finkelstein showed [18] that elementary particles models with internal degrees of freedom can be described on manifolds larger then Minkowski spacetime (homogeneous spaces of the Poincaré group). The quantum field theories on the Poincaré group were discussed in the papers [33,28,5,3,30,51,34,17,23,26]. A consideration of the field models on the homogeneous spaces leads naturally to a generalization of the concept of wave function (fields on the Poincaré group).…”

Relativistic wavefunctions on the Poincaré group

“…Relativistic wavefunctions of the work [5] were introduced without explicit reference to wave equations and for this reason they represent purely group theoretical constructions. However, if we further develop the Poincaré group representations of wavefunctions with reference to such basic notions of QFT as a Lagrangian and wave equations, then we come to a quantum field theory on the Poincaré group 1 (QFTPG) introduced by Lurçat in 1964 [20] (see also [3,18,7,2,19,31,21,8,11,12] and references therein). In contrast to the standard QFT (QFT in the Minkowski spacetime) case, for QFTPG all the notions and quantities are constructed on a ten-dimensional group manifold F of the Poincaré group.…”