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Geometro-stochastically quantized fields with internal spin variables. (English) Zbl 0892.58089
Summary: 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 space $$\overline S$$ is chosen possessing an invariant measure and being able to represent integer as well as half integer spins. $$\overline S$$ is a homogeneous space of the group $$\text{SL} (2,{\mathcal C})$$ parametrized in terms of spinors $$\alpha\in {\mathcal C}_2$$ and their complex conjugates $$\overline\alpha$$. The generalized scalar quantum mechanical wave functions may be reduced to yield irreducible components of definite physical mass and spin $$[m,s]$$, with $$m\geq 0$$ and $$s=0, {1\over 2}, 1,{3 \over 2}, \dots$$, 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 $$\alpha$$ to a compact subspace of $$\overline S$$, i.e., a “spin shell” $$S^2_{r=2s}$$ of radius $$r=2s$$ in $${\mathcal C}_2$$. 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.

##### MSC:
 58Z05 Applications of global analysis to the sciences 81P20 Stochastic mechanics (including stochastic electrodynamics) 81S25 Quantum stochastic calculus 81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
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