# Formulation of Spinors in Terms of Gauge Fields

- 138 Downloads

## Abstract

It is shown in the present paper that the transformation relating a parallel transported vector in a Weyl space to the original one is the product of a multiplicative gauge transformation and a proper orthochronous Lorentz transformation. Such a Lorentz transformation admits a spinor representation, which is obtained and used to deduce the transportation properties of a Weyl spinor, which are then expressed in terms of a composite gauge group defined as the product of a multiplicative gauge group and the spinor group. These properties render a spinor amenable to its treatment as a particle coupled to a multidimensional gauge field in the framework of the Kaluza–Klein formulation extended to multidimensional gauge fields. In this framework, a fiber bundle is constructed with a horizontal, base space and a vertical, gauge space, which is a Lie group manifold, termed its structure group. For the present, the base is the Minkowski spacetime and the vertical space is the composite gauge group mentioned above. The fiber bundle is equipped with a Riemannian structure, which is used to obtain the classical description of motion of a spinor. In its classical picture, a Weyl spinor is found to behave as a spinning charged particle in translational motion. The corresponding quantum description is deduced from the Klein–Gordon equation in the Riemann spaces obtained by the methods of path-integration. This equation in the present fiber bundle reduces to the equation for a Weyl spinor, which is close to but differs somewhat from the squared Dirac equation.

## Keywords

Spinors in Weyl geometry Gauge fields Kaluza–Klein formulation Path-integrals in curved spaces Klein–Gordon equation in Riemannian spaces## References

- 1.Adler, R.J.: Spinors in a Weyl geometry. J. Math. Phys.
**11**, 1185–1191 (1970)ADSCrossRefGoogle Scholar - 2.Macfarlane, A.J.: On the restricted Lorentz group and groups homomorphically related to it. J. Math. Phys.
**3**, 1116–1129 (1962)ADSCrossRefMATHMathSciNetGoogle Scholar - 3.Lee, H.C. (ed.): An Introduction to the Kaluza–Klein Theories. World Scientific, Singapore (1984)Google Scholar
- 4.Kerner, R.: Generalization of the Kaluza–Klein theory for an arbitrary non-Abelian gauge group. In: Appelquist, T. (ed.) Frontiers in Physics, pp. 115–124. Addison-Wesley, Reading (1987)Google Scholar
- 5.Witten, E.: Search for a realistic Kaluza–Klein theory. Nuclear Physics
**B186**, 412–428 (1981)ADSCrossRefMathSciNetGoogle Scholar - 6.Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)MATHGoogle Scholar
- 7.Schulman, L.S.: Techniques and Applications of Path Integration. Wiley, New York (1981)MATHGoogle Scholar
- 8.Barut, A.O., Zanghi, N.: Classical model of the Dirac electron. Phys. Rev. Lett.
**52**, 2009–2012 (1984)ADSCrossRefMathSciNetGoogle Scholar - 9.Barut, A.O., Duru, L.H.: Path-integral derivation of the Dirac propagator. Phys. Rev. Lett.
**53**, 2355–2358 (1984)ADSCrossRefMathSciNetGoogle Scholar - 10.Santamato, E., De Martini, F.: Derivation of the Dirac Equation by conformal differential geometry. Found. Phys.
**43**, 631–641 (2013)ADSCrossRefMATHMathSciNetGoogle Scholar - 11.DeWitt, B.S.: Dynamical theory in curved spaces I. A review of the classical and quantum action principles. Rev. Mod. Phys.
**29**, 377–397 (1957)ADSCrossRefMATHMathSciNetGoogle Scholar - 12.Cheng, K.S.: Quantization of a general dynamical system by Feynman path integral formulation. J. Math. Phys.
**13**, 1723–1726 (1972)ADSCrossRefGoogle Scholar - 13.Vatsya, S.R.: Mechanics of a particle in a Riemannian manifold. Chaos, Solitons & Fractals
**10**, 1391–1397 (1999)ADSCrossRefMATHMathSciNetGoogle Scholar - 14.Wheeler, J.T.: Quantum measurement and geometry. Phys. Rev. D
**41**, 431–441 (1990)ADSCrossRefMathSciNetGoogle Scholar - 15.Messiah, A.: Quantum Mechanics, Vol. II, (trans. Potter, J.). North-Holland, Amsterdam (1962) pp. 880–882; 900–904.Google Scholar
- 16.Weyl, H.: The Theory of Groups and Quantum Mechanics, (trans. H. P. Robertson). Dover, New York (1950) pp. 146–165.Google Scholar
- 17.Vatsya, S.R.: Mechanics of a charged particle on the Kaluza–Klein background. Can. J. Phys.
**73**, 602–607 (1995)ADSCrossRefMathSciNetGoogle Scholar - 18.Wong, S.K.: Field and particle equations for the classical Yang-Mills field and particles with isotopic spin. Nuovo Cimento LXV A, 689–694 (1970).Google Scholar