# Separation of Variables for Systems of First-Order Partial Differential Equations and the Dirac Equation in Two-Dimensional Manifolds

## Abstract

The problem of solving the Dirac equation on two-dimensional manifolds is approached from the point of separation of variables, with the aim of creating a foundation for analysis in higher dimensions. Beginning from a sound definition of multiplicative separation for systems of two first order linear partial differential equations of Dirac type and the characterization of those systems admitting multiplicatively separated solutions in some arbitrarily given coordinate system, more structure is step by step added to the problem by requiring the separation constants are associated with commuting differential operators. Finally, the requirement that the original system coincides with the Dirac equation on a two-dimensional manifold allows the characterization of the orthonormal frames and metrics admitting separation of variables for the equation and of the symmetries associated with the separated coordinates.

## Keywords

Dirac Equation Dirac Operator Separate Equation Symmetry Operator Separation Constant## Preview

Unable to display preview. Download preview PDF.

## References

- [1]W. Miller, Jr.,
*Symmetry and separation of variables*, Addison-Wesley, Reading, 1977.MATHGoogle Scholar - [2]E.G. Kalnins,
*Separation of variables for Riemannian spaces of constant curvature*, Longman, Harlow, 1986.MATHGoogle Scholar - [3]L. Fatibene and M. Francaviglia,
*Natural and gauge natural formalism for classical field theories. A geometric perspective including spinors and gauge theories*, Kluwer, Dordrecht, 2003.MATHGoogle Scholar - [4]D.R. Brill and J.A. Wheeler,
*Interaction of neutrinos and gravitational fields*, Rev. Mod. Phys. 29: 465–479, 1957.MATHCrossRefMathSciNetGoogle Scholar - [5]S. Chandrasekhar,
*The mathematical theory of black holes*, Clarendon, Oxford, 1983, p. 531.MATHGoogle Scholar - [6]B. Carter and R.G. McLenaghan,
*Generalized total angular momentum for the Dirac operator in curved space-time*, Phys. Rev. D 19: 1093–1097, 1979.CrossRefMathSciNetGoogle Scholar - [7]W. Miller, Jr.,
*Mechanism for variable separation in partial differential equations and their relationship to group theory*. In Symmetries and Nonlinear Phenomena, World Scientific, Singapore, 1988, pp. 188–221.Google Scholar - [8]M. Fels and N. Kamran,
*Non-factorizable separable systems and higher-order symmetries of the Dirac operator*, Proc. Roy. Soc. London A**428**: 229–249, 1990.MATHCrossRefMathSciNetGoogle Scholar - [9]L. Fatibene, R.G. McLenaghan, and S. Smith,
*Separation of variables for the Dirac equation on low dimensional spaces. In Advances in general relativity and cosmology*, Pitagora, Bologna, 2003, 109–127.Google Scholar - [10]J.T. Horwood and R.G. McLenaghan,
*Transformation to pseudo-Cartesian coordinates in locally flat pseudo-Riemannian spaces*, preprint, University of Waterloo, 2006.Google Scholar - [11]E.G. Kalnins, W. Miller,
*Separation of variables methods for systems of differential equations in mathematical physics*, Proceedings of the Annual Seminar of the Canadian Mathematical Society, Lie Theory, Differential Equations and Representation Theory, pp. 283–300, 1989.Google Scholar - [12]S. Smith,
*Symmetry operators and separation of variables for the Dirac equation on curved space-times*, PhD thesis, University of Waterloo, 2002.Google Scholar - [13]T. Levi-civita,
*Sulla integrazione della equazione di Hamilton-Jacobi per separazione di variabili*, Math.Ann.**59**: 383–397, 1904.MATHCrossRefMathSciNetGoogle Scholar - [14]
- [15]V.N. Shapovalov and G.G. Ekle,
*Complete sets and integration of a first-order linear system I and II*, Izv. vyssh. ucheb. zaved. Fiz, (2), 83–92, 1974.Google Scholar - [16]S. Benenti, C. Chanu, and G. Rastelli,
*Variable separation theory for the null Hamilton-Jacobi equation*, J.Math. Phys.**46**: 042901/29, 2005.Google Scholar - [17]S. Benenti, C. Chanu, and G. Rastelli,
*Remarks on the connection between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schr’odinger equations. I. The completeness and Robertson condition*, J. Math. Phys.**43**: 5183–5222, 2002.MATHCrossRefMathSciNetGoogle Scholar