Separation of Variables for Systems of First-Order Partial Differential Equations and the Dirac Equation in Two-Dimensional Manifolds
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.
KeywordsDirac Equation Dirac Operator Separate Equation Symmetry Operator Separation Constant
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