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A Modified Dirac Operator in Parameter–Dependent Clifford Algebra: A Physical Realization

Abstract:

In the present paper we introduce a modified Dirac operator and solve the associated Dirichlet boundary value problem in $${\mathbb{R}^3}$$R3 for functions which are q–monogenic when the Clifford algebra depends on parameters. By determining the compatibility conditions we establish theorems of existence and uniqueness of solutions to the modified Dirac equation, and the associated Laplace equation, such that they correspond to a scalar operator. We also discuss the problem using fixedpoint methods and analyze a physical realization were the solutions to the Laplace equation can be interpreted as a set of electric potentials coupled to each other. Our solutions could be relevant in the analysis of new exotic phases of nature, such as topological insulators were the Dirac nature of the charge carriers implies new physical properties which go beyond the standard description of conventional charge carriers in electronic systems by means of the Schrödinger equation.