On Clifford multivector-valued actions, generalized Dirac equation and quantization of branes

Abstract

We explore the construction of a generalized Dirac equation via the introduction of the notion of Clifford multivector-valued actions, which was inspired by the work of Kanatchikov [Kanatchikov IV. De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory. Rept Math Phys. 1999;43:157–170; Kanatchikov IV. De Donder-Weyl Hamiltonian formulation and precanonical quantization of Vielbein gravity. J Phys Conf Ser. 2013;442:012041] on the De Donder-Weyl formulation of field theory. Crucial in this construction is the evaluation of the exponentials of multivectors associated with Clifford (hypercomplex) analysis. Exact matrix solutions (instead of spinors) of the generalized Dirac equation in D = 2 , 3 space–time dimensions are found. This formalism can be extended to curved space–time backgrounds. We conclude by proposing a wave-functional equation governing the quantum dynamics of branes living in C-spaces (Clifford spaces), and which is based on the De Donder–Weyl Hamiltonian formulation of field theory.