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Solutions for the Klein–Gordon and Dirac Equations on the Lattice Based on Chebyshev Polynomials

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Abstract

The main goal of this paper is to adopt a multivector calculus scheme to study finite difference discretizations of Klein–Gordon and Dirac equations for which Chebyshev polynomials of the first kind may be used to represent a set of solutions. The development of a well-adapted discrete Clifford calculus framework based on spinor fields allows us to represent, using solely projection based arguments, the solutions for the discretized Dirac equations from the knowledge of the solutions of the discretized Klein–Gordon equation. Implications of those findings on the interpretation of the lattice fermion doubling problem is briefly discussed.

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Acknowledgments

I would like to thank to Jayme Vaz Jr. (IMECC-UNICAMP, Brazil) for driving my attention at an earlier stage to Rabin’s approach [26] and for his research paper [29], on which the construction of spinor-like spaces, analogue to the ones constructed in Sect. 2, was considered. This paper was further developed in depth after a guest visit to School of Mathematics, University of Leeds (UK), from April 22 till April 30 2014. I would also like to thank to Vladimir V. Kisil for the fruitful discussions around this topic during these days and for his surge of interest on the feasibility of this approach. Last but not least a special acknowledgment to Artan Boriçi (University of Tirana, Albania) and to Paulo A. F. da Veiga (ICMC-USP, Brazil) for the references [3, 10, 12], and to the anonymous referees for the careful readership of the paper.

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  1. Departamento de Matematica Aplicada, Instituto de Matemática, Estatística e Computação Científica Universidade Estadual de Campinas, Campinas, SP, CEP 13083-859, Brasil

    Nelson Faustino

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  1. Nelson Faustino
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Correspondence to Nelson Faustino.

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Communicated by Klaus Guerlebeck.

Research supported by the fellowship 13/07590-8 of FAPESP (S.P., Brazil).

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Faustino, N. Solutions for the Klein–Gordon and Dirac Equations on the Lattice Based on Chebyshev Polynomials. Complex Anal. Oper. Theory 10, 379–399 (2016). https://doi.org/10.1007/s11785-015-0476-5

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