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Clifford-like calculus over lattices

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Abstract

We introduce a calculus over a lattice based on a lattice generalization of the Clifford algebras. We show that Clifford algebras, in contrast to the continuum, are not an adequated algebraic structure for lattice problems. Then we introduce a new algebraic structure, that reduces to a Clifford algebra in the continuum limit, in terms of which we can develop a formalism analogous to the differential geometry of the continuum, also in the sense that we have intrinsic expressions. The differential operator is given by the graded commutator of an operator that generalizes the Dirac operator. We also discuss the applications of this formalism in lattice gauge theories, with particular attention to the fermion doubling problem.

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References

  1. J. Vaz, Jr., inClifford Algebras with Numeric and Symbolic Computations, R. Abłamowicz, P. Lounesto and J. M. Parra (eds), pg. 33–55, Birkhauser (1996).

  2. I. Benn and R. W. Tucker,Introduction to Spinors and Geometry, Adam Hilger (1987).

  3. J. E. Gilbert and M. A. M. Murray,Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge University Press (1991).

  4. Z. Oziewicz, inClifford Algebras and their Applications in Mathematical Physics, J. S. R. Chisholm and A. K. Common (eds), pg. 245–256, D. Reidel Publ. Co. (1986).

  5. J. G. Hocking and G. S. Young,Topology, Addison-Wesley Publ. Co. (1961).

  6. P. Becher and H. Joos,Z. Phys.,C 15, 343 (1982).

    MathSciNet  Google Scholar 

  7. I. Porteous,Clifford Algebras and the Classical Groups, Cambridge University Press (1995).

  8. W. Greub,Multilinear Algebra, Springer-Verlag, 2nd ed. (1978).

  9. A. Crumeyrolle,Orthogonal and Symplectic Clifford Algebras, Kluwer Acad. Publ. (1990).

  10. V. L. Figueiredo, E. C. Oliveira and W. A. Rodrigues, Jr.,Int. J. Theor. Phys.,29, 371 (1990).

    Article  MATH  Google Scholar 

  11. C. Chevalley,The Algebraic Theory of Spinors, Columbia University Press (1954).

  12. E. Kahler, Rend. Mat. e delle sue Applicazione, Ser. V,21, 425 (1962).

    MathSciNet  Google Scholar 

  13. W. Graf,Ann. Inst. H. Poincaré,A29, 85 (1978).

    MATH  MathSciNet  Google Scholar 

  14. D. Ivanenko and L. Landau,Z. Phys. 48, 340 (1928); V. Fock and D. Ivanenko,Z. Phys.,54, 798 (1929).

    Article  ADS  Google Scholar 

  15. A. Proca,C. R. Acad. Sciences Paris,190, 1377 (1930);191, 26 (1930).

    MATH  Google Scholar 

  16. A. Eddington,Fundamental Theory, Cambridge (1948).

  17. M. Schonberg,An. Acad. Brasil. Ciencias,29, 473 (1957).

    MathSciNet  Google Scholar 

  18. A. Maia, Jr., E. Recami, W. A. Rodrigues, Jr., and M. A. F. Rosa,J. Math. Phys.,31, 502 (1990).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  19. W. A. Rodrigues, Jr., Q. A. G. Souza, J. Vaz, Jr. and P. Lounesto, “Dirac-Hestenes spinors on Riemann-Cartan manifolds”, to appear in Int. J. Theor. Phys. (1996).

  20. F. Reese Harvey,Spinors and Calibrations, Academic Press (1990).

  21. J.M. Rabin, Nucl. Phys.B 201, 315 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  22. I. Montvay and G. Münster,Quantum Fields on a Lattice, Cambridge University Press (1994).

  23. K. G. Wilson,Phys. Rev.,D,10, 2445 (1974).

    ADS  Google Scholar 

  24. A. Connes,Noncommutative Geometry, Academic Press (1994).

  25. A. Dimakis and F. Müller-Hoissen,J. Math. Phys.,35, 6703 (1994).

    Article  MATH  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Syracuse University, 13244-1130, Syracuse, NY, USA

    Jayme Vaz

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  1. Jayme Vaz
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Correspondence to Jayme Vaz.

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On leave of absence from Department of Applied Mathematics, State University at Campinas (UNICAMP), 13081-970 Campinas, SP, Brazil — vaz@ime.unicamp.br

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Vaz, J. Clifford-like calculus over lattices. AACA 7, 37–70 (1997). https://doi.org/10.1007/BF03041215

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  • DOI: https://doi.org/10.1007/BF03041215

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