Clifford-like calculus over lattices

@article{Vaz1997CliffordlikeCO,
  title={Clifford-like calculus over lattices},
  author={Jayme Vaz},
  journal={Advances in Applied Clifford Algebras},
  year={1997},
  volume={7},
  pages={37-70}
}
  • J. Vaz
  • Published 1 June 1997
  • Mathematics
  • Advances in Applied Clifford Algebras
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… 
The Dirac Operator over Abelian Finite Groups
In this paper we show how to construct a Dirac operator on a lattice in complete analogy with the continuum. In fact we consider a more general problem, that is, the Dirac operator over an abelian
On a correspondence principle between discrete differential forms, graph structure and multi-vector calculus on symmetric lattices ∗
TLDR
A bijective correspondence between first order differential calculi and the graph structure of the symmetric lattice is introduced that allows one to encode completely the interconnectionructure of the graph in the exterior derivative, which naturally leads to a discrete version of Clifford Analysis.
DIRAC–KÄHLER FERMION FROM CLIFFORD PRODUCT WITH NONCOMMUTATIVE DIFFERENTIAL FORM ON A LATTICE
We formulate Dirac–Kahler fermion action by introducing a new Clifford product with noncommutative differential form on a lattice. Hermiticity of the Dirac–Kahler action requires to choose the
DKP algebra, DKP equation, and differential forms
It is well known that the Clifford algebras and the Dirac equation have a representation in terms of differential forms known as the Kahler-Atiyah algebra and the Dirac-Kahler equation, respectively.
Discrete Clifford analysis
This survey is intended as an overview of discrete Clifford analysis and its current developments. Since in the discrete case one has to replace the partial derivative with two difference operators,
Solutions for the Klein–Gordon and Dirac Equations on the Lattice Based on Chebyshev Polynomials
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
Bosonic symmetries of the massless Dirac equation
The results of spin 1 symmetries of massless Dirac equation [21] are proved completely in the space of 4-component Dirac spinors on the basis of unitary operator in this space connecting this
...
...

References

SHOWING 1-10 OF 14 REFERENCES
Cli ord Algebras and the Classical Groups
The Clifford algebras of real quadratic forms and their complexifications are studied here in detail, and those parts which are immediately relevant to theoretical physics are seen in the proper
From Grassmann to Clifford
The aims of this note are to convince the readers in as elementary as possible way that: 1. Clifford algebra is the particularcase of the Grassmann algebra which is the most fundamental from
Discrete differential calculus graphs, topologies and gauge theory
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a ‘‘reduction’’ of the ‘‘universal
Dirac-Hestenes spinor fields on Riemann-Cartan manifolds
In this paper we study Dirac-Hestenes spinor fields (DHSF) on a four-dimensional Riemann-Cartan spacetime (RCST). We prove that these fields must be defined as certain equivalence classes of even
Confinement of Quarks
A mechanism for total confinement of quarks, similar to that of Schwinger, is defined which requires the existence of Abelian or non-Abelian gauge fields. It is shown how to quantize a gauge field
Covariant, algebraic, and operator spinors
We deal with three different definitions for spinors: (I) thecovariant definition, where a particular kind ofcovariant spinor (c-spinor) is a set of complex variables defined by its transformations
Clifford algebras and their applications in mathematical physics
General Surveys.- A Unified Language for Mathematics and Physics.- Clifford Algebras and Spinors.- Classification of Clifford Algebras.- Pseudo-Euclidean Hurwitz Pairs and Generalized Fueter
Magnetic monopoles without string in the Kähler–Clifford algebra bundle: A geometrical interpretation
In substitution for Dirac monopoles with string (and for topological monopoles), ‘‘monopoles without string’’ have recently been introduced on the basis of a generalized potential, the sum of a
The use of computer algebra and Clifford algebra in teaching mathematical physics
In this paper we give a collection of examples of how computer algebra can be used within Clifford algebras in teaching of Mathematical Physics. These examples cover elementary and advanced topics,
...
...