The authors show how algebra, geometry and differential equations all play a more fundamental role in euclidean fourier analysis than has been fully realized before. To this end one needs to construct discrete dirac operators which factorize the discrete star or crosslaplacian. Any smooth manifold m defines a lie algebroid, where a tm. These operators are discrete versions of the continuous dirac operators and consist in the sum of a discrete free dirac operator with a random potential. Pdf linear and nonlinear theories of discrete analytic. Ams proceedings of the american mathematical society. Clifford algebras and dirac operators in harmonic analysis by. Discrete dirac operators on riemann surfaces and kasteleyn matrices received april 19, 2010 abstract. Abstract two discretizations, linear and nonlinear, of basic notions of the complex analysis are considered.
Postmodern topics in discrete clifford analysis nelson faustino what ive learned from frank the radial algebra approach beyond landauweyl calculus liealgebraic discretization weylheisenberg symmetries appell set formulation su1. As in euclidean clifford analysis, the construction of discrete monogenic functions, that is, functions in the kernel of the discrete dirac operator, is a. We establish dynamical localization for random dirac operators on the ddimensional lattice, with d2f1. Clifford algebras and dirac operators in harmonic analysis. The cauchykovalevskaya extension theorem in discrete. Diractype operators are central to the field of clifford analysis, where recently there has been considerable. Dirac operators, boundary value problems, and the bcalculus paul loya abstract. Lectures on dirac operators and index theory ucsb math. Dirac operators in riemannian geometry thomas friedrich. In this paper we work in the split discrete clifford analysis setting, i. Fourier analysis of discrete dirac operators on the n torus nelson faustino new developments on discrete dirac operators. The kogutsusskind fermion and the staggered fermion actions are derived directly from the dirackahler fermion formulated by the clifford product.
In this paper, we establish a cauchykovalevskaya extension theorem for discrete. Similar frameworks for the dirackahler operator date to the 1980s 3, 35. In order to convey some idea of the kind of results we obtained, consider the following example see corollary 3. Harmonic analysis for general first order differential operators in lipschitz domains emilio. The ones marked may be different from the article in the profile. The relative dirac operator is highly sensitive to.
Dirac operators and nilpotent lie algebra cohomology. The lattice qcd action with dirackahler matter fermion is also derived via an inner product defined by the clifford product. Hermitean clifford analysis focusses on monogenic functions taking values in a complex clifford algebra or in a complex spinor space. Fourier analysis of discrete dirac operators on the n. Topics in clifford analysis special volume in honor of. We show how the space mk of discrete spherical monogenics homogeneous of degree k, is. Calculating the fourier symbol of the boundary operator we construct the corresponding discrete hilbert transforms, the projection operators arising from them, and discuss the notion of discrete hardy spaces. Discrete dirac operators in clifford analysis core. The abstract hodgedirac operator and its stable discretization arxiv. Pal saha institute of nuclear physics 1af bidhannagar, calcutta 700064, india abstract this is a pedagogical article which discusses various kinds of fermion.
Can compatible discretization, fem, and discrete cli ord analysis be fruitfully combined. Here monogenicity is expressed by means of two complex mutually adjoint dirac operators, which are invariant under the action of a representation of the unitary group. Dirac operators, boundary value problems, and the bcalculus. Fourier analysis of discrete dirac operators on the n torus nelson faustino new developments on discrete dirac operators discrete dirac operators the toroidal approach kleingordon type equations the model problem wave type propagators playing around some remarkable connections a spacetime fourier inversion formula the discrete heat semigroup. The laplace and dirac operators are rotation invariant operators which can be neatly expressed in continuous euclidean clifford analysis. On discrete stokes and navierstokes equations in the plane klaus gurlebeck and angela hommel 4. The systematic study of the discrete counterparts to the operators, spaces and. Boundary value problems for dirac operators and maxwells. In this paper, we consider the discrete counterparts of these operators, i. In clifford analysis one also considers differential operators on upper half space, the disc, or hyperbola with respect to the hyperbolic, or poincare metric.
Clifford analysis progress in mathematical physics. Dynamical localization for discrete anderson dirac operators. The discrete dirac operator and the discrete generalized. In the end dirac operators on quadgraphs are constructed. Discrete clifford analysis is a discrete higherdimensional function theory which corresponds simultaneously to a refinement of discrete harmonic analysis and to a discrete counterpart of euclidean clifford analysis. May 24, 2007 read discrete dirac operators in clifford analysis, advances in applied clifford algebras on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. This cited by count includes citations to the following articles in scholar. Siam journal on numerical analysis society for industrial.
Poincare series in clifford analysis rolf soren krausshar 6. Dirac operators, sobolev spaces, besov spaces, maxwells equations. Historically, dirac operator was discovered by dirac who else. Discrete dirac operators in clifford analysis deepdyve.
These discrete objects are then shown to converge to the continuous. Dirac type operators are central to the field of clifford analysis, where recently there has been considerable interest in their discretization. May 24, 2007 we develop a constructive framework to define difference approximations of dirac operators which factorize the discrete laplacian. Pdf dirac operators in representation theory semantic scholar. Dirac himself remarked in one of his talks that his equation was more intelligent than its author. Eigenvalues of dirac and raritaschwinger operators doojin hong 14. Taylor series expansion in discrete clifford analysis. Discrete dirac operators on riemann surfaces and kasteleyn. Researchers at the cli ord research group at ghent university in belgium have also recently published a paper aimed at further development of the theory of discrete cli ord analysis 7. The clifford algebra clv,g is an algebra over r generated by the.
Weyl operator, dirac operator, slice regular functions abstract. In this paper we have further developed the hermitean theory by introducing socalled zonal. All of this provides potential links to atiyahsinger index theory and other aspects of geometric analysis involving dirac type operators. This paper adapts the techniques of finite element exterior calculus to study and discretize the abstract hodgedirac operator, which is a square root of the abstract hodgelaplace operator considered by arnold, falk, and winther bull. Discrete hardy spaces, journal of fourier analysis and. The morera problem in clifford algebras and the heisenberg group. It is easy to verify that these operators have the correct commutation relations. Pdf dirac operators in representation theory semantic. It is centered around the study of clifford algebra valued null solutions, called discrete monogenic functions, of a discrete dirac operator, i. Can compatible discretization, finite element methods, and. Fourier analysis of discrete dirac operators on the ntorus. The spectral geometry of operators of dirac and laplace type. It is wellknown that the index of a dirac operator with augmented atiyahpatodisinger aps boundary conditions on a compact manifold with boundary can be identi.
From a clifford analytic point of view a particular interesting object is a discrete analogue to the class of monogenic or regular functions. Finite element geometric calculus fegc fegc via hodge decomposition a viable approach is to discretize boundary value problems involving the multivectorvalued elds and dirac operators by using hodge decomposition followed by the existing techniques of feec. The dirac equation a great deal more was hidden in the dirac equation than the author had expected when he wrote it down in 1928. The discrete dirac operator and the discrete generalized weierstrass representation in pseudoeuclidean spaces dmitry zakharov in this thesis we consider the problem of. This paper describes work in progress, towards the formulation, implementation and testing of compatible discretization of di erential equations, using a combination of finite element exterior calculus and discrete. The eigenfunction magnitudes of the intrinsic and extrinsic dirac operators, discretized in a uni. We clarify the role of the rhombic condition in both theories. The discrete framework is based on a discrete dirac operator that combines both forward and backward difference operators and on the splitting of the basis elements. The linear theory is based on the discrete cauchyriemann equations, the nonlinear one is based on the notion of circle patterns. We show that an appropriate deformation of the spectral properties of the eigen. Bogdan balcerzak1, antoni pierzchalski2 1institute of mathematics, technical university of lodz, 2faculty of mathematics and computer science, university of lodz bogdan. Pdf we develop a constructive framework to define difference approximations of dirac operators which factorize the discrete laplacian.
The aim of this book is to unite the seemingly disparate topics of clifford algebras, analysis on manifolds and harmonic analysis. Rotations in discrete clifford analysis sciencedirect. Guangbin ren for the recent collaboration research on connections between clifford analysis and umbral calculus. We can also define our original dirac operators expressed in the spin operators. We prove a priori stability and convergence estimates, and show that several of the results in finite element exterior calculus can be recovered as corollaries of these new estimates. We develop a discrete version of clifford analysis, i. Read discrete dirac operators in clifford analysis, advances in applied clifford algebras on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. According to einsteins special relativity, a free particle of mass min r3 with momentum vector p p 1. We develop a constructive framework to define difference approximations of dirac operators which factorize the discrete laplacian. The operator d n extends the dirac and weyl operators to functions that can depend on all the coordinates of the algebra. Discrete clifford analysis is a higher dimensional discrete function theory based on skew weyl relations.
It should be added, however, that it was dirac who found most of the additional insights. We study the boundary behavior of discrete monogenic functions, i. Pdf discrete dirac operators in clifford analysis researchgate. Discrete dirac operators in clifford analysis springerlink. Clifford modulus and the dirac operators a lie algebroid over a manifold m is a vector bundle a over m with a homomorphism of vector bundles u a. The aim of this volume is to provide an overview of modern topics of the clifford analysis presented by specialists in this field and thus to honor the many years of work in and for the clifford analysis by w. Recently, there has been growing interest in developing a theory of discrete clifford analysis, based on lattice discretizations of dirac operators faustino et al. The discrete framework is based on a discrete dirac operator that combines both forward and backward difference operators and on. The machinery of generalized dirac operators makes the whole analysis simple, economical, and elegant. The square of the smallest, in absolute value, nonzero eigenvalue of the clas. Discrete dirac operators in clifford analysis article pdf available in advances in applied clifford algebras 173.
Lectures on dirac operators and index theory xianzhe dai january 7, 2015. N a as a discrete variable, but we can fix this by a simple linear. The underlying lattice is an arbitrary quasicrystallic rhombic tiling of a plane. The relative dirac operator is highly sensitive to features like surface texture. The discrete framework is based on a discrete dirac operator that combines both forward and backward difference operators and on the splitting of the basis elements into forward. A symmetric functional calculus for systems of operators of type. This resulting notion of discrete monogenic functions is compared with the notion of discrete holomorphic functions on quadgraphs. Hermitean clifford analysis then focusses on the simultaneous null solutions of two mutually conjugate hermitean dirac operators, naturally arising in the present context and being invariant under.
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