As in euclidean clifford analysis, the construction of discrete monogenic functions, that is, functions in the kernel of the discrete dirac operator, is a. Lectures on dirac operators and index theory ucsb math. A symmetric functional calculus for systems of operators of type. N a as a discrete variable, but we can fix this by a simple linear. Fourier analysis of discrete dirac operators on the ntorus. The underlying lattice is an arbitrary quasicrystallic rhombic tiling of a plane. The relative dirac operator is highly sensitive to. 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. The discrete dirac operator and the discrete generalized weierstrass representation in pseudoeuclidean spaces dmitry zakharov in this thesis we consider the problem of. Rotations in discrete clifford analysis sciencedirect. The morera problem in clifford algebras and the heisenberg group. This resulting notion of discrete monogenic functions is compared with the notion of discrete holomorphic functions on quadgraphs. All of this provides potential links to atiyahsinger index theory and other aspects of geometric analysis involving dirac type operators. In this paper we work in the split discrete clifford analysis setting, i.
Dirac type operators are central to the field of clifford analysis, where recently there has been considerable interest in their discretization. Boundary value problems for dirac operators and maxwells. 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. From a clifford analytic point of view a particular interesting object is a discrete analogue to the class of monogenic or regular functions. The machinery of generalized dirac operators makes the whole analysis simple, economical, and elegant. The relative dirac operator is highly sensitive to features like surface texture. We show how the space mk of discrete spherical monogenics homogeneous of degree k, is. Dynamical localization for discrete anderson dirac operators.
To this end one needs to construct discrete dirac operators which factorize the discrete star or crosslaplacian. Discrete dirac operators on riemann surfaces and kasteleyn matrices received april 19, 2010 abstract. Discrete dirac operators in clifford analysis springerlink. Weyl operator, dirac operator, slice regular functions abstract. Fourier analysis of discrete dirac operators on the n torus nelson faustino new developments on discrete dirac operators. The square of the smallest, in absolute value, nonzero eigenvalue of the clas. 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. Dirac operators, sobolev spaces, besov spaces, maxwells equations. We show that an appropriate deformation of the spectral properties of the eigen. We study the boundary behavior of discrete monogenic functions, i.
We develop a constructive framework to define difference approximations of dirac operators which factorize the discrete laplacian. Discrete dirac operators in clifford analysis core. Pdf dirac operators in representation theory semantic scholar. The aim of this book is to unite the seemingly disparate topics of clifford algebras, analysis on manifolds and harmonic analysis. 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 kogutsusskind fermion and the staggered fermion actions are derived directly from the dirackahler fermion formulated by the clifford product. Discrete dirac operators on riemann surfaces and kasteleyn. 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. These discrete objects are then shown to converge to the continuous. Discrete hardy spaces, journal of fourier analysis and. Hjbpodbased feedback design for the optimal control of evolution problems reconstruction of surface and stochastic dynamics from a planar projection of trajectories. Pdf dirac operators in representation theory semantic. The lattice qcd action with dirackahler matter fermion is also derived via an inner product defined by the clifford product. 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. Discrete clifford analysis is a higher dimensional discrete function theory based on skew weyl relations. 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. 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. 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.
The linear theory is based on the discrete cauchyriemann equations, the nonlinear one is based on the notion of circle patterns. Siam journal on numerical analysis society for industrial. 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. Similar frameworks for the dirac kahler operator date to the 1980s 3, 35.
It is centered around the study of clifford algebra valued null solutions, called discrete monogenic functions, of a discrete dirac operator, i. Ams proceedings of the american mathematical society. Clifford algebras and dirac operators in harmonic analysis. Poincare series in clifford analysis rolf soren krausshar 6. 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. It provides an overview of different directions of modern clifford analysis. The eigenfunction magnitudes of the intrinsic and extrinsic dirac operators, discretized in a uni. Bogdan balcerzak1, antoni pierzchalski2 1institute of mathematics, technical university of lodz, 2faculty of mathematics and computer science, university of lodz bogdan. Guangbin ren for the recent collaboration research on connections between clifford analysis and umbral calculus. Discrete dirac operators in clifford analysis article pdf available in advances in applied clifford algebras 173. 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. In this paper, we consider the discrete counterparts of these operators, i. The clifford algebra clv,g is an algebra over r generated by the. In the end dirac operators on quadgraphs are constructed.
Dirac himself remarked in one of his talks that his equation was more intelligent than its author. 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. We establish dynamical localization for random dirac operators on the ddimensional lattice, with d2f1. It should be added, however, that it was dirac who found most of the additional insights. The abstract hodgedirac operator and its stable discretization arxiv. Recently, there has been growing interest in developing a theory of discrete clifford analysis, based on lattice discretizations of dirac operators faustino et al. We develop a discrete version of clifford analysis, i. In this paper we have further developed the hermitean theory by introducing socalled zonal. Pdf discrete dirac operators in clifford analysis researchgate. 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. Harmonic analysis for general first order differential operators in lipschitz domains emilio.
The laplace and dirac operators are rotation invariant operators which can be neatly expressed in continuous euclidean clifford analysis. Diractype operators are central to the field of clifford analysis, where recently there has been considerable. In order to convey some idea of the kind of results we obtained, consider the following example see corollary 3. Clifford analysis progress in mathematical physics. The cauchykovalevskaya extension theorem in discrete.
The ones marked may be different from the article in the profile. On discrete stokes and navierstokes equations in the plane klaus gurlebeck and angela hommel 4. 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. Pdf linear and nonlinear theories of discrete analytic. 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. We can also define our original dirac operators expressed in the spin operators. Similar frameworks for the dirackahler operator date to the 1980s 3, 35. 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. 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.
Dirac operators in riemannian geometry thomas friedrich. Eigenvalues of dirac and raritaschwinger operators doojin hong 14. Dirac operators, boundary value problems, and the bcalculus. Lectures on dirac operators and index theory xianzhe dai january 7, 2015. Topics in clifford analysis special volume in honor of. We clarify the role of the rhombic condition in both theories.
Abstract two discretizations, linear and nonlinear, of basic notions of the complex analysis are considered. The discrete framework is based on a discrete dirac operator that combines both forward and backward difference operators and on. In clifford analysis one also considers differential operators on upper half space, the disc, or hyperbola with respect to the hyperbolic, or poincare metric. Dirac operators and nilpotent lie algebra cohomology. It is easy to verify that these operators have the correct commutation relations. The operator d n extends the dirac and weyl operators to functions that can depend on all the coordinates of the algebra. Any smooth manifold m defines a lie algebroid, where a tm. The spectral geometry of operators of dirac and laplace type. Can compatible discretization, finite element methods, and. 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. Hermitean clifford analysis focusses on monogenic functions taking values in a complex clifford algebra or in a complex spinor space. This cited by count includes citations to the following articles in scholar. Historically, dirac operator was discovered by dirac who else.
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. According to einsteins special relativity, a free particle of mass min r3 with momentum vector p p 1. The discrete dirac operator and the discrete generalized. In this text, friedrich examines the dirac operator on riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold. Clifford algebras and dirac operators in harmonic analysis by. Taylor series expansion in discrete clifford analysis. Fourier analysis of discrete dirac operators on the n. Pdf we develop a constructive framework to define difference approximations of dirac operators which factorize the discrete laplacian. 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. In this paper, we establish a cauchykovalevskaya extension theorem for discrete. Discrete dirac operators in clifford analysis deepdyve. The systematic study of the discrete counterparts to the operators, spaces and.
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