Discrete dirac operators in clifford analysis deepdyve. Ams proceedings of the american mathematical society. In this paper we have further developed the hermitean theory by introducing socalled zonal. Pal saha institute of nuclear physics 1af bidhannagar, calcutta 700064, india abstract this is a pedagogical article which discusses various kinds of fermion. Pdf discrete dirac operators in clifford analysis researchgate. 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 abstract hodgedirac operator and its stable discretization arxiv. The relative dirac operator is highly sensitive to. The discrete framework is based on a discrete dirac operator that combines both forward and backward difference operators and on.
May 24, 2007 we develop a constructive framework to define difference approximations of dirac operators which factorize the discrete laplacian. We show how the space mk of discrete spherical monogenics homogeneous of degree k, is. 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. In the end dirac operators on quadgraphs are constructed. The square of the smallest, in absolute value, nonzero eigenvalue of the clas. 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 consider the discrete counterparts of these operators, i. On discrete stokes and navierstokes equations in the plane klaus gurlebeck and angela hommel 4. 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 develop a constructive framework to define difference approximations of dirac operators which factorize the discrete laplacian. The machinery of generalized dirac operators makes the whole analysis simple, economical, and elegant. It is centered around the study of clifford algebra valued null solutions, called discrete monogenic functions, of a discrete dirac operator, i.
The clifford algebra clv,g is an algebra over r generated by the. 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. Rotations in discrete clifford analysis sciencedirect. Hermitean clifford analysis focusses on monogenic functions taking values in a complex clifford algebra or in a complex spinor space. The relative dirac operator is highly sensitive to features like surface texture. Taylor series expansion in discrete clifford analysis. 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. Pdf dirac operators in representation theory semantic. Discrete dirac operators on riemann surfaces and kasteleyn. Fourier analysis of discrete dirac operators on the ntorus.
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. Eigenvalues of dirac and raritaschwinger operators doojin hong 14. Discrete dirac operators in clifford analysis article pdf available in advances in applied clifford algebras 173. Dirac operators, boundary value problems, and the bcalculus. It provides an overview of different directions of modern clifford analysis. The systematic study of the discrete counterparts to the operators, spaces and. 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 eigenfunction magnitudes of the intrinsic and extrinsic dirac operators, discretized in a uni. Boundary value problems for dirac operators and maxwells. To this end one needs to construct discrete dirac operators which factorize the discrete star or crosslaplacian.
It should be added, however, that it was dirac who found most of the additional insights. Guangbin ren for the recent collaboration research on connections between clifford analysis and umbral calculus. The linear theory is based on the discrete cauchyriemann equations, the nonlinear one is based on the notion of circle patterns. Can compatible discretization, fem, and discrete cli ord analysis be fruitfully combined. 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. Clifford algebras and dirac operators in harmonic analysis by. 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. Diractype operators are central to the field of clifford analysis, where recently there has been considerable. Discrete dirac operators on riemann surfaces and kasteleyn matrices received april 19, 2010 abstract. Pdf we develop a constructive framework to define difference approximations of dirac operators which factorize the discrete laplacian. These discrete objects are then shown to converge to the continuous. Pdf dirac operators in representation theory semantic scholar. Hjbpodbased feedback design for the optimal control of evolution problems reconstruction of surface and stochastic dynamics from a planar projection of trajectories.
Similar frameworks for the dirac kahler operator date to the 1980s 3, 35. We show that an appropriate deformation of the spectral properties of the eigen. 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. 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. It is easy to verify that these operators have the correct commutation relations. We clarify the role of the rhombic condition in both theories. The morera problem in clifford algebras and the heisenberg group.
The aim of this book is to unite the seemingly disparate topics of clifford algebras, analysis on manifolds and harmonic analysis. 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. 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. 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. Harmonic analysis for general first order differential operators in lipschitz domains emilio. 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. 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. We develop a discrete version of clifford analysis, i. Discrete dirac operators in clifford analysis core. Dirac operators, sobolev spaces, besov spaces, maxwells equations. Poincare series in clifford analysis rolf soren krausshar 6. Clifford algebras and dirac operators in harmonic analysis. The underlying lattice is an arbitrary quasicrystallic rhombic tiling of a plane.
Lectures on dirac operators and index theory xianzhe dai january 7, 2015. The discrete dirac operator and the discrete generalized weierstrass representation in pseudoeuclidean spaces dmitry zakharov in this thesis we consider the problem of. Fourier analysis of discrete dirac operators on the n torus nelson faustino new developments on discrete dirac operators. 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.
From a clifford analytic point of view a particular interesting object is a discrete analogue to the class of monogenic or regular functions. All of this provides potential links to atiyahsinger index theory and other aspects of geometric analysis involving dirac type operators. Discrete hardy spaces, journal of fourier analysis and. We study the boundary behavior of discrete monogenic functions, i. We can also define our original dirac operators expressed in the spin operators. Topics in clifford analysis special volume in honor of. 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. According to einsteins special relativity, a free particle of mass min r3 with momentum vector p p 1. Lectures on dirac operators and index theory ucsb math. 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. Clifford analysis progress in mathematical physics. Any smooth manifold m defines a lie algebroid, where a tm.
N a as a discrete variable, but we can fix this by a simple linear. Can compatible discretization, finite element methods, and. Dirac himself remarked in one of his talks that his equation was more intelligent than its author. Bogdan balcerzak1, antoni pierzchalski2 1institute of mathematics, technical university of lodz, 2faculty of mathematics and computer science, university of lodz bogdan. In this paper we work in the split discrete clifford analysis setting, i.
Weyl operator, dirac operator, slice regular functions abstract. Historically, dirac operator was discovered by dirac who else. The laplace and dirac operators are rotation invariant operators which can be neatly expressed in continuous euclidean clifford analysis. Dynamical localization for discrete anderson dirac operators. Discrete dirac operators in clifford analysis springerlink. Dirac operators in riemannian geometry thomas friedrich. Dirac operators, boundary value problems, and the bcalculus paul loya abstract. 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.
In this text, friedrich examines the dirac operator on riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold. The discrete dirac operator and the discrete generalized. This resulting notion of discrete monogenic functions is compared with the notion of discrete holomorphic functions on quadgraphs. The lattice qcd action with dirackahler matter fermion is also derived via an inner product defined by the clifford product. 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. Pdf linear and nonlinear theories of discrete analytic.
Dirac operators and nilpotent lie algebra cohomology. In clifford analysis one also considers differential operators on upper half space, the disc, or hyperbola with respect to the hyperbolic, or poincare metric. The ones marked may be different from the article in the profile. Similar frameworks for the dirackahler operator date to the 1980s 3, 35. 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.
Discrete clifford analysis is a higher dimensional discrete function theory based on skew weyl relations. In order to convey some idea of the kind of results we obtained, consider the following example see corollary 3. The cauchykovalevskaya extension theorem in discrete. As in euclidean clifford analysis, the construction of discrete monogenic functions, that is, functions in the kernel of the discrete dirac operator, is a. A symmetric functional calculus for systems of operators of type. The operator d n extends the dirac and weyl operators to functions that can depend on all the coordinates of the algebra. This cited by count includes citations to the following articles in scholar. Siam journal on numerical analysis society for industrial.
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