Clifford analysis
Encyclopedia
Clifford analysis, using Clifford algebra
s named after William Kingdon Clifford
, is the study of Dirac operator
s, and Dirac type operators in analysis and geometry, together with their applications. Examples of Dirac type operators include, but are not limited to, the Hodge-Dirac operator,
on a Riemannian manifold
, the Dirac operator in euclidean space and its inverse on
and their conformal equivalents on the sphere, the Laplacian in euclidean n-space and the Atiyah
-Singer-Dirac operator on a spin manifold, Rarita-Schwinger/ Stein-Weiss type operators, conformal Laplacians, spinorial Laplacians and Dirac operators on Spinc manifolds, systems of Dirac operators, the Paneitz operator
, Dirac operators on hyperbolic space
, the hyperbolic Laplacian and Weinstein equations.
where e1...en is an orthonormal basis for Rn, and Rn is considered to be embedded in a complex Clifford algebra
, Cℓn(C) so that ej2 = -1.
This gives
where Δn is the Laplacian in n-euclidean space.
The fundamental solution
to the euclidean Dirac operator is
where ωn is the surface area of the unit sphere Sn-1.
Note that
where
is the fundamental solution
to Laplace's equation
for n ≥ 3.
The most basic example of a Dirac operator is the Cauchy-Riemann operator
in the complex plane. Indeed many basic properties of one variable complex analysis
follow through for many first order Dirac type operators. In euclidean space this includes a Cauchy Theorem
, a Cauchy integral formula, Morera's Theorem
, Taylor series
, Laurent series
and Liouville Theorem
. In this case the Cauchy kernel is
. The proof of the Cauchy integral formula is the same as in one complex variable and makes use of the fact that each non-zero vector 
in euclidean space has a multiplicative inverse in the Clifford algebra, namely 
. Up to a sign this inverse is the Kelvin inverse of 
. Solutions to the euclidean Dirac equation 
are called (left) monogenic functions. Monogenic functions are special cases of harmonic spinors on a spin manifold.
In 3 and 4 dimensions Clifford analysis is sometimes referred to as quaternion
ic analysis. When n=4 the Dirac operator is sometimes referred to as the Cauchy-Riemann-Fueter operator. Further some aspects of Clifford analysis are referred to as hypercomplex analysis.
Clifford analysis has analogues of Cauchy transforms, Bergman kernel
s, Szegő kernel
s, Plemelj operators, Hardy spaces, a Kerzman-Stein formula and a Π, or Beurling-Ahlfors, transform. These have all found applications in solving boundary value problem
s, including moving boundary value problems, singular integral
s and classic harmonic analysis. In particular Clifford analysis has been used to solve, in certain Sobolev space
s, the full water wave problem in 3-D. This method works in all dimensions greater than 2.
Much of Clifford analysis works if we replace the complex Clifford algebra
by a real Clifford algebra
,
. This is not the case though when we need to deal with the interaction between the Dirac operator
and the Fourier transform
.
with boundary 
, the span of 
, under the Fourier transform
the symbol of the Dirac operator
is 
where 
.
In this setting the Plemelj formulas are
and the symbols for these operators are, up to a sign, 
. These are projection operators, otherwise known as mutually annihilating idempotents, on the space of Cln(C) valued square integrable functions on Rn-1.
Note that
where 
is the 
-th Riesz potential, 
. As the symbol of 
is 
it is easily determined from the Clifford multiplication that 
. So the convolution operator 
is a natural generalization to euclidean space of the Hilbert transform
.
Suppose
is a domain in 
and 
is a 
valued real analytic function. Then 
has a Cauchy-Kovalevskaia extension to the Dirac equation
on some neighborhood of
in 
. The extension is explicitly given by 
.
When this extension is applied to the variable
in 
we get that 
is the restriction to 
of 
where 
is a monogenic function in upper half space and 
is a monogenic function in lower half space.
There is also a Paley-Weiner theorem in n-euclidean space arising in Clifford analysis.
or stereographic projection
from
to the unit sphere 
transforms the euclidean Dirac operator to a spherical Dirac operator 
. Explicitly
where
is the spherical Beltrami-Dirac operator
and
.
The Cayley transform
over n-space is
where
. Its inverse is
.
For a function
defined on a domain 
in n-euclidean space and a solution to the Dirac equation
, then
is annihilated by 
, on 
where 
Further
, the conformal Laplacian or Yamabe operator on 
. Explicitly 
where 
is the Laplace-Beltrami operator
on
. The operator 
is, via the Cayley transform, conformally equivalent to the euclidean Laplacian. Also 
is the Paneitz operator, 
, on the n-sphere. Via the Cayley transform this operator is conformally equivalent to the bi-Laplacian, 
. These are all examples of operators of Dirac type.
where 
, 
, 
and 
∈ Cln and satisfy certain constraints. The associated 2 × 2 matrix is called an Ahlfors-Vahlen matrix. If 
and 
then 
is a solution to the Dirac equation where 
and 
is a basic antiautomorphism acting on the Clifford algebra
. The operators
, or Δnk/2 when 
is even, exhibit similar covariances under Moebius transform including the Cayley transform
.
When
and 
are non-zero they are both members of the Clifford group.
As
then we have a choice in sign in defining 
. This means that for a conformally flat manifold 
we need a spin structure
on
in order to define a spinor bundle on whose sections we can allow a Dirac operator to act. Explicit simple examples include the n-cylinder, the Hopf manifold obtained from n-euclidean space minus the origin, and generalizations of k-handled toruses obtained from upper half space by factoring it out by actions of generalized modular groups acting on upper half space totally discontinuosly. A Dirac operator
can be introduced in these contexts. These Dirac operators are special examples of Atiyah-Singer-Dirac operators.
, with a spinor bundle 
then given a smooth section 
in 
then in terms of a local orthonormal basis e1(x)... en(x) of the tangent bundle of 
the Atiyah-Singer-Dirac operator acting on 
is defined to be
where
is the lifting to 
of the Levi-Civita connection
on
. When 
is n-euclidean space we return to the euclidean Dirac operator
.
From an Atiyah-Singer-Dirac operator
we have the Lichnerowicz formula
where
is the scalar curvature on the manifold
, and
is the adjoint of 
. The operator 
is known as the spinorial Laplacian.
If
is compact and 
and 
somewhere then there are no non-trivial harmonic spinors on the manifold. This is Lichnerowicz' Theorem. It is readily seen that that Lichnerowicz' Theorem is a generalization of Liouville's Theorem from one variable complex analysis. This allows us to note that over the space of smooth spinor sections
the operator
is invertible for such a manifold.
In the cases where the Atiyah-Singer-Dirac operator is invertible on the space of smooth spinor sections with compact support one may introduce
where
with 
and 
is the Dirac delta function
evaluated at
. This gives rise to a Cauchy kernel, which is the fundamental solution
to this Dirac operator. From this one may obtain a Cauchy integral formula for harmonic spinors. With this kernel much of what is described in the first section of this entry carries through for invertible Atiyah-Singer-Dirac operators.
Using Stokes' theorem
, or otherwise, one can further determine that under a conformal change of metric the Dirac operators associated to each metric are proportional to each other, and consequently so are their inverses, if they exist.
All of this provides potential links to Atiyah-Singer index theory and other aspects of geometric analysis involving Dirac type operators.
metric.
For upper half space one splits the Clifford algebra
,
into 
. So for 
one may express 
as 
with 
, 
. One then has projection operators 
and 
defined as follows 
and 
. The Hodge-Dirac operator acting on a function 
with respect to the hyperbolic metric in upper half space is now defined to be
.
In this case
.
The operator
is the Laplacian with respect to the Poincaré metric
while the other operator is an example of a Weinstein operator.
The hyperbolic Laplacian is invariant under actions of the conformal group, while the hyperbolic Dirac operator is covariant under such actions.
is a conformally covariant first order differential operator. Here 
. When 
then the Rarita-Schwinger operator is just the Dirac operator. In representation theory for the orthogonal group, 
it is common to consider functions taking values in spaces of homogeneous harmonic
polynomials. When one refines this representation theory to the double covering
of 
one replaces spaces of homogeneous harmonic polynomials by spaces of k homogeneous polynomial solutions to the Dirac equation, otherwise known as 
monogenic polynomials. One considers a function 
where 
a domain in 
and 
varies over 
. Further 
is a k-monogenic polynomial in 
. Now apply the Dirac operator 
in 
to 
. Now as the Clifford algebra is not commutative 
then this function is no longer 
monogenic but is a homogeneous harmonic polynomial in 
. Now for each harmonic polynomial 
homogeneous of degree 
there is an Almansi-Fischer decomposition 
where 
and 
are respectively 
and 
monogenic polynomials. Let 
be the projection of 
to 
then the Rarita Schwinger operator is defined to be 
, and it is denoted by 
. Using Euler's Lemma one may determine that 
. So 
.
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
s named after William Kingdon Clifford
William Kingdon Clifford
William Kingdon Clifford FRS was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour, with interesting applications in contemporary mathematical physics...
, is the study of Dirac operator
Dirac operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian...
s, and Dirac type operators in analysis and geometry, together with their applications. Examples of Dirac type operators include, but are not limited to, the Hodge-Dirac operator,
on a Riemannian manifoldRiemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
, the Dirac operator in euclidean space and its inverse on
and their conformal equivalents on the sphere, the Laplacian in euclidean n-space and the AtiyahAtiyah
Atiyah is the surname of several notable people including:*Sir Michael Atiyah, British mathematician, brother of Patrick*Patrick Atiyah, English barrister and legal writer, brother of Michael*Edward Atiyah, Lebanese born writer, father of Michael and Patrick...
-Singer-Dirac operator on a spin manifold, Rarita-Schwinger/ Stein-Weiss type operators, conformal Laplacians, spinorial Laplacians and Dirac operators on Spinc manifolds, systems of Dirac operators, the Paneitz operator
Paneitz operator
In the mathematical field of differential geometry, the Paneitz operator is a fourth-order differential operator defined on a Riemannian manifold of dimension n. It is named after Stephen Paneitz, who discovered it in 1983, and whose preprint was later published posthumously in...
, Dirac operators on hyperbolic space
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
, the hyperbolic Laplacian and Weinstein equations.
Euclidean space
In Euclidean space the Dirac operator has the formwhere e1...en is an orthonormal basis for Rn, and Rn is considered to be embedded in a complex Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
, Cℓn(C) so that ej2 = -1.
This gives
where Δn is the Laplacian in n-euclidean space.
The fundamental solution
Fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function...
to the euclidean Dirac operator is
where ωn is the surface area of the unit sphere Sn-1.
Note that
where
is the fundamental solutionFundamental solution
In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function...
to Laplace's equation
Laplace's equation
In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...
for n ≥ 3.
The most basic example of a Dirac operator is the Cauchy-Riemann operator
in the complex plane. Indeed many basic properties of one variable complex analysisComplex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
follow through for many first order Dirac type operators. In euclidean space this includes a Cauchy Theorem
Cauchy theorem
Several theorems are named after Augustin-Louis Cauchy. Cauchy theorem may mean:*Cauchy's integral theorem in complex analysis, also Cauchy's integral formula*Cauchy's mean value theorem in real analysis, an extended form of the mean value theorem...
, a Cauchy integral formula, Morera's Theorem
Morera's theorem
In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic....
, Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
, Laurent series
Laurent series
In mathematics, the Laurent series of a complex function f is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where...
and Liouville Theorem
Liouville's theorem (complex analysis)
In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that |f| ≤ M for all z in C is constant.The theorem is considerably improved by...
. In this case the Cauchy kernel is
. The proof of the Cauchy integral formula is the same as in one complex variable and makes use of the fact that each non-zero vector in euclidean space has a multiplicative inverse in the Clifford algebra, namely . Up to a sign this inverse is the Kelvin inverse of . Solutions to the euclidean Dirac equation are called (left) monogenic functions. Monogenic functions are special cases of harmonic spinors on a spin manifold.In 3 and 4 dimensions Clifford analysis is sometimes referred to as quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
ic analysis. When n=4 the Dirac operator is sometimes referred to as the Cauchy-Riemann-Fueter operator. Further some aspects of Clifford analysis are referred to as hypercomplex analysis.
Clifford analysis has analogues of Cauchy transforms, Bergman kernel
Bergman kernel
In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is a reproducing kernel for the Hilbert space of all square integrable holomorphic functions on a domain D in Cn....
s, Szegő kernel
Szegő kernel
In the mathematical study of several complex variables, the Szegő kernel is an integral kernel that gives rise to a reproducing kernel on a natural Hilbert space of holomorphic functions...
s, Plemelj operators, Hardy spaces, a Kerzman-Stein formula and a Π, or Beurling-Ahlfors, transform. These have all found applications in solving boundary value problem
Boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions...
s, including moving boundary value problems, singular integral
Singular integral
In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator...
s and classic harmonic analysis. In particular Clifford analysis has been used to solve, in certain Sobolev space
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...
s, the full water wave problem in 3-D. This method works in all dimensions greater than 2.
Much of Clifford analysis works if we replace the complex Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
by a real Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
,
. This is not the case though when we need to deal with the interaction between the Dirac operatorDirac operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian...
and the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
.
The Fourier Transform
When we consider upper half space with boundary , the span of , under the Fourier transformFourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
the symbol of the Dirac operator
is where .In this setting the Plemelj formulas are
and the symbols for these operators are, up to a sign, . These are projection operators, otherwise known as mutually annihilating idempotents, on the space of Cln(C) valued square integrable functions on Rn-1.Note that
where is the -th Riesz potential, . As the symbol of is it is easily determined from the Clifford multiplication that . So the convolution operator is a natural generalization to euclidean space of the Hilbert transformHilbert transform
In mathematics and in signal processing, the Hilbert transform is a linear operator which takes a function, u, and produces a function, H, with the same domain. The Hilbert transform is named after David Hilbert, who first introduced the operator in order to solve a special case of the...
.
Suppose
is a domain in and is a valued real analytic function. Then has a Cauchy-Kovalevskaia extension to the Dirac equationDirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
on some neighborhood of
in . The extension is explicitly given by .When this extension is applied to the variable
in we get that is the restriction to of where is a monogenic function in upper half space and is a monogenic function in lower half space.There is also a Paley-Weiner theorem in n-euclidean space arising in Clifford analysis.
Conformal Structure
Many Dirac type operators have a covariance under conformal change in metric. This is true for the Dirac operator in euclidean space, and the Dirac operator on the sphere under Moebius transformations. Consequently this holds true for Dirac operators on conformally flat manifolds and conformal manifolds which are simultaneously spin manifolds.Cayley transform (stereographic projection)
The Cayley transformCayley transform
In mathematics, the Cayley transform, named after Arthur Cayley, has a cluster of related meanings. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. In complex analysis, the Cayley transform is a conformal mapping in...
or stereographic projection
Stereographic projection
The stereographic projection, in geometry, is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it...
from
to the unit sphere transforms the euclidean Dirac operator to a spherical Dirac operator . Explicitlywhere
is the spherical Beltrami-Dirac operatorand
.The Cayley transform
Cayley transform
In mathematics, the Cayley transform, named after Arthur Cayley, has a cluster of related meanings. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. In complex analysis, the Cayley transform is a conformal mapping in...
over n-space is
where
. Its inverse is.For a function
defined on a domain in n-euclidean space and a solution to the Dirac equationDirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
, then
is annihilated by , on where Further
, the conformal Laplacian or Yamabe operator on . Explicitly where is the Laplace-Beltrami operatorLaplace-Beltrami operator
In differential geometry, the Laplace operator, named after Pierre-Simon Laplace, can be generalized to operate on functions defined on surfaces in Euclidean space and, more generally, on Riemannian and pseudo-Riemannian manifolds. This more general operator goes by the name Laplace–Beltrami...
on
. The operator is, via the Cayley transform, conformally equivalent to the euclidean Laplacian. Also is the Paneitz operator, , on the n-sphere. Via the Cayley transform this operator is conformally equivalent to the bi-Laplacian, . These are all examples of operators of Dirac type.Moebius transform
A Moebius transform over n-euclidean space can be expressed as where , , and ∈ Cln and satisfy certain constraints. The associated 2 × 2 matrix is called an Ahlfors-Vahlen matrix. If and then is a solution to the Dirac equation where and is a basic antiautomorphism acting on the Clifford algebraClifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
. The operators
, or Δnk/2 when is even, exhibit similar covariances under Moebius transform including the Cayley transformCayley transform
In mathematics, the Cayley transform, named after Arthur Cayley, has a cluster of related meanings. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. In complex analysis, the Cayley transform is a conformal mapping in...
.
When
and are non-zero they are both members of the Clifford group.As
then we have a choice in sign in defining . This means that for a conformally flat manifold we need a spin structureSpin structure
In differential geometry, a spin structure on an orientable Riemannian manifold \,allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry....
on
in order to define a spinor bundle on whose sections we can allow a Dirac operator to act. Explicit simple examples include the n-cylinder, the Hopf manifold obtained from n-euclidean space minus the origin, and generalizations of k-handled toruses obtained from upper half space by factoring it out by actions of generalized modular groups acting on upper half space totally discontinuosly. A Dirac operatorDirac operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian...
can be introduced in these contexts. These Dirac operators are special examples of Atiyah-Singer-Dirac operators.
The Atiyah-Singer-Dirac Operator
Given a spin manifold,, with a spinor bundle then given a smooth section in then in terms of a local orthonormal basis e1(x)... en(x) of the tangent bundle of the Atiyah-Singer-Dirac operator acting on is defined to bewhere
is the lifting to of the Levi-Civita connectionLevi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
on
. When is n-euclidean space we return to the euclidean Dirac operatorDirac operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian...
.
From an Atiyah-Singer-Dirac operator
we have the Lichnerowicz formulaLichnerowicz formula
The Lichnerowicz formula is a fundamental equation in the analysis of spinors on pseudo-Riemannian manifolds. In dimension 4, it forms a piece of Seiberg–Witten theory and other aspects of gauge theory. It is named after noted mathematician André Lichnerowicz who proved it in 1963...
where
is the scalar curvature on the manifoldManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
, and
is the adjoint of . The operator is known as the spinorial Laplacian.If
is compact and and somewhere then there are no non-trivial harmonic spinors on the manifold. This is Lichnerowicz' Theorem. It is readily seen that that Lichnerowicz' Theorem is a generalization of Liouville's Theorem from one variable complex analysis. This allows us to note that over the space of smooth spinor sectionsthe operator
is invertible for such a manifold.In the cases where the Atiyah-Singer-Dirac operator is invertible on the space of smooth spinor sections with compact support one may introduce
where
with and is the Dirac delta functionDirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
evaluated at
. This gives rise to a Cauchy kernel, which is the fundamental solutionFundamental solution
In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function...
to this Dirac operator. From this one may obtain a Cauchy integral formula for harmonic spinors. With this kernel much of what is described in the first section of this entry carries through for invertible Atiyah-Singer-Dirac operators.
Using Stokes' theorem
Stokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
, or otherwise, one can further determine that under a conformal change of metric the Dirac operators associated to each metric are proportional to each other, and consequently so are their inverses, if they exist.
All of this provides potential links to Atiyah-Singer index theory and other aspects of geometric analysis involving Dirac type operators.
Hyperbolic Dirac Type Operators
In Clifford analysis one also considers differential operators on upper half space, the disc, or hyperbola with respect to the hyperbolic, or PoincaréPoincaré
Several members of the French Poincaré family have been successful in public and scientific life:* Henri Poincaré , physicist, mathematician and philosopher of science* Lucien Poincaré , physicist, brother of Raymond and cousin of Henri...
metric.
For upper half space one splits the Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
,
into . So for one may express as with , . One then has projection operators and defined as follows and . The Hodge-Dirac operator acting on a function with respect to the hyperbolic metric in upper half space is now defined to be.In this case
.The operator
is the Laplacian with respect to the Poincaré metricPoincaré metric
In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces.There are three equivalent...
while the other operator is an example of a Weinstein operator.
The hyperbolic Laplacian is invariant under actions of the conformal group, while the hyperbolic Dirac operator is covariant under such actions.
Rarita-Schwinger/ Stein-Weiss Operators
Rarita-Schwinger operators, also known as Stein-Weiss operators, arise in representation theory for the Spin and Pin groups. The operator is a conformally covariant first order differential operator. Here . When then the Rarita-Schwinger operator is just the Dirac operator. In representation theory for the orthogonal group, it is common to consider functions taking values in spaces of homogeneous harmonicpolynomials. When one refines this representation theory to the double covering
of one replaces spaces of homogeneous harmonic polynomials by spaces of k homogeneous polynomial solutions to the Dirac equation, otherwise known as monogenic polynomials. One considers a function where a domain in and varies over . Further is a k-monogenic polynomial in . Now apply the Dirac operator in to . Now as the Clifford algebra is not commutative then this function is no longer monogenic but is a homogeneous harmonic polynomial in . Now for each harmonic polynomial homogeneous of degree there is an Almansi-Fischer decomposition where and are respectively and monogenic polynomials. Let be the projection of to then the Rarita Schwinger operator is defined to be , and it is denoted by . Using Euler's Lemma one may determine that . So .See also
- Clifford algebraClifford algebraIn mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
- Complex spin structure
- Conformal manifold
- Conformally flat manifold
- Dirac operatorDirac operatorIn mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian...
- Poincaré metricPoincaré metricIn mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces.There are three equivalent...
- Spin structureSpin structureIn differential geometry, a spin structure on an orientable Riemannian manifold \,allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry....
- Spinor bundle