Clifford analysis
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Clifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of
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. The original case which concerned Paul Dirac was to factorise forma ...
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, d+d on a Riemannian manifold, the Dirac operator in euclidean space and its inverse on C_^(\mathbf^) and their conformal equivalents on the sphere, the Laplacian in euclidean ''n''-space and the
Atiyah Atiyyah ( ar, عطية ''‘aṭiyyah''), which generally implies "something (money or goods given as regarded) received as a gift" or also means "present, gift, benefit, boon, favor, granting, giving"''.'' The name is also spelt Ateah, Atiyeh, ...
–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 In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
, the hyperbolic Laplacian and Weinstein equations.


Euclidean space

In Euclidean space the Dirac operator has the form :D=\sum_^e_\frac where ''e''1, ..., ''e''''n'' is an orthonormal basis for R''n'', and R''n'' is considered to be embedded in a complex Clifford algebra, Cl''n''(C) so that . This gives :D^ = -\Delta_ where Δ''n'' is the Laplacian in ''n''-euclidean space. The
fundamental solution In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not a ...
to the euclidean Dirac operator is :G(x-y):=\frac\frac where ω''n'' is the surface area of the unit sphere ''S''''n''−1. Note that :D\frac=G(x-y) where :\frac is the
fundamental solution In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not a ...
to Laplace's equation for . The most basic example of a Dirac operator is the Cauchy–Riemann operator :\frac+i\frac in the complex plane. Indeed, many basic properties of one variable
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
follow through for many first order Dirac type operators. In euclidean space this includes a Cauchy Theorem, a
Cauchy integral formula In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
,
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. Morera's theorem states that a continuous, complex-valued function ''f'' defined ...
,
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
, Laurent series and Liouville Theorem. In this case the
Cauchy kernel In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
is ''G''(''x''−''y''). The proof of the
Cauchy integral formula In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
is the same as in one complex variable and makes use of the fact that each non-zero vector ''x'' in euclidean space has a multiplicative inverse in the Clifford algebra, namely :-\frac\in\mathbf^. Up to a sign this inverse is the
Kelvin inverse The Kelvin transform is a device used in classical potential theory to extend the concept of a harmonic function, by allowing the definition of a function which is 'harmonic at infinity'. This technique is also used in the study of subharmonic and s ...
of ''x''. Solutions to the euclidean Dirac equation ''Df'' = 0 are called (left) monogenic functions. Monogenic functions are special cases of
harmonic spinor 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. The original case which concerned Paul Dirac was to factorise formally ...
s on a spin manifold. In 3 and 4 dimensions Clifford analysis is sometimes referred to as quaternionic analysis. When , 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 transform Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
s,
Bergman kernel In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is the reproducing kernel for the Hilbert space ( RKHS) of all square integrable holomorphic functions on a domain ''D'' in C''n''. In de ...
s, Szegő kernels,
Plemelj operator Josip Plemelj (December 11, 1873 – May 22, 1967) was a Slovene mathematician, whose main contributions were to the theory of analytic functions and the application of integral equations to potential theory. He was the first chancellor of t ...
s,
Hardy spaces In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain Space (mathematics), spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, becaus ...
, a Kerzman–Stein formula and a Π, or Beurling–Ahlfors, transform. These have all found applications in solving boundary value problems, including moving boundary value problems,
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 : T(f)(x) = \int K(x,y)f(y) \, dy, who ...
s and classic harmonic analysis. In particular Clifford analysis has been used to solve, in certain
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 together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s, the full water wave problem in 3D. 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, Cl''n''. This is not the case though when we need to deal with the interaction between the
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. The original case which concerned Paul Dirac was to factorise forma ...
and the Fourier transform.


The Fourier transform

When we consider upper half space R''n'',+ with boundary R''n''−1, the span of ''e''1, ..., ''e''''n''−1, under the Fourier transform the symbol of the Dirac operator :D_ = \sum_^ \frac \partial is ''iζ'' where :\zeta=\zeta_1 e_1 +\cdots+ \zeta_e_. In this setting the Plemelj formulas are :\pm\tfrac+G(x-y), _ and the symbols for these operators are, up to a sign, :\frac \left (1\pm i\frac \right ). These are projection operators, otherwise known as mutually annihilating idempotents, on the space of Cl''n''(C) valued square integrable functions on R''n''−1. Note that :G, _=\sum_^ e_j R_j where ''Rj'' is the ''j''-th Riesz potential, :\frac. As the symbol of G, _ is :\frac it is easily determined from the Clifford multiplication that :\sum_^ R_j^2=1. So the
convolution operator In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
G, _ is a natural generalization to euclidean space of the
Hilbert transform In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the functi ...
. Suppose ''U''′ is a domain in R''n''−1 and ''g''(''x'') is a Cl''n''(C) valued
real analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
. Then ''g'' has a Cauchy–Kovalevskaia extension to the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
on some neighborhood of ''U''′ in R''n''. The extension is explicitly given by :\sum_^\infty \left (x_n e_n^D_ \right )^j g(x). When this extension is applied to the variable ''x'' in :e^ \left (\tfrac \left (1\pm i\frac \right ) \right ) we get that :e^ is the restriction to R''n''−1 of ''E''+ + ''E'' where ''E''+ is a monogenic function in upper half space and ''E'' is a monogenic function in lower half space. There is also a
Paley–Wiener theorem In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (189 ...
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 Möbius transformations. Consequently, this holds true for Dirac operators on
conformally flat manifold A (pseudo-) Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation. In practice, the metric g of the manifold M has to be conformal to the flat metric \eta, i.e., the ...
s and
conformal manifold In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two di ...
s which are simultaneously spin manifolds.


Cayley transform (stereographic projection)

The
Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform ...
or stereographic projection from R''n'' to the unit sphere ''S''''n'' transforms the euclidean Dirac operator to a spherical Dirac operator ''DS''. Explicitly :D_S=x \left(\Gamma_n + \frac n 2 \right) where Γ''n'' is the spherical Beltrami–Dirac operator :\sum\nolimits_e_e_ \left (x_\frac-x_\frac \right ) and ''x'' in ''S''''n''. The
Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform ...
over ''n''-space is :y=C(x)=(e_x+1)(x+e_)^, \qquad x \in \mathbf^n. Its inverse is :x=(-e_+1)(y-e_)^. For a function ''f''(''x'') defined on a domain ''U'' in ''n''-euclidean space and a solution to the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
, then :J(C^,y) f(C^(y)) is annihilated by ''DS'', on ''C''(''U'') where :J(C^,y)=\frac. Further :D_S(D_S-x)=\triangle_S, the conformal Laplacian or Yamabe operator on ''S''''n''. Explicitly :\triangle_S = -\triangle_+\tfrac 1 4 n(n-2) where \triangle_ is the
Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named af ...
on ''S''''n''. The operator \triangle_S is, via the Cayley transform, conformally equivalent to the euclidean Laplacian. Also :D_s(D_S-x)(D_S-x)(D_S-2x) is the Paneitz operator, :-\triangle_S(\triangle_S+2), on the ''n''-sphere. Via the Cayley transform this operator is conformally equivalent to the bi-Laplacian, \triangle_n^2. These are all examples of operators of Dirac type.


Möbius transform

A
Möbius transform Moebius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Theodor Möbius (1821–1890), German philologist * Karl Möbius (1825–1908), German zoologist and ecologist * Pau ...
over ''n''-euclidean space can be expressed as :\frac, where ''a'', ''b'', ''c'' and ''d'' ∈ Cl''n'' and satisfy certain constraints. The associated matrix is called an Ahlfors–Vahlen matrix. If :y=M(x)+\frac and ''Df''(''y'') = 0 then J(M,x)f(M(x)) is a solution to the Dirac equation where :J(M,x)=\frac and ~ is a basic
antiautomorphism In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is a bijective antihomomorphism, i.e. an antiisomorphism, from a set to itself. From ...
acting on the Clifford algebra. The operators ''Dk'', or Δ''n''''k''/2 when ''k'' is even, exhibit similar covariances under
Möbius transform Moebius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Theodor Möbius (1821–1890), German philologist * Karl Möbius (1825–1908), German zoologist and ecologist * Pau ...
including the
Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform ...
. When ''ax''+''b'' and ''cx''+''d'' are non-zero they are both members of the
Clifford group In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomp ...
. As :\frac=\frac then we have a choice in sign in defining ''J''(''M'', ''x''). This means that for a
conformally flat manifold A (pseudo-) Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation. In practice, the metric g of the manifold M has to be conformal to the flat metric \eta, i.e., the ...
''M'' we need a
spin 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. Spin structures have wide applications to mathematical ...
on ''M'' in order to define a
spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...
on whose sections we can allow a Dirac operator to act. Explicit simple examples include the ''n''-cylinder, the
Hopf manifold In complex geometry, a Hopf manifold is obtained as a quotient of the complex vector space (with zero deleted) (^n\backslash 0) by a free action of the group \Gamma \cong of integers, with the generator \gamma of \Gamma acting by holomorphic co ...
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 discontinuously. A
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. The original case which concerned Paul Dirac was to factorise forma ...
can be introduced in these contexts. These Dirac operators are special examples of Atiyah–Singer–Dirac operators.


Atiyah–Singer–Dirac operator

Given a spin manifold ''M'' with a
spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...
''S'' and a smooth section ''s''(''x'') in ''S'' then, in terms of a local orthonormal basis ''e''1(''x''), ..., ''e''''n''(''x'') of the tangent bundle of ''M'', the Atiyah–Singer–Dirac operator acting on ''s'' is defined to be :Ds(x)=\sum_^e_(x)\tilde_s(x) , where \widetilde is the
spin connection In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz tr ...
, the lifting to ''S'' of the Levi-Civita connection on ''M''. When ''M'' is ''n''-euclidean space we return to the euclidean
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. The original case which concerned Paul Dirac was to factorise forma ...
. From an Atiyah–Singer–Dirac operator ''D'' we have the
Lichnerowicz formula The Lichnerowicz formula (also known as the Lichnerowicz–Weitzenböck 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 ...
:D^=\Gamma^\Gamma+\tfrac , where ''τ'' is the
scalar curvature In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometr ...
on the
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, and Γ is the adjoint of Γ. The operator ''D''2 is known as the spinorial Laplacian. If ''M'' is compact and and somewhere then there are no non-trivial
harmonic spinor 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. The original case which concerned Paul Dirac was to factorise formally ...
s on the manifold. This is Lichnerowicz' theorem. It is readily seen 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 ''D'' is invertible 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 :C(x,y):=D^*\delta_, \qquad x \neq y \in M, where ''δ''''y'' is the Dirac delta function evaluated at ''y''. This gives rise to a
Cauchy kernel In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
, which is the
fundamental solution In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not a ...
to this Dirac operator. From this one may obtain a
Cauchy integral formula In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary o ...
for
harmonic spinor 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. The original case which concerned Paul Dirac was to factorise formally ...
s. 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.


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
Poincaré 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 ...
. For upper half space one splits the Clifford algebra, Cl''n'' into Cl''n''−1 + Cl''n''−1''en''. So for ''a'' in Cl''n'' one may express ''a'' as ''b'' + ''cen'' with ''a'', ''b'' in Cl''n''−1. One then has projection operators ''P'' and ''Q'' defined as follows ''P''(''a'') = ''b'' and ''Q''(''a'') = ''c''. The Hodge–Dirac operator acting on a function ''f'' with respect to the hyperbolic metric in upper half space is now defined to be :Mf=Df+\fracQ(f). In this case :M^f=-\triangle_P(f)+\frac\frac- \left (\triangle_Q(f)-\frac\frac+ \fracQ(f) \right )e_. The operator :\triangle_-\frac\frac is the Laplacian with respect to the
Poincaré 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 ...
while the other operator is an example of a Weinstein operator. The
hyperbolic Laplacian Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...
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 group The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies. History and shareholding The PIN Group originally traded under ...
s. The operator ''Rk'' is a conformally covariant first order differential operator. Here ''k'' = 0, 1, 2, .... When ''k'' = 0, the Rarita–Schwinger operator is just the Dirac operator. In representation theory for the orthogonal group, O(''n'') it is common to consider functions taking values in spaces of homogeneous
harmonic polynomial In mathematics, in abstract algebra, a multivariate polynomial over a field such that the Laplacian of is zero is termed a harmonic polynomial. The harmonic polynomials form a vector subspace of the vector space of polynomials over the field. In ...
s. When one refines this
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
to the double covering Pin(''n'') of O(''n'') one replaces spaces of homogeneous harmonic polynomials by spaces of ''k''
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
solutions to the Dirac equation, otherwise known as ''k'' monogenic polynomials. One considers a function ''f''(''x'', ''u'') where ''x'' in ''U'', a domain in R''n'', and ''u'' varies over R''n''. Further ''f''(''x'', ''u'') is a ''k''-monogenic polynomial in ''u''. Now apply the Dirac operator ''Dx'' in ''x'' to ''f''(''x'', ''u''). Now as the Clifford algebra is not commutative ''Dxf''(''x'', ''u'') then this function is no longer ''k'' monogenic but is a homogeneous harmonic polynomial in ''u''. Now for each harmonic polynomial ''hk'' homogeneous of degree ''k'' there is an Almansi–Fischer decomposition : h_(x)=p_(x)+xp_(x) where ''p''''k'' and ''p''''k''−1 are respectively ''k'' and ''k''−1 monogenic polynomials. Let ''P'' be the projection of ''h''''k'' to ''p''''k'' then the Rarita–Schwinger operator is defined to be ''PDk'', and it is denoted by ''Rk''. Using Euler's Lemma one may determine that :D_up_(u)=(-n-2k+2)p_. So :R_=\left(I+\fracuD_\right)D_.


Conferences and Journals

There is a vibrant and interdisciplinary community around Clifford and Geometric Algebras with a wide range of applications. The main conferences in this subject include th
International Conference on Clifford Algebras and their Applications in Mathematical Physics (ICCA)
an
Applications of Geometric Algebra in Computer Science and Engineering (AGACSE)
series. A main publication outlet is the Springer journal
Advances in Applied Clifford Algebras ''Advances in Applied Clifford Algebras'' is a peer-reviewed scientific journal that publishes original research papers and also notes, expository and survey articles, book reviews, reproduces abstracts and also reports on conferences and workshops ...
.


See also

* Clifford algebra *
Complex spin 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. Spin structures have wide applications to mathematica ...
*
Conformal manifold In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two di ...
*
Conformally flat manifold A (pseudo-) Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation. In practice, the metric g of the manifold M has to be conformal to the flat metric \eta, i.e., the ...
*
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. The original case which concerned Paul Dirac was to factorise forma ...
*
Poincaré 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 ...
*
Spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a ...
*
Spin 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. Spin structures have wide applications to mathematical ...
*
Spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...


References

*. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *.


External links


Lecture notes on Dirac operators in analysis and geometry
* {{Industrial and applied mathematics Differential geometry