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| | I like Golf. Seems boring? Not at all!<br>I also to learn Danish in my spare time. <br>xunjie 核分裂の増加となりましたブランドは急速に主要な2番目と3番目の層の都市をつかむ手助けするために、 |
| '''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, <math>d+*d*</math> on a [[Riemannian manifold]], the Dirac operator in euclidean space and its inverse on <math>C_{0}^{\infty}(\mathbf{R}^{n})</math> and their conformal equivalents on the sphere, the [[Laplacian]] in euclidean ''n''-space and the [[Michael Atiyah|Atiyah]]–Singer–Dirac operator on a [[spin manifold]], Rarita–Schwinger/Stein–Weiss type operators, conformal Laplacians, spinorial Laplacians and Dirac operators on [[Complex spin structure|Spin<sup>''c''</sup>]] manifolds, systems of Dirac operators, the [[Paneitz operator]], Dirac operators on [[hyperbolic space]], the hyperbolic Laplacian and Weinstein equations.
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|
| |
|
| == Euclidean space ==
| | Here is my webpage ... [http://www.horseshop-online.ch/gallery/list/bottega/ ボッテガヴェネタ ボストン] |
| In Euclidean space the Dirac operator has the form
| |
| :<math>D=\sum_{j=1}^{n}e_{j}\frac{\partial}{\partial x_{j}}</math>
| |
| where ''e''<sub>1</sub>, ..., ''e''<sub>''n''</sub> is an orthonormal basis for '''R'''<sup>''n''</sup>, and '''R'''<sup>''n''</sup> is considered to be embedded in a complex [[Clifford algebra]], ''C''ℓ<sub>''n''</sub>('''C''') so that {{nowrap|1=''e''<sub>''j''</sub><sup>2</sup> = −1}}.
| |
| | |
| This gives
| |
| :<math>D^{2} = -\Delta_{n}</math>
| |
| where Δ<sub>''n''</sub> is the [[Laplacian]] in ''n''-euclidean space.
| |
| | |
| The [[fundamental solution]] to the euclidean Dirac operator is
| |
| :<math>G(x-y):=\frac{1}{\omega_{n}}\frac{x-y}{\|x-y\|^n}</math>
| |
| where ω<sub>''n''</sub> is the surface area of the unit sphere '''S'''<sup>''n''−1</sup>.
| |
| | |
| Note that
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| :<math>D\frac{1}{(n-2)\omega_{n}\|x-y\|^{n-2}}=G(x-y)</math>
| |
| where
| |
| :<math>\frac{1}{(n-2)\;\omega_{n}\;\|x-y\|^{n-2}}</math>
| |
| is the [[fundamental solution]] to [[Laplace's equation]] for {{nowrap|''n'' ≥ 3}}.
| |
| | |
| The most basic example of a Dirac operator is the [[Cauchy–Riemann operator]]
| |
| :<math>\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}</math>
| |
| 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's theorem (geometry)|Cauchy Theorem]], a [[Cauchy integral formula]], [[Morera's Theorem]], [[Taylor series]], [[Laurent series]] and [[Liouville's theorem (complex analysis)|Liouville Theorem]]. In this case the [[Cauchy kernel]] is ''G''(''x''−''y''). 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 ''x'' in euclidean space has a multiplicative inverse in the Clifford algebra, namely
| |
| :<math>-\frac{x}{\|x\|^{2}}\in\mathbf{R}^{n}.</math>
| |
| Up to a sign this inverse is the [[Kelvin inverse]] of ''x''. Solutions to the euclidean Dirac equation ''Df'' = 0 are called (left) monogenic functions. Monogenic functions are special cases of [[harmonic spinor]]s on a [[spin manifold]].
| |
| | |
| In 3 and 4 dimensions Clifford analysis is sometimes referred to as [[quaternion]]ic analysis. When {{nowrap|1=''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 transform]]s, [[Bergman kernel]]s, [[Szegő kernel]]s, [[Plemelj operator]]s, [[Hardy spaces]], a [[Kerzman–Stein formula]] and a Π, or [[Beurling–Ahlfors transform|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 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]], ''C''ℓ<sub>''n''</sub>. This is not the case though when we need to deal with the interaction between the [[Dirac operator]] and the [[Fourier transform]].
| |
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| ==The Fourier transform==
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| When we consider upper half space '''R'''<sup>''n'',+</sup> with boundary '''R'''<sup>''n''−1</sup>, the span of ''e''<sub>1</sub>, ..., ''e''<sub>''n''−1</sub>, under the [[Fourier transform]] the symbol of the Dirac operator
| |
| :<math>D_{n-1}=\sum_{j=1}^{n-1}\frac{\partial}{\partial x_{j}}</math>
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| is ''i''ζ where
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| :<math>\zeta=\zeta_{1}e_{1}+\ldots+\zeta_{n-1}e_{n-1}.</math>
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| In this setting the Plemelj formulas are
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| :<math>\pm\tfrac{1}{2}+G(x-y)|_{\mathbf{R}^{n-1}}</math>
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| and the symbols for these operators are, up to a sign,
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| :<math>\frac{1}{2} \left (1\pm i\frac{\zeta}{\|\zeta\|} \right ).</math>
| |
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| These are projection operators, otherwise known as mutually annihilating idempotents, on the space of ''C''ℓ<sub>''n''</sub>('''C''') valued square integrable functions on '''R'''<sup>''n''−1</sup>.
| |
| | |
| Note that
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| :<math>G|_{\mathbf{R}^{n}}=\sum_{j=1}^{n-1}e_{j}R_{j}</math>
| |
| where ''R<sub>j</sub>'' is the ''j''-th Riesz potential,
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| :<math>\frac{x_{j}}{\|x\|^{n}}.</math>
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| As the symbol of <math>G|_{\mathbf{R}^{n}}</math> is
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| :<math>\frac{i\zeta}{\|\zeta\|}</math>
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| it is easily determined from the Clifford multiplication that
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| :<math>\sum_{j=1}^{n-1}R_{j}^{2}=1.</math>
| |
| | |
| So the [[convolution operator]] <math>G|_{\mathbf{R}^{n}}</math> is a natural generalization to euclidean space of the [[Hilbert transform]].
| |
| | |
| Suppose ''U''′ is a domain in '''R'''<sup>''n''−1</sup> and ''g''(''x'') is a ''C''ℓ<sub>''n''</sub>('''C''') valued [[real analytic function]]. Then ''g'' has a Cauchy–Kovalevskaia extension to the [[Dirac equation]] on some neighborhood of ''U''′ in '''R'''<sup>''n''</sup>. The extension is explicitly given by
| |
| | |
| :<math>\sum_{j=0}^{\infty} \left (x_{n}e_{n}^{-1}D_{n-1} \right )^{j}g(x).</math>
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| | |
| When this extension is applied to the variable ''x'' in
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| :<math>e^{-i\langle x,\zeta\rangle} \left (\tfrac{1}{2} \left (1\pm i\frac{\zeta}{\|\zeta\|} \right ) \right )</math>
| |
| we get that
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| :<math>e^{-i\langle x,\zeta\rangle}</math>
| |
| is the restriction to '''R'''<sup>''n''−1</sup> of ''E''<sub>+</sub>+''E''<sub>−</sub> where ''E''<sub>+</sub> is a monogenic function in upper half space and ''E''<sub>−</sub> is a monogenic function in lower half space.
| |
| | |
| There is also a [[Paley–Wiener theorem]] in ''n''-euclidean space arising in Clifford analysis.
| |
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| ==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]]s and [[conformal manifold]]s which are simultaneously [[spin manifold]]s.
| |
| | |
| === Cayley transform (stereographic projection) ===
| |
| The [[Cayley transform]] or [[stereographic projection]] from '''R'''<sup>''n''</sup> to the unit sphere '''S'''<sup>''n''</sup> transforms the euclidean Dirac operator to a spherical Dirac operator ''D<sub>S</sub>''. Explicitly
| |
| :<math>D_{S}=x(\Gamma_{n}+\frac{n}{2})</math>
| |
| where Γ<sub>''n''</sub> is the spherical Beltrami–Dirac operator
| |
| :<math>\sum\nolimits_{1\leq i<j\leq n+1}e_{i}e_{j} \left (x_{i}\frac{\partial}{\partial x_{j}}-x_{j}\frac{\partial}{\partial x_{i}} \right )</math>
| |
| and ''x'' in '''S'''<sup>''n''</sup>.
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| | |
| The [[Cayley transform]] over ''n''-space is
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| :<math>y=C(x)=(e_{n+1}x+1)(x+e_{n+1})^{-1}, \qquad x \in \mathbf{R}^n.</math>
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| Its inverse is
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| :<math>x=(-e_{n+1}+1)(y-e_{n+1})^{-1}.</math>
| |
| For a function ''f''(''x'') defined on a domain ''U'' in ''n''-euclidean space and a solution to the [[Dirac equation]], then
| |
| :<math>J \left (C^{-1},y \right ) f \left (C^{-1}(y) \right )</math>
| |
| is annihilated by ''D<sub>S</sub>'', on ''C''(''U'') where
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| :<math>J(C^{-1},y)=\frac{y-e_{n+1}}{\|y-e_{n+1}\|^{n}}.</math>
| |
| | |
| Further
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| :<math>D_{S}(D_{S}-x)=\triangle_{S},</math>
| |
| the conformal Laplacian or Yamabe operator on '''S'''<sup>''n''</sup>. Explicitly
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| :<math>\triangle_{S}=-\triangle_{LB}+\tfrac{1}{4}n(n-2)</math>
| |
| where <math>\triangle_{LB}</math> is the [[Laplace–Beltrami operator]] on '''S'''<sup>''n''</sup>. The operator <math>\triangle_{S}</math> is, via the Cayley transform, conformally equivalent to the euclidean Laplacian. Also
| |
| :<math>D_s(D_S-x)(D_S-x)(D_S-2x)</math>
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| is the Paneitz operator,
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| :<math>-\triangle_{S}(\triangle_{S}+2),</math>
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| on the ''n''-sphere. Via the Cayley transform this operator is conformally equivalent to the bi-Laplacian, <math>\triangle_{n}^{2}</math>. These are all examples of operators of Dirac type.
| |
| | |
| === Möbius transform ===
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| A [[Möbius transform]] over ''n''-euclidean space can be expressed as
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| :<math>\frac{ax+b}{cx+d},</math>
| |
| where ''a'', ''b'', ''c'' and ''d'' ∈ ''C''ℓ<sub>''n''</sub> and satisfy certain constraints. The associated {{nowrap|2 × 2}} matrix is called an Ahlfors–Vahlen matrix. If
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| :<math>y=M(x)+\frac{ax+b}{cx+d}</math>
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| and ''Df''(''y'') = 0 then <math>J(M,x)f(M(x))</math> is a solution to the Dirac equation where
| |
| :<math>J(M,x)=\frac{\widetilde{cx+d}}{\|cx+d\|^{n}}</math>
| |
| and ~ is a basic [[antiautomorphism]] acting on the [[Clifford algebra]]. The operators ''D<sup>k</sup>'', or Δ<sub>''n''</sub><sup>''k''/2</sup> when ''k'' is even, exhibit similar covariances under [[Möbius transform]] including the [[Cayley transform]].
| |
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| When ''ax''+''b'' and ''cx''+''d'' are non-zero they are both members of the [[Clifford group]].
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| As
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| :<math>\frac{ax+b}{cx+d}=\frac{-ax-b}{-cx-d}</math>
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| then we have a choice in sign in defining ''J''(''M'', ''x''). This means that for a [[conformally flat manifold]] ''M'' we need a [[spin structure]] on ''M'' 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 discontinuously. A [[Dirac operator]] can be introduced in these contexts. These Dirac operators are special examples of Atiyah–Singer–Dirac operators.
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| ==The Atiyah–Singer–Dirac operator==
| |
| Given a [[spin manifold]], ''M'', with a [[spinor bundle]] ''S'' then given a smooth section ''s''(''x'') in ''S'' then in terms of a local orthonormal basis ''e''<sub>1</sub>(''x''), ..., ''e''<sub>''n''</sub>(''x'') of the tangent bundle of ''M'' the Atiyah–Singer–Dirac operator acting on ''s'' is defined to be
| |
| :<math>Ds(x)=\sum_{j=1}^{n}e_{j}(x)\tilde{\Gamma}_{e_{j}(x)}s(x)</math>
| |
| where <math>\widetilde{\Gamma}</math> is the lifting to ''S'' of the [[Levi-Civita connection]] on ''M''. When ''M'' is ''n''-euclidean space we return to the euclidean [[Dirac operator]].
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| | |
| From an Atiyah–Singer–Dirac operator ''D'' we have the [[Lichnerowicz formula]]
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| :<math>D^{2}=\Gamma^{*}\Gamma+\tfrac{\tau}{4}</math>
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| where τ is the scalar curvature on the [[manifold]], and Γ* is the adjoint of Γ. The operator ''D''<sup>2</sup> is known as the spinorial Laplacian.
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| If ''M'' is compact and τ ≥ 0 and τ > 0 somewhere then there are no non-trivial [[harmonic spinor]]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.
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| In the cases where the Atiyah–Singer–Dirac operator is invertible on the space of smooth spinor sections with compact support one may introduce
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| :<math>C(x,y):=D^{-1}*\delta_{y}, \qquad x \neq y \in M,</math>
| |
| where δ<sub>''y''</sub> is the [[Dirac delta function]] evaluated at ''y''. 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 spinor]]s. With this kernel much of what is described in the first section of this entry carries through for invertible Atiyah–Singer–Dirac operators.
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| | |
| 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.
| |
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| All of this provides potential links to Atiyah–Singer index theory and other aspects of geometric analysis involving Dirac type operators.
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| ==Hyperbolic Dirac type operators==
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| In Clifford analysis one also considers differential operators on upper half space, the disc, or hyperbola with respect to the hyperbolic, or [[Poincaré metric|Poincaré]] metric.
| |
| | |
| For upper half space one splits the [[Clifford algebra]], ''C''ℓ<sub>''n''</sub> into ''C''ℓ<sub>''n''−1</sub>+''C''ℓ<sub>''n''</sub>''e<sub>n</sub>''. So for ''a'' in ''C''ℓ<sub>''n''</sub> one may express ''a'' as ''b''+''ce<sub>n</sub>'' with ''a'', ''b'' in ''C''ℓ<sub>''n''−1</sub>. 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
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| :<math>Mf=Df+\frac{n-2}{x_{n}}Q(f)</math>.
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| In this case
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| :<math>M^{2}f=-\triangle_{n}P(f)+\frac{n-2}{x_{n}}\frac{\partial P(f)}{\partial x_{n}}- \left (\triangle_{n}Q(f)-\frac{n-2}{x_{n}}\frac{\partial Q(f)}{\partial x_{n}}+ \frac{n-2}{x_{n}^{2}}Q(f) \right )e_{n}</math>.
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| The operator
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| :<math>\triangle_{n}-\frac{n-2}{x_{n}}\frac{\partial}{\partial x_{n}}</math>
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| is the [[Laplacian]] with respect to the [[Poincaré metric]] while the other operator is an example of a Weinstein operator.
| |
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| The [[hyperbolic Laplacian]] is invariant under actions of the conformal group, while the hyperbolic Dirac operator is covariant under such actions.
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| | |
| ==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 ''R<sub>k</sub>'' 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 polynomials. When one refines this representation theory to the double covering Pin(''n'') of O(''n'') one replaces spaces of homogeneous harmonic polynomials by spaces of ''k'' homogeneous polynomial 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'''<sup>''n''</sup>, and ''u'' varies over '''R'''<sup>''n''</sup>. Further ''f''(''x'', ''u'') is a ''k''-monogenic polynomial in ''u''. Now apply the Dirac operator ''D<sub>x</sub>'' in ''x'' to ''f''(''x'', ''u''). Now as the Clifford algebra is not commutative ''D<sub>x</sub>f''(''x'', ''u'') then this function is no longer ''k'' monogenic but is a homogeneous harmonic polynomial in ''u''. Now for each harmonic polynomial ''h<sub>k</sub>'' homogeneous of degree ''k'' there is an Almansi–Fischer decomposition
| |
| :<math> h_{k}(x)=p_{k}(x)+xp_{k-1}(x) </math>
| |
| where ''p<sub>k</sub>'' and ''p<sub>k''−1</sub> are respectively ''k'' and ''k''−1 monogenic polynomials. Let ''P'' be the projection of ''h<sub>k</sub>'' to ''p<sub>k</sub>'' then the Rarita–Schwinger operator is defined to be ''PD<sub>k</sub>'', and it is denoted by ''R<sub>k</sub>''. Using Euler's Lemma one may determine that
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| :<math>D_{u}up_{k-1}(u)=(-n-2k+2)p_{k-1}.</math>
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| So
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| :<math>R_{k}=\left(I+\frac{1}{n+2k-2}uD_{u}\right)D_{x}.</math>
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| | |
| ==See also==
| |
| *[[Clifford algebra]]
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| *[[Complex spin structure]]
| |
| *[[Conformal manifold]]
| |
| *[[Conformally flat manifold]]
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| *[[Dirac operator]]
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| *[[Poincaré metric]]
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| *[[Spin group]]
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| *[[Spin structure]]
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| *[[Spinor bundle]]
| |
| | |
| ==References==
| |
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| *{{Citation | last1=Stein | first1=E. | last2=Weiss | first2=G. |title=Generalizations of the [[Cauchy Riemann equations]] and representations of the rotation group | journal=American Journal of Mathematics| volume=90 |pages= 163–196 | year=1968 | doi=10.2307/2373431 | issue=1 | publisher=The Johns Hopkins University Press | jstor=2373431}}.
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| *{{Citation | last=Sudbery | first=A. | title=[[Quaternion]]ic analysis | journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=85 | issue=02 |pages=199–225 | year=1979 | doi=10.1017/S0305004100055638|bibcode = 1979MPCPS..85..199S }}.
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| *{{citation| last=Tao| first=T. | title=[[Convolution operator]]s on Lipschitz graphs with harmonic kernels | journal=Advances in Applied Clifford Algebras |volume=6 |pages=207–218 | year=1996}}.
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| *{{Citation | last=Wu | first=S. | title=Well-posedness in [[Sobolev space]]s of the full water wave problem in 3-D | journal=[[Journal of the American Mathematical Society]] |volume=12 | issue=02 |pages=445–495 | year=1999 | doi=10.1090/S0894-0347-99-00290-8}}.
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| ==External links==
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| *[http://comp.uark.edu/~jryan/notes.doc Lecture notes on Dirac operators in analysis and geometry]
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| *[http://bib.mathematics.dk/preprint.php?lang=en&id=IMADA-PP-1997-53 Dirac operators and Clifford analysis on manifolds with boundary, by David Calderbank]
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| {{DEFAULTSORT:Clifford Analysis}}
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| [[Category:Differential geometry]]
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