Clifford analysis

<h2 id="euclidean-space">Euclidean space</h2>
In Euclidean space the Dirac operator has the form

D
        =
        
          ∑
          
            j
            =
            1
          
          
            n
          
        
        
          e
          
            j
          
        
        
          
            ∂
            
              ∂
              
                x
                
                  j
                
              
            
          
        
      
    
    {\displaystyle D=\sum _{j=1}^{n}e_{j}{\frac {\partial }{\partial x_{j}}}}

where e1, ..., en is an orthonormal basis for Rn, and Rn is considered to be embedded in a complex <a href="/facts/Clifford_algebra/qnqSCz5h">Clifford algebra</a>, Cln(C) so that ej2 = −1.
This gives

D
          
            2
          
        
        =
        −
        
          Δ
          
            n
          
        
      
    
    {\displaystyle D^{2}=-\Delta _{n}}

where Δn is the <a href="/facts/Laplacian/kov9u3PT">Laplacian</a> in n-euclidean space.
The <a href="/facts/Fundamental_solution/hdu8vXJa">fundamental solution</a> to the euclidean Dirac operator is

G
        (
        x
        −
        y
        )
        :=
        
          
            1
            
              ω
              
                n
              
            
          
        
        
          
            
              x
              −
              y
            
            
              ‖
              x
              −
              y
              
                ‖
                
                  n
                
              
            
          
        
      
    
    {\displaystyle G(x-y):={\frac {1}{\omega _{n}}}{\frac {x-y}{\|x-y\|^{n}}}}

where ωn is the surface area of the unit sphere Sn−1.
Note that

D
        
          
            1
            
              (
              n
              −
              2
              )
              
                ω
                
                  n
                
              
              ‖
              x
              −
              y
              
                ‖
                
                  n
                  −
                  2
                
              
            
          
        
        =
        G
        (
        x
        −
        y
        )
      
    
    {\displaystyle D{\frac {1}{(n-2)\omega _{n}\|x-y\|^{n-2}}}=G(x-y)}

where

1
            
              (
              n
              −
              2
              )
              
              
                ω
                
                  n
                
              
              
              ‖
              x
              −
              y
              
                ‖
                
                  n
                  −
                  2
                
              
            
          
        
      
    
    {\displaystyle {\frac {1}{(n-2)\;\omega _{n}\;\|x-y\|^{n-2}}}}

is the <a href="/facts/Fundamental_solution/hdu8vXJa">fundamental solution</a> to <a href="/facts/Laplace%2527s_equation/8n1YkmFS">Laplace's equation</a> for n ≥ 3.
The most basic example of a Dirac operator is the <a href="/facts/Cauchy%25E2%2580%2593Riemann_operator/SWkIHflU">Cauchy–Riemann operator</a>

∂
            
              ∂
              x
            
          
        
        +
        i
        
          
            ∂
            
              ∂
              y
            
          
        
      
    
    {\displaystyle {\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}}

in the complex plane. Indeed, many basic properties of one variable <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a> follow through for many first order Dirac type operators. In euclidean space this includes a <a href="/facts/Cauchy%2527s_theorem_(geometry)/JrvSbBtn">Cauchy Theorem</a>, a <a href="/facts/Cauchy_integral_formula/4B35nkvV">Cauchy integral formula</a>, <a href="/facts/Morera%2527s_theorem/vqw1Pvu8">Morera's theorem</a>, <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a>, <a href="/facts/Laurent_series/sSMevFU8">Laurent series</a> and <a href="/facts/Liouville%2527s_theorem_(complex_analysis)/I97twMBQ">Liouville Theorem</a>. In this case the <a href="/facts/Cauchy_kernel/4B35nkvV">Cauchy kernel</a> is G(x−y). The proof of the <a href="/facts/Cauchy_integral_formula/4B35nkvV">Cauchy integral formula</a> 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

−
        
          
            x
            
              ‖
              x
              
                ‖
                
                  2
                
              
            
          
        
        ∈
        
          
            R
          
          
            n
          
        
        .
      
    
    {\displaystyle -{\frac {x}{\|x\|^{2}}}\in \mathbf {R} ^{n}.}

Up to a sign this inverse is the <a href="/facts/Kelvin_inverse/WjW2wwY4">Kelvin inverse</a> of x. Solutions to the euclidean Dirac equation Df = 0 are called (left) monogenic functions. Monogenic functions are special cases of <a href="/facts/Harmonic_spinor/1kdGnoTt">harmonic spinors</a> on a <a href="/facts/Spin_manifold/BF78kFz8">spin manifold</a>.
In 3 and 4 dimensions Clifford analysis is sometimes referred to as <a href="/facts/Quaternion/T3tnu7mX">quaternionic</a> 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, <a href="/facts/Bergman_kernel/IeSkXbGR">Bergman kernels</a>, <a href="/facts/Szeg%25C5%2591_kernel/519k3t4d">Szegő kernels</a>, Plemelj operators, <a href="/facts/Hardy_spaces/aN45dgx7">Hardy spaces</a>, a Kerzman–Stein formula and a Π, or <a href="/facts/Beurling%25E2%2580%2593Ahlfors_transform/uz0QuIfD">Beurling–Ahlfors</a>, transform. These have all found applications in solving <a href="/facts/Boundary_value_problem/QEfmUqmP">boundary value problems</a>, including moving boundary value problems, <a href="/facts/Singular_integral/MIMjXeIw">singular integrals</a> and <a href="/facts/Classic_harmonic_analysis/5HaSt00U">classic harmonic analysis</a>. In particular Clifford analysis has been used to solve, in certain <a href="/facts/Sobolev_space/34qxWCyr">Sobolev spaces</a>, 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 <a href="/facts/Clifford_algebra/qnqSCz5h">Clifford algebra</a> by a real <a href="/facts/Clifford_algebra/qnqSCz5h">Clifford algebra</a>, Cln. This is not the case though when we need to deal with the interaction between the <a href="/facts/Dirac_operator/1kdGnoTt">Dirac operator</a> and the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a>.

<h2 id="the-fourier-transform">The Fourier transform</h2>
When we consider upper half space Rn,+ with boundary Rn−1, the span of e1, ..., en−1, under the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a> the symbol of the Dirac operator

D
          
            n
            −
            1
          
        
        =
        
          ∑
          
            j
            =
            1
          
          
            n
            −
            1
          
        
        
          
            ∂
            
              ∂
              
                x
                
                  j
                
              
            
          
        
      
    
    {\displaystyle D_{n-1}=\sum _{j=1}^{n-1}{\frac {\partial }{\partial x_{j}}}}

is iζ where

ζ
        =
        
          ζ
          
            1
          
        
        
          e
          
            1
          
        
        +
        ⋯
        +
        
          ζ
          
            n
            −
            1
          
        
        
          e
          
            n
            −
            1
          
        
        .
      
    
    {\displaystyle \zeta =\zeta _{1}e_{1}+\cdots +\zeta _{n-1}e_{n-1}.}

In this setting the <a href="/facts/Sokhotski%25E2%2580%2593Plemelj_theorem/xoxGrXR6">Plemelj formulas</a> are

±
        
          
            
              1
              2
            
          
        
        +
        G
        (
        x
        −
        y
        )
        
          
            |
          
          
            
              
                R
              
              
                n
                −
                1
              
            
          
        
      
    
    {\displaystyle \pm {\tfrac {1}{2}}+G(x-y)|_{\mathbf {R} ^{n-1}}}

and the symbols for these operators are, up to a sign,

1
            2
          
        
        
          (
          
            1
            ±
            i
            
              
                ζ
                
                  ‖
                  ζ
                  ‖
                
              
            
          
          )
        
        .
      
    
    {\displaystyle {\frac {1}{2}}\left(1\pm i{\frac {\zeta }{\|\zeta \|}}\right).}

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

G
        
          
            |
          
          
            
              
                R
              
              
                n
              
            
          
        
        =
        
          ∑
          
            j
            =
            1
          
          
            n
            −
            1
          
        
        
          e
          
            j
          
        
        
          R
          
            j
          
        
      
    
    {\displaystyle G|_{\mathbf {R} ^{n}}=\sum _{j=1}^{n-1}e_{j}R_{j}}

where Rj is the j-th Riesz potential,

x
              
                j
              
            
            
              ‖
              x
              
                ‖
                
                  n
                
              
            
          
        
        .
      
    
    {\displaystyle {\frac {x_{j}}{\|x\|^{n}}}.}

As the symbol of 
 
 
 
 G
 
 
 |
 
 
 
 
 R
 
 
 n
 
 
 
 
 
 
 {\displaystyle G|_{\mathbf {R} ^{n}}}
 
 is

i
              ζ
            
            
              ‖
              ζ
              ‖
            
          
        
      
    
    {\displaystyle {\frac {i\zeta }{\|\zeta \|}}}

it is easily determined from the Clifford multiplication that

∑
          
            j
            =
            1
          
          
            n
            −
            1
          
        
        
          R
          
            j
          
          
            2
          
        
        =
        1.
      
    
    {\displaystyle \sum _{j=1}^{n-1}R_{j}^{2}=1.}

So the <a href="/facts/Convolution_operator/PVPrdz9J">convolution operator</a> 
 
 
 
 G
 
 
 |
 
 
 
 
 R
 
 
 n
 
 
 
 
 
 
 {\displaystyle G|_{\mathbf {R} ^{n}}}
 
 is a natural generalization to euclidean space of the <a href="/facts/Hilbert_transform/RoQaC8MS">Hilbert transform</a>.
Suppose U′ is a domain in Rn−1 and g(x) is a Cln(C) valued <a href="/facts/Real_analytic_function/JhL3z8FP">real analytic function</a>. Then g has a <a href="/facts/Cauchy%25E2%2580%2593Kowalevski_theorem/TBqUWjxU">Cauchy–Kovalevskaia extension</a> to the <a href="/facts/Dirac_equation/DzFaXS4p">Dirac equation</a> on some neighborhood of U′ in Rn. The extension is explicitly given by

∑
          
            j
            =
            0
          
          
            ∞
          
        
        
          
            (
            
              
                x
                
                  n
                
              
              
                e
                
                  n
                
                
                  −
                  1
                
              
              
                D
                
                  n
                  −
                  1
                
              
            
            )
          
          
            j
          
        
        g
        (
        x
        )
        .
      
    
    {\displaystyle \sum _{j=0}^{\infty }\left(x_{n}e_{n}^{-1}D_{n-1}\right)^{j}g(x).}

When this extension is applied to the variable x in

e
          
            −
            i
            ⟨
            x
            ,
            ζ
            ⟩
          
        
        
          (
          
            
              
                
                  1
                  2
                
              
            
            
              (
              
                1
                ±
                i
                
                  
                    ζ
                    
                      ‖
                      ζ
                      ‖
                    
                  
                
              
              )
            
          
          )
        
      
    
    {\displaystyle e^{-i\langle x,\zeta \rangle }\left({\tfrac {1}{2}}\left(1\pm i{\frac {\zeta }{\|\zeta \|}}\right)\right)}

we get that

e
          
            −
            i
            ⟨
            x
            ,
            ζ
            ⟩
          
        
      
    
    {\displaystyle e^{-i\langle x,\zeta \rangle }}

is the restriction to Rn−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 <a href="/facts/Paley%25E2%2580%2593Wiener_theorem/02pKjt9F">Paley–Wiener theorem</a> in n-Euclidean space arising in Clifford analysis.

<h2 id="conformal-structure">Conformal structure</h2>
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 <a href="/facts/Conformally_flat_manifold/QNAk4QA0">conformally flat manifolds</a> and <a href="/facts/Conformal_manifold/tuU181cE">conformal manifolds</a> which are simultaneously <a href="/facts/Spin_manifold/BF78kFz8">spin manifolds</a>.

<h3>Cayley transform (stereographic projection)</h3>
The <a href="/facts/Cayley_transform/2jbbGdA9">Cayley transform</a> or <a href="/facts/Stereographic_projection/oVyMrWqR">stereographic projection</a> from Rn to the unit sphere Sn transforms the euclidean Dirac operator to a spherical Dirac operator DS. Explicitly

D
          
            S
          
        
        =
        x
        
          (
          
            
              Γ
              
                n
              
            
            +
            
              
                n
                2
              
            
          
          )
        
      
    
    {\displaystyle D_{S}=x\left(\Gamma _{n}+{\frac {n}{2}}\right)}

where Γn is the spherical Beltrami–Dirac operator

∑
 
 1
 ≤
 i
 <
 j
 ≤
 n
 +
 1
 
 
 
 e
 
 i
 
 
 
 e
 
 j
 
 
 
 (
 
 
 x
 
 i
 
 
 
 
 ∂
 
 ∂
 
 x
 
 j
 
 
 
 
 
 −
 
 x
 
 j
 
 
 
 
 ∂
 
 ∂
 
 x
 
 i
 
 
 
 
 
 
 )
 
 
 
 {\displaystyle \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)}

and x in Sn.
The <a href="/facts/Cayley_transform/2jbbGdA9">Cayley transform</a> over n-space is

y
        =
        C
        (
        x
        )
        =
        (
        
          e
          
            n
            +
            1
          
        
        x
        +
        1
        )
        (
        x
        +
        
          e
          
            n
            +
            1
          
        
        
          )
          
            −
            1
          
        
        ,
        
        x
        ∈
        
          
            R
          
          
            n
          
        
        .
      
    
    {\displaystyle y=C(x)=(e_{n+1}x+1)(x+e_{n+1})^{-1},\qquad x\in \mathbf {R} ^{n}.}

Its inverse is

x
        =
        (
        −
        
          e
          
            n
            +
            1
          
        
        +
        1
        )
        (
        y
        −
        
          e
          
            n
            +
            1
          
        
        
          )
          
            −
            1
          
        
        .
      
    
    {\displaystyle x=(-e_{n+1}+1)(y-e_{n+1})^{-1}.}

For a function f(x) defined on a domain U in n-euclidean space and a solution to the <a href="/facts/Dirac_equation/DzFaXS4p">Dirac equation</a>, then

J
        (
        
          C
          
            −
            1
          
        
        ,
        y
        )
        f
        (
        
          C
          
            −
            1
          
        
        (
        y
        )
        )
      
    
    {\displaystyle J(C^{-1},y)f(C^{-1}(y))}

is annihilated by DS, on C(U) where

J
        (
        
          C
          
            −
            1
          
        
        ,
        y
        )
        =
        
          
            
              y
              −
              
                e
                
                  n
                  +
                  1
                
              
            
            
              ‖
              y
              −
              
                e
                
                  n
                  +
                  1
                
              
              
                ‖
                
                  n
                
              
            
          
        
        .
      
    
    {\displaystyle J(C^{-1},y)={\frac {y-e_{n+1}}{\|y-e_{n+1}\|^{n}}}.}

Further

D
          
            S
          
        
        (
        
          D
          
            S
          
        
        −
        x
        )
        =
        
          △
          
            S
          
        
        ,
      
    
    {\displaystyle D_{S}(D_{S}-x)=\triangle _{S},}

the conformal Laplacian or Yamabe operator on Sn. Explicitly

△
          
            S
          
        
        =
        −
        
          △
          
            L
            B
          
        
        +
        
          
            
              1
              4
            
          
        
        n
        (
        n
        −
        2
        )
      
    
    {\displaystyle \triangle _{S}=-\triangle _{LB}+{\tfrac {1}{4}}n(n-2)}

where 
 
 
 
 
 △
 
 L
 B
 
 
 
 
 {\displaystyle \triangle _{LB}}
 
 is the <a href="/facts/Laplace%25E2%2580%2593Beltrami_operator/AJmyUpWz">Laplace–Beltrami operator</a> on Sn. The operator 
 
 
 
 
 △
 
 S
 
 
 
 
 {\displaystyle \triangle _{S}}
 
 is, via the Cayley transform, conformally equivalent to the euclidean Laplacian. Also

D
          
            s
          
        
        (
        
          D
          
            S
          
        
        −
        x
        )
        (
        
          D
          
            S
          
        
        −
        x
        )
        (
        
          D
          
            S
          
        
        −
        2
        x
        )
      
    
    {\displaystyle D_{s}(D_{S}-x)(D_{S}-x)(D_{S}-2x)}

is the Paneitz operator,

−
        
          △
          
            S
          
        
        (
        
          △
          
            S
          
        
        +
        2
        )
        ,
      
    
    {\displaystyle -\triangle _{S}(\triangle _{S}+2),}

on the n-sphere. Via the Cayley transform this operator is conformally equivalent to the bi-Laplacian, 
 
 
 
 
 △
 
 n
 
 
 2
 
 
 
 
 {\displaystyle \triangle _{n}^{2}}
 
. These are all examples of operators of Dirac type.

<h3>Möbius transform</h3>
A <a href="/facts/M%25C3%25B6bius_transform/xDnSrPp6">Möbius transform</a> over n-euclidean space can be expressed as

a
              x
              +
              b
            
            
              c
              x
              +
              d
            
          
        
        ,
      
    
    {\displaystyle {\frac {ax+b}{cx+d}},}

where a, b, c and d ∈ Cln and satisfy certain constraints. The associated 2 × 2 matrix is called an Ahlfors–Vahlen matrix. If

y
        =
        M
        (
        x
        )
        +
        
          
            
              a
              x
              +
              b
            
            
              c
              x
              +
              d
            
          
        
      
    
    {\displaystyle y=M(x)+{\frac {ax+b}{cx+d}}}

and Df(y) = 0 then 
 
 
 
 J
 (
 M
 ,
 x
 )
 f
 (
 M
 (
 x
 )
 )
 
 
 {\displaystyle J(M,x)f(M(x))}
 
 is a solution to the Dirac equation where

J
        (
        M
        ,
        x
        )
        =
        
          
            
              
                
                  c
                  x
                  +
                  d
                
                ~
              
            
            
              ‖
              c
              x
              +
              d
              
                ‖
                
                  n
                
              
            
          
        
      
    
    {\displaystyle J(M,x)={\frac {\widetilde {cx+d}}{\|cx+d\|^{n}}}}

and ~ is a basic <a href="/facts/Antiautomorphism/nIRWxknY">antiautomorphism</a> acting on the <a href="/facts/Clifford_algebra/qnqSCz5h">Clifford algebra</a>. The operators Dk, or Δnk/2 when k is even, exhibit similar covariances under <a href="/facts/M%25C3%25B6bius_transform/xDnSrPp6">Möbius transform</a> including the <a href="/facts/Cayley_transform/2jbbGdA9">Cayley transform</a>.
When ax+b and cx+d are non-zero they are both members of the <a href="/facts/Clifford_group/qnqSCz5h">Clifford group</a>.
As

a
              x
              +
              b
            
            
              c
              x
              +
              d
            
          
        
        =
        
          
            
              −
              a
              x
              −
              b
            
            
              −
              c
              x
              −
              d
            
          
        
      
    
    {\displaystyle {\frac {ax+b}{cx+d}}={\frac {-ax-b}{-cx-d}}}

then we have a choice in sign in defining J(M, x). This means that for a <a href="/facts/Conformally_flat_manifold/QNAk4QA0">conformally flat manifold</a> M we need a <a href="/facts/Spin_structure/BF78kFz8">spin structure</a> on M in order to define a <a href="/facts/Spinor_bundle/xjXIskf5">spinor bundle</a> on whose sections we can allow a Dirac operator to act. Explicit simple examples include the n-cylinder, the <a href="/facts/Hopf_manifold/Kxu2wX9l">Hopf manifold</a> 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 <a href="/facts/Dirac_operator/1kdGnoTt">Dirac operator</a> can be introduced in these contexts. These Dirac operators are special examples of Atiyah–Singer–Dirac operators.

<h2 id="atiyahsingerdirac-operator">Atiyah–Singer–Dirac operator</h2>
Given a <a href="/facts/Spin_manifold/BF78kFz8">spin manifold</a> M with a <a href="/facts/Spinor_bundle/xjXIskf5">spinor bundle</a> S and a smooth section s(x) in S then, in terms of a local orthonormal basis e1(x), ..., en(x) of the tangent bundle of M, the Atiyah–Singer–Dirac operator acting on s is defined to be

D
        s
        (
        x
        )
        =
        
          ∑
          
            j
            =
            1
          
          
            n
          
        
        
          e
          
            j
          
        
        (
        x
        )
        
          
            
              
                Γ
                ~
              
            
          
          
            
              e
              
                j
              
            
            (
            x
            )
          
        
        s
        (
        x
        )
        ,
      
    
    {\displaystyle Ds(x)=\sum _{j=1}^{n}e_{j}(x){\tilde {\Gamma }}_{e_{j}(x)}s(x),}

where 
 
 
 
 
 
 
 Γ
 ~
 
 
 
 
 
 {\displaystyle {\widetilde {\Gamma }}}
 
 is the <a href="/facts/Spin_connection/GdnYxk3f">spin connection</a>, the lifting to S of the <a href="/facts/Levi-Civita_connection/16KGSQUY">Levi-Civita connection</a> on M. When M is n-euclidean space we return to the euclidean <a href="/facts/Dirac_operator/1kdGnoTt">Dirac operator</a>.
From an Atiyah–Singer–Dirac operator D we have the <a href="/facts/Lichnerowicz_formula/FYYz1kTr">Lichnerowicz formula</a>

D
          
            2
          
        
        =
        
          Γ
          
            ∗
          
        
        Γ
        +
        
          
            
              τ
              4
            
          
        
        ,
      
    
    {\displaystyle D^{2}=\Gamma ^{*}\Gamma +{\tfrac {\tau }{4}},}

where τ is the <a href="/facts/Scalar_curvature/tfgm3AuK">scalar curvature</a> on the <a href="/facts/Manifold/rXPERJtu">manifold</a>, and Γ∗ is the adjoint of Γ. The operator D2 is known as the spinorial Laplacian.
If M is compact and τ ≥ 0 and τ > 0 somewhere then there are no non-trivial <a href="/facts/Harmonic_spinor/1kdGnoTt">harmonic spinors</a> on the manifold. This is Lichnerowicz' theorem. It is readily seen that Lichnerowicz' theorem is a generalization of <a href="/facts/Liouville%2527s_theorem_(complex_analysis)/I97twMBQ">Liouville's theorem</a> 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
          
            −
            1
          
        
        ∗
        
          δ
          
            y
          
        
        ,
        
        x
        ≠
        y
        ∈
        M
        ,
      
    
    {\displaystyle C(x,y):=D^{-1}*\delta _{y},\qquad x\neq y\in M,}

where δy is the <a href="/facts/Dirac_delta_function/V8PjRP8T">Dirac delta function</a> evaluated at y. This gives rise to a <a href="/facts/Cauchy_kernel/4B35nkvV">Cauchy kernel</a>, which is the <a href="/facts/Fundamental_solution/hdu8vXJa">fundamental solution</a> to this Dirac operator. From this one may obtain a <a href="/facts/Cauchy_integral_formula/4B35nkvV">Cauchy integral formula</a> for <a href="/facts/Harmonic_spinor/1kdGnoTt">harmonic spinors</a>. With this kernel much of what is described in the first section of this entry carries through for invertible Atiyah–Singer–Dirac operators.
Using <a href="/facts/Stokes%2527_theorem/z0pAoj2x">Stokes' theorem</a>, 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.

<h2 id="hyperbolic-dirac-type-operators">Hyperbolic Dirac type operators</h2>
In Clifford analysis one also considers differential operators on upper half space, the disc, or hyperbola with respect to the hyperbolic, or <a href="/facts/Poincar%25C3%25A9_metric/BVRGJuvL">Poincaré metric</a>.
For upper half space one splits the <a href="/facts/Clifford_algebra/qnqSCz5h">Clifford algebra</a>, Cln into Cln−1 + Cln−1en. So for a in Cln one may express a as b + cen with a, b in Cln−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

M
 f
 =
 D
 f
 +
 
 
 
 n
 −
 2
 
 
 x
 
 n
 
 
 
 
 Q
 (
 f
 )
 
 
 {\displaystyle Mf=Df+{\frac {n-2}{x_{n}}}Q(f)}
 
.
In this case

M
 
 2
 
 
 f
 =
 −
 
 △
 
 n
 
 
 P
 (
 f
 )
 +
 
 
 
 n
 −
 2
 
 
 x
 
 n
 
 
 
 
 
 
 
 ∂
 P
 (
 f
 )
 
 
 ∂
 
 x
 
 n
 
 
 
 
 
 −
 
 (
 
 
 △
 
 n
 
 
 Q
 (
 f
 )
 −
 
 
 
 n
 −
 2
 
 
 x
 
 n
 
 
 
 
 
 
 
 ∂
 Q
 (
 f
 )
 
 
 ∂
 
 x
 
 n
 
 
 
 
 
 +
 
 
 
 n
 −
 2
 
 
 x
 
 n
 
 
 2
 
 
 
 
 Q
 (
 f
 )
 
 )
 
 
 e
 
 n
 
 
 
 
 {\displaystyle 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}}
 
.
The operator

△
          
            n
          
        
        −
        
          
            
              n
              −
              2
            
            
              x
              
                n
              
            
          
        
        
          
            ∂
            
              ∂
              
                x
                
                  n
                
              
            
          
        
      
    
    {\displaystyle \triangle _{n}-{\frac {n-2}{x_{n}}}{\frac {\partial }{\partial x_{n}}}}

is the <a href="/facts/Laplacian/kov9u3PT">Laplacian</a> with respect to the <a href="/facts/Poincar%25C3%25A9_metric/BVRGJuvL">Poincaré metric</a> while the other operator is an example of a Weinstein operator.
The <a href="/facts/Hyperbolic_Laplacian/BVRGJuvL">hyperbolic Laplacian</a> is invariant under actions of the conformal group, while the hyperbolic Dirac operator is covariant under such actions.

<h2 id="raritaschwingersteinweiss-operators">Rarita–Schwinger/Stein–Weiss operators</h2>
<a href="/facts/Rarita%25E2%2580%2593Schwinger_equation/rkWc0UEX">Rarita–Schwinger operators</a>, also known as Stein–Weiss operators, arise in representation theory for the Spin and <a href="/facts/Pin_group/1lHT68gC">Pin groups</a>. 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 <a href="/facts/Orthogonal_group/jKWM1KDM">orthogonal group</a>, O(n) it is common to consider functions taking values in spaces of homogeneous <a href="/facts/Harmonic_polynomial/LDo2rwos">harmonic polynomials</a>. When one refines this <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a> to the double covering Pin(n) of O(n) one replaces spaces of homogeneous harmonic polynomials by spaces of k <a href="/facts/Homogeneous_polynomial/WUHL7Kct">homogeneous polynomial</a> 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 Rn, and u varies over Rn. 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
          
            k
          
        
        (
        x
        )
        =
        
          p
          
            k
          
        
        (
        x
        )
        +
        x
        
          p
          
            k
            −
            1
          
        
        (
        x
        )
      
    
    {\displaystyle h_{k}(x)=p_{k}(x)+xp_{k-1}(x)}

where pk and pk−1 are respectively k and k−1 <a href="/facts/Monic_polynomial/4JzuIJ5R">monogenic polynomials</a>. Let P be the projection of hk to pk 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
          
            u
          
        
        u
        
          p
          
            k
            −
            1
          
        
        (
        u
        )
        =
        (
        −
        n
        −
        2
        k
        +
        2
        )
        
          p
          
            k
            −
            1
          
        
        .
      
    
    {\displaystyle D_{u}up_{k-1}(u)=(-n-2k+2)p_{k-1}.}

So

R
          
            k
          
        
        =
        
          (
          
            I
            +
            
              
                1
                
                  n
                  +
                  2
                  k
                  −
                  2
                
              
            
            u
            
              D
              
                u
              
            
          
          )
        
        
          D
          
            x
          
        
        .
      
    
    {\displaystyle R_{k}=\left(I+{\frac {1}{n+2k-2}}uD_{u}\right)D_{x}.}

<h2 id="conferences-and-journals">Conferences and Journals</h2>
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 the <a href="http://www.smartchair.org/hp/ICCA2020/">International Conference on Clifford Algebras and their Applications in Mathematical Physics (ICCA)</a> and <a href="http://agacse2021.fme.vutbr.cz/main.php">Applications of Geometric Algebra in Computer Science and Engineering (AGACSE)</a> series. A main publication outlet is the Springer journal <a href="/facts/Advances_in_Applied_Clifford_Algebras/6VVIEvjR">Advances in Applied Clifford Algebras</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Clifford_algebra/qnqSCz5h">Clifford algebra</a></li>
<li><a href="/facts/Complex_spin_structure/BF78kFz8">Complex spin structure</a></li>
<li><a href="/facts/Conformal_manifold/tuU181cE">Conformal manifold</a></li>
<li><a href="/facts/Conformally_flat_manifold/QNAk4QA0">Conformally flat manifold</a></li>
<li><a href="/facts/Dirac_operator/1kdGnoTt">Dirac operator</a></li>
<li><a href="/facts/Poincar%25C3%25A9_metric/BVRGJuvL">Poincaré metric</a></li>
<li><a href="/facts/Spin_group/tfvR9cqM">Spin group</a></li>
<li><a href="/facts/Spin_structure/BF78kFz8">Spin structure</a></li>
<li><a href="/facts/Spinor_bundle/xjXIskf5">Spinor bundle</a></li></ul>

<ul><li>Ahlfors, L.V. (1981), Möbius Transformations in Several Dimensions, Ordway professorship lectures in mathematics, University of Minnesota, <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/2027%2Fmdp.39015015619276">2027/mdp.39015015619276</a>, <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/681384835">681384835</a>.</li>
<li>Ahlfors, L. (1986), "Mobius transformations in Rn expressed through 2 × 2 matrices of Clifford numbers", Complex Variables, 5 (2–4): 215–224, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1080%2F17476938608814142">10.1080/17476938608814142</a>.</li>
<li>Brackx, F.; Delanghe, R.; Sommen, F. (1982), Clifford Analysis, Pitman Research Notes in Mathematics, <a href="/facts/Longman/l5JQ1NGz">Longman</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-273-08535-2.</li>
<li>Bures, J.; Sommen, F.; Soucek, V.; VanLancker, P. (2001), "Rarita–Schwinger type operators in Clifford analysis", Journal of Functional Analysis, 185 (2): 425–455, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1006%2Fjfan.2001.3781">10.1006/jfan.2001.3781</a>.</li>
<li>Colombo, F.; <a href="/facts/Irene_Sabadini/EH791Jt6">Sabadini, I.</a>; Sommen, F.; Struppa, D. (2004), Analysis of Dirac Systems and Computational Algebra, Progress in Mathematical Physics, Birkhauser Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8176-4255-2.</li>
<li>Eastwood, M.; Ryan, J. (2007), "Aspects of Dirac operators in analysis", Milan Journal of Mathematics, 75 (1): 91–116, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fs00032-007-0077-5">10.1007/s00032-007-0077-5</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120593186">120593186</a>.</li>
<li>Friedrich, T. (2000), <a href="https://books.google.com/books?id=epLpBwAAQBAJ">Dirac Operators in Riemannian Geometry</a>, Graduate Studies in Mathematics, vol. 25, <a href="/facts/American_Mathematical_Society/fr5gotCt">American Mathematical Society</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780821820551.</li>
<li>Jefferies, B. (2004), Spectral Properties of Noncommuting Operators, Lecture Notes in Mathematics, vol. 1843, <a href="/facts/Springer_Verlag/nAesf6nT">Springer Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-21923-4.</li>
<li>Krausshar, R. S. (2004), Generalized Analytic Automorphic Forms in Hypercomplex Space, Frontiers in Mathematics, Birkhauser Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-7643-7059-9.</li>
<li>Lawson, H. B.; Michelsohn, M.-L. (1989), Spin Geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-691-08542-0.</li>
<li>McIntosh, A. (1996), "Clifford algebras, Fourier theory, singular integrals, and harmonic functions on Lipschitz domains", in Ryan, J. (ed.), Clifford Algebras in Analysis and Related Topics, Studies in Advanced Mathematics, <a href="/facts/CRC_Press/duTuH4hz">CRC Press</a>, pp. 33–87, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8493-8481-8.</li>
<li>Mitrea, M. (1994), Singular Integrals, Hardy Spaces and Clifford Wavelets, Lecture Notes in Mathematics, vol. 1575, <a href="/facts/Springer_Verlag/nAesf6nT">Springer Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-57884-6.</li>
<li>Roe, J. (1998), Elliptic Operators, Topology and Asymptotic Methods, Pitman Research Notes in Mathematics, vol. 395 <a href="/facts/Longman/l5JQ1NGz">Longman</a> (2nd ed.), Harlow, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-582-32502-1{{citation}}: CS1 maint: location missing publisher (link).</li>
<li>Ryan, J. (1996), <a href="/facts/Clifford_Algebra/qnqSCz5h">Clifford Algebras</a> in Analysis and Related Topics, Studies in Advanced Mathematics, <a href="/facts/CRC_Press/duTuH4hz">CRC Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8493-8481-8.</li>
<li>Stein, E.; Weiss, G. (1968), "Generalizations of the <a href="/facts/Cauchy_Riemann_equations/SWkIHflU">Cauchy Riemann equations</a> and representations of the rotation group", American Journal of Mathematics, 90 (1): 163–196, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2373431">10.2307/2373431</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2373431">2373431</a>.</li>
<li>Sudbery, A. (1979), "<a href="/facts/Quaternion/T3tnu7mX">Quaternionic</a> analysis", <a href="/facts/Mathematical_Proceedings_of_the_Cambridge_Philosophical_Society/KUUzcS7S">Mathematical Proceedings of the Cambridge Philosophical Society</a>, 85 (2): 199–225, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1979MPCPS..85..199S">1979MPCPS..85..199S</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FS0305004100055638">10.1017/S0305004100055638</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:7606387">7606387</a>.</li>
<li>Tao, T. (1996), "<a href="/facts/Convolution_operator/PVPrdz9J">Convolution operators</a> on Lipschitz graphs with harmonic kernels", Advances in Applied Clifford Algebras, 6: 207–218, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0188-7009">0188-7009</a>.</li>
<li><a href="/facts/Sijue_Wu/4skoRXba">Wu, S.</a> (1999), "Well-posedness in <a href="/facts/Sobolev_space/34qxWCyr">Sobolev spaces</a> of the full water wave problem in 3-D", <a href="/facts/Journal_of_the_American_Mathematical_Society/qR98drCm">Journal of the American Mathematical Society</a>, 12 (2): 445–495, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0894-0347-99-00290-8">10.1090/S0894-0347-99-00290-8</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://wayback.archive-it.org/all/20111004195500/http://comp.uark.edu/%7Ejryan/notes.doc">Lecture notes on Dirac operators in analysis and geometry</a></li>
<li>Calderbank, David M.J. (1997-12-19), <a href="https://web.archive.org/web/20090813180506/http://bib.mathematics.dk/preprint.php?lang=en&id=IMADA-PP-1997-53">Dirac operators and Clifford analysis on manifolds with boundary</a>, Danish Mathematical Society, DMF-1997-12-007 PP-1997-53, archived from <a href="http://bib.mathematics.dk/preprint.php?lang=en&id=IMADA-PP-1997-53">the original</a> on 2009-08-13</li></ul>

Clifford analysis open-in-new

Clifford analysis