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This article describes several equations named after [http://en.wikipedia.org/wiki/Paul_Dirac Paul Dirac]. 
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==Dirac operator==

Given a Clifford algebra ''Cℓ''<sub>''p'',''q''</sub>('''C''') spanned by Dirac matrices <math>\,\gamma_i\!</math> such that

<center><math>\, \gamma_i\gamma_j+\gamma_j\gamma_i=2\eta_{ij}\!</math></center>

being <math>\, \eta_{ij}\!</math> the matrix of a quadratic form with signature (p,q), Dirac operator is given by

<center><math>\, D=i\eta_{ij}\gamma_i\partial_j. \!</math></center>

With a gauge connection <math>\,A\!</math>this becomes

<center><math>\, D_A=i\eta_{ij}\gamma_i(\partial_j+iA_j). \!</math></center>

==The Maxwell-Dirac equation==

[More info on this equation would be greatly appreciated. - Ed.]

This equation essentially reads

<center><math>\,D_A  y  =  - y \! </math></center>

<center><math>\Box A +  \nabla (\nabla_{x,t} A)= \underline{y} y </math></center>

where  <math>\, y\!</math>  is a spinor field (solving a coupled massive Dirac equation), and <math>\, D\!</math> is the Dirac operator with connection A. We put  <math>y</math>  in <math>H^{s_1}</math> and <math>A</math> in <math>H^{s_2} \times H^{s_2 - 1}</math>.

* Scaling is <math>(s_1, s_2) = (n/2-3/2, n/2-1)</math>.
* When <math>n=1</math>, there is GWP for small smooth data [[Chd1973]]
* When <math>n=3</math> there is LWP for <math>(s_1, s_2) = (1, 1)</math> in the Coulomb gauge [[Bou1999]], and for <math>(s_1, s_2) = (1/2+, 1+)</math> in the Lorenz gauge [[Bou1996]]
** For <math>(s_1, s_2) = (1,2)</math> in the Coulomb gauge this is in [[Bou1996]]
** This has recently been improved by Selberg to <math>(1/4+, 1)</math>. Note that for technical reasons, lower-regularity results do not automatically imply higher ones when the regularity of one field (e.g. <math>A</math>) is kept fixed.
** LWP for smooth data was obtained in [[Grs1966]]
** GWP for small smooth data was obtained in [[Ge1991]]
* When <math>n=4</math>, GWP for small smooth data is known (Psarelli?)

In the nonrelativistic limit this equation converges to a Maxwell-Poisson system for data in the energy space [[BecMauSb-p2]]; furthermore one has a local well-posedness result which grows logarithmically in the asymptotic parameter. Earlier work appears in [[MasNa2003]].

==Dirac-Klein-Gordon equation==

[More info on this equation would be greatly appreciated. - Ed.]

This equation essentially reads

<center><math>\, D \psi = \phi \psi - \psi\!</math></center>
<center><math>\Box  \phi  = \overline{\psi} \psi </math></center>

where  <math>\psi</math>  is a spinor field (solving a coupled massive Dirac equation), <math>D</math> is the Dirac operator and  <math>\phi</math>  is a scalar (real) field. We put <math>\psi</math>  in <math>H^{s_1}</math> and <math>( \phi,  \phi_t)</math> in <math>H^{s_2} \times H^{s_2 - 1}</math>.

The energy class is essentially <math>(s_1,s_2) = (1/2,1)</math>, but the energy density is not positive. However, the <math>L^2</math> norm of  <math>y</math>  is also positive and conserved..

* Scaling is <math>(s_1, s_2) = (d/2-3/2, d/2-1)</math>.
* When <math>n=1</math> there is GWP for <math>(s_1,s_2) = (1,1)</math> [[Chd1973]], [[Bou2000]] and LWP for <math>(s_1, s_2) = (0, 1/2)</math> [[Bou2000]].
* When <math>n=2</math> there are some LWP results in [[Bou2001]]

==Nonlinear Dirac equation==

This equation essentially reads

<center><math>\, D \psi -  m \psi = \lambda (\gamma \psi, \psi) \psi\!</math></center>

where  <math>\psi</math>  is a spinor field, <math>m > 0</math> is the mass, <math>\lambda</math> is a complex parameter,  <math>\gamma</math> is the zeroth Pauli matrix, and <math>(,)</math> is the spinor inner product.

* Scaling is <math>s_c =1</math> (at least in the massless case <math>m=0</math>).
* In <math>R^3</math>, LWP is known for <math>H^s</math> when <math>s > 1</math> [[EscVe1997]]
** This can be improved to LWP in <math>H^1</math> (and GWP for small <math>H^1</math> data) if an epsilon of additional regularity as assumed in the radial variable [[MacNkrNaOz-p]]; in particular one has GWP for radial <math>H^1</math> data.
* In <math>R^3</math>, GWP is known for small <math>H^s</math> data when <math>s > 1</math> [[MacNaOz-p2]]. Some results on the [[nonrelativistic limit]] of this equation are also obtained in that paper.

[[Category:Wave]]
[[Category:Equations]]