Dirac equations: Difference between revisions

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(Defined Dirac operator)
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This article describes several equations named after [http://en.wikipedia.org/wiki/Paul_Dirac Paul Dirac].  
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==
==The Maxwell-Dirac equation==


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<center><math>\Box A +  \nabla (\nabla_{x,t} A)= \underline{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>.
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>.
* Scaling is <math>(s_1, s_2) = (n/2-3/2, n/2-1)</math>.

Revision as of 07:27, 19 June 2009

This article describes several equations named after Paul Dirac.

Dirac operator

Given a Clifford algebra Cℓp,q(C) spanned by Dirac matrices such that

being the matrix of a quadratic form with signature (p,q), Dirac operator is given by

With a gauge connection this becomes

The Maxwell-Dirac equation

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

This equation essentially reads

where is a spinor field (solving a coupled massive Dirac equation), and is the Dirac operator with connection A. We put in and in .

  • Scaling is .
  • When , there is GWP for small smooth data Chd1973
  • When there is LWP for in the Coulomb gauge Bou1999, and for in the Lorenz gauge Bou1996
    • For in the Coulomb gauge this is in Bou1996
    • This has recently been improved by Selberg to . Note that for technical reasons, lower-regularity results do not automatically imply higher ones when the regularity of one field (e.g. ) is kept fixed.
    • LWP for smooth data was obtained in Grs1966
    • GWP for small smooth data was obtained in Ge1991
  • When , 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

where is a spinor field (solving a coupled massive Dirac equation), is the Dirac operator and is a scalar (real) field. We put in and in .

The energy class is essentially , but the energy density is not positive. However, the norm of is also positive and conserved..

  • Scaling is .
  • When there is GWP for Chd1973, Bou2000 and LWP for Bou2000.
  • When there are some LWP results in Bou2001

Nonlinear Dirac equation

This equation essentially reads

where is a spinor field, is the mass, is a complex parameter, is the zeroth Pauli matrix, and is the spinor inner product.

  • Scaling is (at least in the massless case ).
  • In , LWP is known for when EscVe1997
    • This can be improved to LWP in (and GWP for small data) if an epsilon of additional regularity as assumed in the radial variable MacNkrNaOz-p; in particular one has GWP for radial data.
  • In , GWP is known for small data when MacNaOz-p2. Some results on the nonrelativistic limit of this equation are also obtained in that paper.