Difference between revisions of "Dirac equations"

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====The Maxwell-Dirac equation====
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This article describes several equations named after [http://en.wikipedia.org/wiki/Paul_Dirac Paul Dirac].
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==The Maxwell-Dirac equation==
  
 
[More info on this equation would be greatly appreciated. - Ed.]
 
[More info on this equation would be greatly appreciated. - Ed.]
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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]].
 
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]].
 
   
 
   
 
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==Dirac-Klein-Gordon equation==
====Dirac-Klein-Gordon equation====
 
  
 
[More info on this equation would be greatly appreciated. - Ed.]
 
[More info on this equation would be greatly appreciated. - Ed.]
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* When <math>n=2</math> there are some LWP results in [[Bou2001]]
 
* When <math>n=2</math> there are some LWP results in [[Bou2001]]
 
   
 
   
 
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==Nonlinear Dirac equation==
====Nonlinear Dirac equation====
 
  
 
This equation essentially reads
 
This equation essentially reads
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* 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.
 
* 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.
  
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[[Category:Wave]]
[[Category:wave]]
 
 
[[Category:Equations]]
 
[[Category:Equations]]

Revision as of 22:11, 3 September 2007

This article describes several equations named after Paul Dirac.

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 Lorentz 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.