Dirac equations

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This article describes several equations named after Paul Dirac.

1 The Maxwell-Dirac equation
2 Dirac-Klein-Gordon equation
3 Nonlinear Dirac equation

The Maxwell-Dirac equation

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

This equation essentially reads

D A y = - y

$\Box A + \nabla (\nabla_{x,t} A)= \underline{y} y$

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

Scaling is $(s 1, s 2) = (n / 2 - 3 / 2, n / 2 - 1)$ .
When $n = 1$ , there is GWP for small smooth data Chd1973
When n = 3 there is LWP for (s₁,s₂) = (1,1) in the Coulomb gauge Bou1999, and for (s₁,s₂) = (1 / 2 + ,1 + ) in the Lorentz gauge Bou1996
- For $(s 1, s 2) = (1,2)$ in the Coulomb gauge this is in Bou1996
- This has recently been improved by Selberg to $(1 / 4 + ,1)$ . Note that for technical reasons, lower-regularity results do not automatically imply higher ones when the regularity of one field (e.g. $A$ ) is kept fixed.
- LWP for smooth data was obtained in Grs1966
- GWP for small smooth data was obtained in Ge1991
When $n = 4$ , 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.

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Dirac-Klein-Gordon equation

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

This equation essentially reads

D ψ = φψ - ψ

$\Box \phi = \overline{\psi} \psi$

where $ψ$ is a spinor field (solving a coupled massive Dirac equation), $D$ is the Dirac operator and $φ$ is a scalar (real) field. We put $ψ$ in $H^{s_1}$ and $(φ,φ t)$ in $H^{s_2} \times H^{s_2 - 1}$ .

The energy class is essentially $(s 1, s 2) = (1 / 2,1)$ , but the energy density is not positive. However, the $L 2$ norm of $y$ is also positive and conserved..

Scaling is $(s 1, s 2) = (d / 2 - 3 / 2, d / 2 - 1)$ .
When $n = 1$ there is GWP for $(s 1, s 2) = (1,1)$ Chd1973, Bou2000 and LWP for $(s 1, s 2) = (0,1 / 2)$ Bou2000.
When $n = 2$ there are some LWP results in Bou2001

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Nonlinear Dirac equation

This equation essentially reads

D ψ - m ψ = λ(γψ,ψ)ψ

where $ψ$ is a spinor field, $m > 0$ is the mass, $λ$ is a complex parameter, $γ$ is the zeroth Pauli matrix, and $(,)$ is the spinor inner product.

Scaling is $s c = 1$ (at least in the massless case $m = 0$ ).
In R³, LWP is known for H^s when s > 1 EscVe1997
- This can be improved to LWP in $H 1$ (and GWP for small $H 1$ data) if an epsilon of additional regularity as assumed in the radial variable MacNkrNaOz-p; in particular one has GWP for radial $H 1$ data.
In $R 3$ , GWP is known for small $H s$ data when $s > 1$ MacNaOz-p2. Some results on the nonrelativistic limit of this equation are also obtained in that paper.