Dirac equations

This article describes several equations named after Paul Dirac.

Dirac operator

Given a Clifford algebra $\,C\ell _{p,q}({\mathbb {C} })\!$ spanned by Dirac matrices $\,\gamma _{i}\!$ such that

$\,\gamma _{i}\gamma _{j}+\gamma _{j}\gamma _{i}=2\eta _{ij}\!$

being $\,\eta _{ij}\!$ the matrix of a quadratic form with signature (p,q), Dirac operator is given by

$\,D=i\eta _{ij}\gamma _{i}\partial _{j}.\!$

With a gauge connection $\,A\!$this becomes

$\,D_{A}=i\eta _{ij}\gamma _{i}(\partial _{j}+iA_{j}).\!$

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 $(s_{1},s_{2})=(1,1)$ in the Coulomb gauge Bou1999, and for $(s_{1},s_{2})=(1/2+,1+)$ in the Lorenz 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.

Dirac-Klein-Gordon equation

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

This equation essentially reads

$\,D\psi =\phi \psi -\psi \!$ $\Box \phi ={\overline {\psi }}\psi$

where $\psi$ is a spinor field (solving a coupled massive Dirac equation), $D$ is the Dirac operator and $\phi$ is a scalar (real) field. We put $\psi$ in $H^{s_{1}}$ and $(\phi ,\phi _{t})$ 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

Nonlinear Dirac equation

This equation essentially reads

$\,D\psi -m\psi =\lambda (\gamma \psi ,\psi )\psi \!$

where $\psi$ is a spinor field, $m>0$ is the mass, $\lambda$ is a complex parameter, $\gamma$ 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 $R^{3}$, 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.