Difference between revisions of "Dirac equations"

The Maxwell-Dirac equation

${\displaystyle D_{A}y=-y}$
${\displaystyle \Box A+\nabla (\nabla _{x,t}A)={\underline {y}}y}$

where ${\displaystyle y}$ is a spinor field (solving a coupled massive Dirac equation), and ${\displaystyle D}$ is the Dirac operator with connection A. We put ${\displaystyle y}$ in ${\displaystyle H^{s_{1}}}$ and ${\displaystyle A}$ in ${\displaystyle H^{s_{2}}\times H^{s_{2}-1}}$.

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

${\displaystyle D\psi =\phi \psi -\psi }$
${\displaystyle \Box \phi ={\overline {\psi }}\psi }$

where ${\displaystyle \psi }$ is a spinor field (solving a coupled massive Dirac equation), ${\displaystyle D}$ is the Dirac operator and ${\displaystyle \phi }$ is a scalar (real) field. We put ${\displaystyle \psi }$ in ${\displaystyle H^{s_{1}}}$ and ${\displaystyle (\phi ,\phi _{t})}$ in ${\displaystyle H^{s_{2}}\times H^{s_{2}-1}}$.

The energy class is essentially ${\displaystyle (s_{1},s_{2})=(1/2,1)}$, but the energy density is not positive. However, the ${\displaystyle L^{2}}$ norm of ${\displaystyle y}$ is also positive and conserved..

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

Nonlinear Dirac equation

${\displaystyle D\psi -m\psi =\lambda (\gamma \psi ,\psi )\psi }$
where ${\displaystyle \psi }$ is a spinor field, ${\displaystyle m>0}$ is the mass, ${\displaystyle \lambda }$ is a complex parameter, ${\displaystyle \gamma }$ is the zeroth Pauli matrix, and ${\displaystyle (,)}$ is the spinor inner product.
• Scaling is ${\displaystyle s_{c}=1}$ (at least in the massless case ${\displaystyle m=0}$).
• In ${\displaystyle R^{3}}$, LWP is known for ${\displaystyle H^{s}}$ when ${\displaystyle s>1}$ EscVe1997
• This can be improved to LWP in ${\displaystyle H^{1}}$ (and GWP for small ${\displaystyle H^{1}}$ data) if an epsilon of additional regularity as assumed in the radial variable MacNkrNaOz-p; in particular one has GWP for radial ${\displaystyle H^{1}}$ data.
• In ${\displaystyle R^{3}}$, GWP is known for small ${\displaystyle H^{s}}$ data when ${\displaystyle s>1}$ MacNaOz-p2. Some results on the nonrelativistic limit of this equation are also obtained in that paper.