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

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

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

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

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?)

There are no exact solutions known for this equation. Small perturbation theory is the only approach to solve them used so far.

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

$$\, D \psi = \phi \psi - \psi\!$$ $$\Box \phi = \overline{\psi} \psi +m^2\phi$$

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

Dirac-Klein-Gordon equation and non-linear Dirac equation

It is interesting to show how the set of equations of the Dirac-Klein-Gordon system can be reduced to a single non-linear Dirac equation in a proper limit. This is easily accomplished by noting that the equation

$$\Box \phi = \overline{\psi} \psi+m^2\phi$$

can be immediately integrated giving

$$\phi = \int d^Dy\Delta(x-y)\overline{\psi}(y) \psi(y)$$

being $\Delta(x-y)$ the Green function for the Klein-Gordon equation. Then, one has the integro-differential equation for the Dirac spinor

$$D \psi = \int d^Dy\Delta(x-y)\overline{\psi}(y) \psi(y) \psi - \psi.$$

For distances large enough one can substitute $\Delta(x-y)$ with $\delta^D(x-y)$ multiplied by a constant reducing the starting set of equations to a non-linear Dirac equation. In physics this non-linear equation is named as Nambu-Jona-Lasinio model and has the property to generate a mass gap equation. This means that one can start with a massless Dirac equation and ends with a massive solution.

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.