Dirac equations
This article describes several equations named after Paul Dirac.
Dirac operator
Given a Clifford algebra Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \,C\ell_{p,q}({\mathbb C}) \!} spanned by Dirac matrices Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \,\gamma_i\!} such that
being Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \, \eta_{ij}\!} the matrix of a quadratic form with signature (p,q), Dirac operator is given by
With a gauge connection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \,A\!} this becomes
Maxwell-Dirac equation
[More info on this equation would be greatly appreciated. - Ed.]
This equation essentially reads
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \, y\!} is a spinor field (solving a coupled massive Dirac equation), and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \, D\!} is the Dirac operator with connection A. We put Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{s_1}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^{s_2} \times H^{s_2 - 1}} .
- Scaling is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (s_1, s_2) = (n/2-3/2, n/2-1)} .
- When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n=1} , there is GWP for small smooth data Chd1973
- When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n=3}
there is LWP for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (s_1, s_2) = (1, 1)}
in the Coulomb gauge Bou1999, and for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (s_1, s_2) = (1/2+, 1+)}
in the Lorenz gauge Bou1996
- For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (s_1, s_2) = (1,2)} in the Coulomb gauge this is in Bou1996
- This has recently been improved by Selberg to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} ) is kept fixed.
- LWP for smooth data was obtained in Grs1966
- GWP for small smooth data was obtained in Ge1991
- When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \psi} is a spinor field, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m > 0} is the mass, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda} is a complex parameter, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \gamma} is the zeroth Pauli matrix, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (,)} is the spinor inner product.
- Scaling is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c =1} (at least in the massless case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m=0} ).
- In Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^3}
, LWP is known for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s}
when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 1}
EscVe1997
- This can be improved to LWP in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1} (and GWP for small Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^1} data.
- In Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R^3} , GWP is known for small Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} data when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 1} MacNaOz-p2. Some results on the nonrelativistic limit of this equation are also obtained in that paper.
