Nonlinear Dirac equation

Two common examples are the massive Thirring model and the Soler model.

In Einstein-Cartan theory the Lagrangian density for a Dirac spinor field is given by (${\displaystyle c=\hbar =1}$)

${\displaystyle {\mathcal {L}}={\sqrt {-g}}{\bigl (}{\overline {\psi }}\left(i\gamma ^{\mu }D_{\mu }-m\right)\psi {\bigr )},}$

where

${\displaystyle D_{\mu }=\partial _{\mu }+{\frac {1}{4}}\omega _{\nu \rho \mu }\gamma ^{\nu }\gamma ^{\rho }}$

is the Fock-Ivanenko covariant derivative of a spinor with respect to the affine connection, ${\displaystyle \omega _{\mu \nu \rho }}$ is the spin connection, ${\displaystyle g}$ is the determinant of the metric tensor ${\displaystyle g_{\mu \nu }}$, and the Dirac matrices satisfy

${\displaystyle \gamma ^{\mu }\gamma ^{\nu }+\gamma ^{\nu }\gamma ^{\mu }=2g^{\mu \nu }I.}$

The Einstein-Cartan field equations for the spin connection yield an algebraic constraint between the spin connection and the spinor field rather than a partial differential equation, which allows the spin connection to be explicitly eliminated from the theory. The final result is a nonlinear Dirac equation containing an effective "spin-spin" self-interaction,

${\displaystyle i\gamma ^{\mu }D_{\mu }\psi -m\psi =i\gamma ^{\mu }\nabla _{\mu }\psi +{\frac {3\kappa }{8}}({\overline {\psi }}\gamma _{\mu }\gamma ^{5}\psi )\gamma ^{\mu }\gamma ^{5}\psi -m\psi =0,}$

where ${\displaystyle \nabla _{\mu }}$ is the general-relativistic covariant derivative of a spinor. The cubic term in this equation becomes significant at densities on the order of ${\displaystyle {\frac {m^{2}}{\kappa }}}$.

• Dirac equation
• Dirac equation in the algebra of physical space
• Gross–Neveu model
• Higher-dimensional gamma matrices
• Nonlinear Schrödinger equation
• Pokhozhaev's identity for the stationary nonlinear Dirac equation
• Soler model
• Thirring model