See Ricci calculus and Van der Waerden notation for the notation.
In quantum field theory, the nonlinear Dirac equation is a model of self-interacting Dirac fermions. This model is widely considered in quantum physics as a toy model of self-interacting electrons. The nonlinear Dirac equation appears in the Einstein–Cartan–Sciama–Kibble theory of gravity, which extends general relativity to matter with intrinsic angular momentum (spin). This theory removes a constraint of the symmetry of the affine connection and treats its antisymmetric part, the torsion tensor, as a variable in varying the action. In the resulting field equations, the torsion tensor is a homogeneous, linear function of the spin tensor. The minimal coupling between torsion and Dirac spinors thus generates an axial-axial, spin–spin interaction in fermionic matter, which becomes significant only at extremely high densities. Consequently, the Dirac equation becomes nonlinear (cubic) in the spinor field, which causes fermions to be spatially extended and may remove the ultraviolet divergence in quantum field theory.
Models Two common examples are the massive Thirring model and the Soler model.
Thirring model The Thirring model was originally formulated as a model in (1 + 1) space-time dimensions and is characterized by the Lagrangian density
L = ψ ¯ ( i ∂ / − m ) ψ − g 2 ( ψ ¯ γ μ ψ ) ( ψ ¯ γ μ ψ ) , {\displaystyle {\mathcal {L}}={\overline {\psi }}(i\partial \!\!\!/-m)\psi -{\frac {g}{2}}\left({\overline {\psi }}\gamma ^{\mu }\psi \right)\left({\overline {\psi }}\gamma _{\mu }\psi \right),}
where ψ ∈ C2 is the spinor field, ψ = ψ*γ0 is the Dirac adjoint spinor,
∂ / = ∑ μ = 0 , 1 γ μ ∂ ∂ x μ , {\displaystyle \partial \!\!\!/=\sum _{\mu =0,1}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\,,}
(Feynman slash notation is used), g is the coupling constant, m is the mass, and γμ are the two-dimensional gamma matrices, finally μ = 0, 1 is an index.
Soler model The Soler model was originally formulated in (3 + 1) space-time dimensions. It is characterized by the Lagrangian density
L = ψ ¯ ( i ∂ / − m ) ψ + g 2 ( ψ ¯ ψ ) 2 , {\displaystyle {\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +{\frac {g}{2}}\left({\overline {\psi }}\psi \right)^{2},}
using the same notations above, except
∂ / = ∑ μ = 0 3 γ μ ∂ ∂ x μ , {\displaystyle \partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\,,}
is now the four-gradient operator contracted with the four-dimensional Dirac gamma matrices γμ, so therein μ = 0, 1, 2, 3.
Einstein–Cartan theory In Einstein–Cartan theory the Lagrangian density for a Dirac spinor field is given by ( c = ℏ = 1 {\displaystyle c=\hbar =1} )
L = − g ( ψ ¯ ( i γ μ D μ − m ) ψ ) , {\displaystyle {\mathcal {L}}={\sqrt {-g}}\left({\overline {\psi }}\left(i\gamma ^{\mu }D_{\mu }-m\right)\psi \right),}
where
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