Nonlinear Dirac equation

See Ricci calculus and Van der Waerden notation for the notation.

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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.^{[1][2][3][4][5]}

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).^[6][7] 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,^[8][9] which causes fermions to be spatially extended and may remove the ultraviolet divergence in quantum field theory.^[10]

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

The Thirring model^[11] was originally formulated as a model in (1 + 1) space-time dimensions and is characterized by the Lagrangian density

${\displaystyle {\mathcal {L}}={\overline {\psi }}(i\partial \!\!\!/-m)\psi -{\frac {g}{2}}\left({\overline {\psi }}\gamma ^{\mu }\psi \right)\left({\overline {\psi }}\gamma _{\mu }\psi \right),}$

where ψ ∈ C² is the spinor field, ψ = ψ*γ⁰ is the Dirac adjoint spinor,

${\displaystyle \partial \!\!\!/=\sum _{\mu =0,1}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\,,}$

(Feynman slash notation is used), g is the coupling constant, m is the mass, and γ^μ are the two-dimensional gamma matrices, finally μ = 0, 1 is an index.

The Soler model^[12] was originally formulated in (3 + 1) space-time dimensions. It is characterized by the Lagrangian density

${\displaystyle {\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +{\frac {g}{2}}\left({\overline {\psi }}\psi \right)^{2},}$

using the same notations above, except

${\displaystyle \partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\,,}$

is now the four-gradient operator contracted with the four-dimensional Dirac gamma matrices γ^μ, so therein μ = 0, 1, 2, 3.

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

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

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,

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

where ${\displaystyle \nabla _{\mu }}$ is the general-relativistic covariant derivative of a spinor, and ${\displaystyle \kappa }$ is the Einstein gravitational constant, ${\textstyle {\frac {8\pi G}{c^{4}}}}$. The cubic term in this equation becomes significant at densities on the order of ${\textstyle {\frac {m^{2}}{\kappa }}}$.

In a more general theory in which torsion is propagating, when torsion is taken in the effective approximation, the non-linearity in the Dirac equation will have the same structure, but with the constant ${\textstyle {\frac {3\kappa }{8}}}$ replaced in terms of the constant ${\textstyle -{\frac {X^{2}}{M^{2}}}}$ where X is the spinor-torsion coupling constant and M the mass of torsion: in this theory, then, the self-interaction is repulsive, exactly like in the Nambu--Jona-Lasinio model, and with non-linearities manifested at the energy scale given by the torsion mass. ^[13]