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]

Models

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

Thirring model

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

\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 $ψ \in ℂ 2$ is the spinor field, $ψ = ψ * γ 0$ is the Dirac adjoint spinor,

\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.

Soler model

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

\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

\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 $μ = 0, 1, 2, 3$ .

Einstein-Cartan theory

The Lagrangian density for a Dirac spinor field is given by ( $c=\hbar=1$ )

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

where

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, $\omega_{\mu\nu\rho}$ is the spin connection, $g$ is the determinant of the metric tensor $g_{\mu\nu}$ , and the Dirac matrices satisfy

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

The resulting Dirac equation is

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

References