[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.
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.
The Thirring model[11] was originally formulated as a model in (1 + 1) space-time dimensions and is characterized by the Lagrangian density where ψ ∈ C2 is the spinor field, ψ = ψ*γ0 is the Dirac adjoint spinor, (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.
It is characterized by the Lagrangian density using the same notations above, except is now the four-gradient operator contracted with the four-dimensional Dirac gamma matrices γμ, so therein μ = 0, 1, 2, 3.
[14][15] The models reviewed by Rañada are meant to be entirely classical in nature and should properly be regarded as having nothing to do with quantum mechanics, but the dependent variable in the Dirac equation is still typically taken as a spinor.
When a purely classical model of this nature is to be considered, the use of a spinor as the dependent variable seems inappropriate.
If a minor modification of the underlying Dirac equation is used, the problem can be avoided in a relatively straightforward way.
[16] Instead of using the usual column vector as the dependent variable in Dirac’s equation, one can use a 4 × 4 matrix.
In this case one simply allows the dependent variable to lie in a different left ideal when there is a transformation in space-time.
The final result is a nonlinear Dirac equation containing an effective "spin-spin" self-interaction, where