Nonlinear Dirac equation

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

: \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, 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 are the two-dimensional gamma matrices, finally is an index.

Soler model

The Soler model 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, so therein .

Einstein–Cartan theory

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

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

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 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,

: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 \nabla_\mu is the general-relativistic covariant derivative of a spinor, and \kappa is the Einstein gravitational constant, \frac{8 \pi G}{c^4}. The cubic term in this equation becomes significant at densities on the order of \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 \frac{3\kappa}{8} replaced in terms of the constant -\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.

References

References

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