Nonlinear Dirac equation

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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.[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.

Thirring model[edit]

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 ∈ ℂ2 is the spinor field, ψ = ψ*γ0 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 γμ are the two-dimensional gamma matrices, finally μ = 0, 1 is an index.

Soler model[edit]

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 γμ, so therein μ = 0, 1, 2, 3.

Einstein-Cartan theory[edit]

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),


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}.

See also[edit]


  1. ^ D.D. Ivanenko (1938). "Notes to the theory of interaction via particles". Zh. Eksp. Teor. Fiz. 8: 260–266. 
  2. ^ R. Finkelstein, R. LeLevier, and M. Ruderman (1951). "Nonlinear spinor fields". Phys. Rev. 83: 326–332. Bibcode:1951PhRv...83..326F. doi:10.1103/PhysRev.83.326. 
  3. ^ R. Finkelstein, C. Fronsdal, and P. Kaus (1956). "Nonlinear Spinor Field". Phys. Rev. 103 (5): 1571–1579. Bibcode:1956PhRv..103.1571F. doi:10.1103/PhysRev.103.1571. 
  4. ^ W. Heisenberg (1957). "Quantum Theory of Fields and Elementary Particles". Rev. Mod. Phys. 29 (3): 269–278. Bibcode:1957RvMP...29..269H. doi:10.1103/RevModPhys.29.269. 
  5. ^ Gross, David J. and Neveu, André (1974). "Dynamical symmetry breaking in asymptotically free field theories". Phys. Rev. D 10 (10): 3235–3253. Bibcode:1974PhRvD..10.3235G. doi:10.1103/PhysRevD.10.3235. 
  6. ^ Dennis W. Sciama, "The physical structure of general relativity". Rev. Mod. Phys. 36, 463-469 (1964).
  7. ^ Tom W. B. Kibble, "Lorentz invariance and the gravitational field". J. Math. Phys. 2, 212-221 (1961).
  8. ^ F. W. Hehl and B. K. Datta (1971). "Nonlinear spinor equation and asymmetric connection in general relativity". J. Math. Phys. 12: 1334–1339. Bibcode:1971JMP....12.1334H. doi:10.1063/1.1665738. 
  9. ^ Friedrich W. Hehl, Paul von der Heyde, G. David Kerlick, and James M. Nester (1976). "General relativity with spin and torsion: Foundations and prospects". Rev. Mod. Phys. 48: 393–416. Bibcode:1976RvMP...48..393H. doi:10.1103/RevModPhys.48.393. 
  10. ^ Nikodem J. Popławski (2010). "Nonsingular Dirac particles in spacetime with torsion". Phys. Lett. B 690: 73–77. arXiv:0910.1181. Bibcode:2010PhLB..690...73P. doi:10.1016/j.physletb.2010.04.073. 
  11. ^ Walter Thirring (1958). "A soluble relativistic field theory". Annals of Physics 3 (1): 91–112. Bibcode:1958AnPhy...3...91T. doi:10.1016/0003-4916(58)90015-0. 
  12. ^ Mario Soler (1970). "Classical, Stable, Nonlinear Spinor Field with Positive Rest Energy". Phys. Rev. D 1 (10): 2766–2769. Bibcode:1970PhRvD...1.2766S. doi:10.1103/PhysRevD.1.2766.