Nonlinear Dirac equation

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]

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

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

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.

Soler model

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.

Einstein–Cartan theory

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 ${\textstyle {\frac {m^{2}}{\kappa }}}$.

Formally, it is possible to work out the Dirac stress-energy tensor (and in particular its time-time component and its trace) to show that the Dirac field behaves in some ways as if it was a van der Waals gas: in this analogy, the torsional nonlinear terms account for the van der Waals extra pressure. ^[13] In the limit in which torsion vanishes, the Dirac field would behave as a perfect gas.

See also

• Dirac equation
• Dirac equation in the algebra of physical space
• Dirac–Kähler equation
• Gross–Neveu model
• Higher-dimensional gamma matrices
• Nonlinear Schrödinger equation
• Pokhozhaev's identity for the stationary nonlinear Dirac equation
• Soler model
• Thirring model

References

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2. R. Finkelstein; R. LeLevier & M. Ruderman (1951). "Nonlinear spinor fields". Phys. Rev. 83 (2): 326–332. Bibcode:1951PhRv...83..326F. doi:10.1103/PhysRev.83.326.
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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 & B. K. Datta (1971). "Nonlinear spinor equation and asymmetric connection in general relativity". J. Math. Phys. 12 (7): 1334–1339. Bibcode:1971JMP....12.1334H. doi:10.1063/1.1665738.
9. Friedrich W. Hehl; Paul von der Heyde; G. David Kerlick & James M. Nester (1976). "General relativity with spin and torsion: Foundations and prospects". Rev. Mod. Phys. 48 (3): 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 (1): 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.
13. Luca Fabbri (2024). "Dirac field, van der Waals gas, Weyssenhoff fluid, Newton particle". Foundations. 4 (2): 134–145. doi: 10.3390/foundations4020010 .