Nonlinear Dirac equation
Dirac equation for self-interacting fermions
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

L = ψ ¯ ( i ∂ / − m ) ψ − g 2 ( ψ ¯ γ μ ψ ) ( ψ ¯ γ μ ψ ) , {\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,

∂ / = ∑ μ = 0 , 1 γ μ ∂ ∂ x μ , {\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 γμ 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

L = ψ ¯ ( i ∂ / − m ) ψ + g 2 ( ψ ¯ ψ ) 2 , {\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

∂ / = ∑ μ = 0 3 γ μ ∂ ∂ x μ , {\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 γμ, 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 ( c = ℏ = 1 {\displaystyle c=\hbar =1} )

L = − g ( ψ ¯ ( i γ μ D μ − m ) ψ ) , {\displaystyle {\mathcal {L}}={\sqrt {-g}}\left({\overline {\psi }}\left(i\gamma ^{\mu }D_{\mu }-m\right)\psi \right),}

where

D μ = ∂ μ + 1 4 ω ν ρ μ γ ν γ ρ {\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, ω μ ν ρ {\displaystyle \omega _{\mu \nu \rho }} is the spin connection, g {\displaystyle g} is the determinant of the metric tensor g μ ν {\displaystyle g_{\mu \nu }} , and the Dirac matrices satisfy

γ μ γ ν + γ ν γ μ = 2 g μ ν I . {\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,

i γ μ D μ ψ − m ψ = i γ μ ∇ μ ψ + 3 κ 8 ( ψ ¯ γ μ γ 5 ψ ) γ μ γ 5 ψ − m ψ = 0 , {\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 ∇ μ {\displaystyle \nabla _{\mu }} is the general-relativistic covariant derivative of a spinor, and κ {\displaystyle \kappa } is the Einstein gravitational constant, 8 π G c 4 {\textstyle {\frac {8\pi G}{c^{4}}}} . The cubic term in this equation becomes significant at densities on the order of m 2 κ {\textstyle {\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 3 κ 8 {\textstyle {\frac {3\kappa }{8}}} replaced in terms of the constant − X 2 M 2 {\textstyle -{\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. [13]

See also

References

  1. ^ Д.Д. Иваненко (1938). "Замечание к теории взаимодействия через частицы" [translated in: D.D. Ivanenko, Notes to the theory of interaction via particles, Sov. Phys. JETP 13 (1938), 141)] (PDF). ЖЭТФ. 8: 260–266.
  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.
  3. ^ R. Finkelstein; C. Fronsdal & 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.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  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 (2023). "Torsionally-Induced Stability in Spinors". Universe. 9: 73. arXiv:2303.12085. doi:10.3390/universe9020073.
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