Nonlinear Dirac equation

{{Short description|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.{{cite journal| author=Д.Д. Иваненко| title=Замечание к теории взаимодействия через частицы| trans-title=translated in: D.D. Ivanenko, Notes to the theory of interaction via particles, Sov. Phys. JETP 13 (1938), 141)| journal=ЖЭТФ| volume=8| year=1938| pages=260–266| url=http://istina.msu.ru/media/publications/articles/079/c1a/1049479/Ivanenko-nonlinear.pdf}}{{cite journal| author1=R. Finkelstein |author2=R. LeLevier |author3=M. Ruderman |name-list-style=amp |title=Nonlinear spinor fields| journal=Phys. Rev.| volume=83| issue=2 |pages=326–332| year=1951| doi=10.1103/PhysRev.83.326| bibcode = 1951PhRv...83..326F}}{{cite journal| author1=R. Finkelstein |author2=C. Fronsdal |author3=P. Kaus |name-list-style=amp |title=Nonlinear Spinor Field| journal=Phys. Rev.| volume=103| issue=5| pages= 1571–1579| year=1956| doi=10.1103/PhysRev.103.1571| bibcode=1956PhRv..103.1571F}}{{cite journal| author=W. Heisenberg| title=Quantum Theory of Fields and Elementary Particles| journal = Rev. Mod. Phys.| volume = 29| issue = 3| pages = 269–278| year = 1957| doi = 10.1103/RevModPhys.29.269 | bibcode = 1957RvMP...29..269H |author-link=Werner Heisenberg}}{{cite journal| author=Gross, David J. and Neveu, André| title=Dynamical symmetry breaking in asymptotically free field theories| journal=Phys. Rev. D| volume=10| issue=10| pages=3235–3253| year=1974| doi=10.1103/PhysRevD.10.3235| bibcode = 1974PhRvD..10.3235G }}

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).Dennis W. Sciama, [http://rmp.aps.org/abstract/RMP/v36/i1/p463_1 "The physical structure of general relativity"]. Rev. Mod. Phys. 36, 463-469 (1964).Tom W. B. Kibble, [https://dx.doi.org/10.1063/1.1703702 "Lorentz invariance and the gravitational field"]. J. Math. Phys. 2, 212-221 (1961). 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,{{cite journal |author1=F. W. Hehl |author2=B. K. Datta |name-list-style=amp |year=1971 |title=Nonlinear spinor equation and asymmetric connection in general relativity |journal=J. Math. Phys. |volume=12 |issue=7 |pages=1334–1339 |doi=10.1063/1.1665738|bibcode = 1971JMP....12.1334H }}{{cite journal |author1=Friedrich W. Hehl |author2=Paul von der Heyde |author3=G. David Kerlick |author4=James M. Nester |name-list-style=amp |year=1976 |title=General relativity with spin and torsion: Foundations and prospects |journal=Rev. Mod. Phys. |volume=48 |issue=3 |pages=393–416 |doi=10.1103/RevModPhys.48.393|bibcode = 1976RvMP...48..393H }} which causes fermions to be spatially extended and may remove the ultraviolet divergence in quantum field theory.{{cite journal |author=Nikodem J. Popławski |year=2010 |title=Nonsingular Dirac particles in spacetime with torsion |journal=Phys. Lett. B |volume=690 |issue=1 |pages=73–77 |doi=10.1016/j.physletb.2010.04.073|arxiv = 0910.1181|bibcode = 2010PhLB..690...73P |author-link=Nikodem Popławski }}

Models

Two common examples are the massive Thirring model and the Soler model.

=Thirring model=

The Thirring model{{ cite journal| author=Walter Thirring| title=A soluble relativistic field theory| journal=Annals of Physics| volume=3| issue=1| pages=91–112| year=1958| doi=10.1016/0003-4916(58)90015-0| bibcode=1958AnPhy...3...91T }} 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 {{math|ψ ∈ C²}} is the spinor field, {{math|1={{overline|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), {{math|g}} is the coupling constant, {{math|m}} is the mass, and {{math|γ{{i sup|μ}}}} are the two-dimensional gamma matrices, finally {{math|1=μ = 0, 1}} is an index.

=Soler model=

The Soler model{{ cite journal| author=Mario Soler| title=Classical, Stable, Nonlinear Spinor Field with Positive Rest Energy| journal=Phys. Rev. D| volume=1| issue=10| pages=2766–2769| year=1970| doi=10.1103/PhysRevD.1.2766| bibcode=1970PhRvD...1.2766S }} 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 {{math|γ{{i sup|μ}}}}, so therein {{math|1=μ = 0, 1, 2, 3}}.

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

References

Category:Quantum field theory

Category:Dirac equation