View source for Nonlinear Dirac equation

{{Short description|Dirac equation for self-interacting fermions}}
:''See [[Ricci calculus]] and [[Van der Waerden notation]] for the notation.''
{{quantum field theory}}
In [[quantum field theory]], the '''nonlinear Dirac equation''' is a model of self-interacting [[Dirac fermion]]s. This model is widely considered in [[quantum physics]] as a [[toy model]] of self-interacting [[electron]]s.<ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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}}</ref><ref>{{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}}</ref><ref>{{cite journal| author=[[David Gross|Gross, David J.]] and [[André Neveu|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 }}</ref>

The nonlinear Dirac equation appears in the [[Einstein–Cartan theory|Einstein–Cartan]]–Sciama–Kibble theory of gravity, which extends [[general relativity]] to matter with intrinsic angular momentum ([[Spin (physics)|spin]]).<ref>[[Dennis Sciama|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).</ref><ref>[[Tom Kibble|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).</ref>  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 spinor]]s thus generates an axial-axial, spin–spin interaction in [[fermion]]ic matter, which becomes significant only at extremely high densities.  Consequently, the Dirac equation becomes nonlinear (cubic) in the spinor field,<ref>{{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 }}</ref><ref>{{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 }}</ref> which causes fermions to be spatially extended and may remove the [[ultraviolet divergence]] in quantum field theory.<ref>{{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 }}</ref>

==Models==

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

===Thirring model===
The Thirring model<ref>{{ 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 }}</ref> was originally formulated as a model in (1 + 1) [[space-time]] dimensions and is characterized by the [[Lagrangian density]]

:<math>
\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),</math>

where {{math|''ψ'' ∈ '''C'''<sup>2</sup>}} is the [[spinor]] field, {{math|1={{overline|''ψ''}} = ''ψ''*''γ''<sup>0</sup>}} is the [[Dirac adjoint]] spinor,

:<math>\partial\!\!\!/=\sum_{\mu=0,1}\gamma^\mu\frac{\partial}{\partial x^\mu}\,,</math>

([[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 matrix|gamma matrices]]<!---Pauli matrices?--->, finally {{math|1=''μ'' = 0, 1}} is an [[index notation|index]].

===Soler model===

The Soler model<ref>{{ 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 }}</ref> was originally formulated in (3 + 1) space-time dimensions. It is characterized by the Lagrangian density

:<math>\mathcal{L} = \overline{\psi} \left(i\partial\!\!\!/-m \right) \psi + \frac{g}{2} \left(\overline{\psi} \psi\right)^2,</math>

using the same notations above, except

:<math>\partial\!\!\!/=\sum_{\mu=0}^3\gamma^\mu\frac{\partial}{\partial x^\mu}\,,</math>

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}}.

===Other models===

Besides the Soler model, extensive work has been done where nonlinear versions of Dirac’s equation are used to describe purely classical, nonlinear particle-like solutions (PLS) in (3 + 1) space-time dimensions. Rañada has given a review of the subject.<ref>{{cite encyclopedia| last=Rañada | first=A.F.|title=Classical nonlinear Dirac field models of extended particles| encyclopedia=Quantum Theory, Groups, Fields and Particles| publisher=Reidel | location=Amsterdam | year=1983| pages= 271–291 | isbn=1-4020-0324-2}}</ref>  Although a more recent review specifically devoted to purely classical, nonlinear PLS has apparently not appeared, pertinent references are available in various more recent publications.<ref>{{cite journal | last1=Esteban | first1=M.J. | last2=Lewin | first2=M. | last3=Séré | first3=E. | title=Variational methods in relativistic quantum mechanics | journal=Bull. Amer. Math. Soc. (N.S.) | volume=45 | pages=535–593 | date=2008| issue=4 | doi=10.1090/S0273-0979-08-01212-3 | s2cid=1686536 | doi-access=free | arxiv=0706.3309 }}</ref><ref>{{cite journal | last=Dudnikova | first=T.V. | title=Virial identities and energy-momentum relation for solitary waves of nonlinear Dirac equations | journal=Lobachevskii Journal of Mathematics | volume=41 | pages=956–981 | date=2020| issue=6 | doi=10.1134/S1995080220060074 | s2cid=226199105 }}</ref>

The models reviewed by Rañada are meant to be entirely classical in nature and should properly be regarded as having nothing to do with quantum mechanics, but the dependent variable in the Dirac equation is still typically taken as a spinor.{{Dubious|date=February 2025}}  When a purely classical model of this nature is to be considered, the use of a spinor as the dependent variable seems inappropriate.{{speculation inline|date=February 2025}}

If a minor modification of the underlying Dirac equation is used, the problem can be avoided in a relatively straightforward way{{Dubious|date=February 2025}}.<ref>{{cite journal | last=Luke | first=J.C. | title=A Note on the Use of the Nonlinear Dirac Equation for Soliton Particle Models. | website=ScholarWorks.IUPUI.edu | url=https://scholarworks.iupui.edu/handle/1805/29157 | date=May 2022 | access-date= June 2, 2022}}</ref>  Instead of using the usual column vector as the dependent variable in Dirac’s equation, one can use a 4 × 4 matrix.  When there is no transformation of coordinates, the leftmost column of the matrix is used in Dirac’s equation in the usual manner, but when there is to be a transformation in space-time, the four components of the dependent variable are sometimes allowed to appear in various different positions in the 4 × 4 matrix.

The result can be understood in terms of a [[Clifford algebra]] since the dependent variable in Dirac’s equation can be represented as a 4 dimensional left ideal of a Clifford algebra.  In this case one simply allows the dependent variable to lie in a ''different'' left ideal when there is a transformation in space-time.

==Einstein–Cartan theory==

In [[Einstein–Cartan theory]] the Lagrangian density for a Dirac spinor field is given by (<math>c = \hbar = 1</math>)

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

where

:<math>D_\mu=\partial_\mu + \frac{1}{4}\omega_{\nu\rho\mu}\gamma^\nu \gamma^\rho</math>

is the Fock–Ivanenko [[covariant derivative]] of a spinor with respect to the affine connection, <math>\omega_{\mu\nu\rho}</math> is the [[spin connection]], <math>g</math> is the determinant of the [[metric tensor]] <math>g_{\mu\nu}</math>, and the Dirac matrices satisfy

:<math>\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2g^{\mu\nu}I.</math>

The [[Einstein–Cartan theory#Field equations|Einstein–Cartan field equations]] for the spin connection yield an [[Algebraic operation|algebraic]] [[Constraint (classical mechanics)|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,

:<math>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,</math>

where <math>\nabla_\mu</math> is the general-relativistic covariant derivative of a spinor, and <math>\kappa</math> is the Einstein gravitational constant, <math display="inline">\frac{8 \pi G}{c^4}</math>.  The cubic term in this equation becomes significant at densities on the order of <math display="inline">\frac{m^2}{\kappa}</math>.

==See also==
{{div col|colwidth=25em}}
* [[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#The Pokhozhaev identity for the stationary nonlinear Dirac equation|Pokhozhaev's identity for the stationary nonlinear Dirac equation]]
* [[Soler model]]
* [[Thirring model]]
{{div col end}}

==References==
<references />

[[Category:Quantum field theory]]
[[Category:Dirac equation]]