Jump to content

Soler model

From Wikipedia, the free encyclopedia

The soler model is a quantum field theory model of Dirac fermions interacting via four fermion interactions in 3 spatial and 1 time dimension. It was introduced in 1938 by Dmitri Ivanenko [1] and re-introduced and investigated in 1970 by Mario Soler[2] as a toy model of self-interacting electron.

This model is described by the Lagrangian density

where is the coupling constant, in the Feynman slash notations, . Here , , are Dirac gamma matrices.

The corresponding equation can be written as

,

where , , and are the Dirac matrices. In one dimension, this model is known as the massive Gross–Neveu model.[3][4]

Generalizations

[edit]

A commonly considered generalization is

with , or even

,

where is a smooth function.

Features

[edit]

Internal symmetry

[edit]

Besides the unitary symmetry U(1), in dimensions 1, 2, and 3 the equation has SU(1,1) global internal symmetry.[5]

Renormalizability

[edit]

The Soler model is renormalizable by the power counting for and in one dimension only, and non-renormalizable for higher values of and in higher dimensions.

Solitary wave solutions

[edit]

The Soler model admits solitary wave solutions of the form where is localized (becomes small when is large) and is a real number.[6]

Reduction to the massive Thirring model

[edit]

In spatial dimension 2, the Soler model coincides with the massive Thirring model, due to the relation , with the relativistic scalar and the charge-current density. The relation follows from the identity , for any .[7]

See also

[edit]

References

[edit]
  1. ^ Dmitri Ivanenko (1938). "Notes to the theory of interaction via particles" (PDF). Zh. Eksp. Teor. Fiz. 8: 260–266.
  2. ^ 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.
  3. ^ 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)
  4. ^ S.Y. Lee & A. Gavrielides (1975). "Quantization of the localized solutions in two-dimensional field theories of massive fermions". Phys. Rev. D. 12 (12): 3880–3886. Bibcode:1975PhRvD..12.3880L. doi:10.1103/PhysRevD.12.3880.
  5. ^ Galindo, A. (1977). "A remarkable invariance of classical Dirac Lagrangians". Lettere al Nuovo Cimento. 20 (6): 210–212. doi:10.1007/BF02785129. S2CID 121750127.
  6. ^ Thierry Cazenave & Luis Vàzquez (1986). "Existence of localized solutions for a classical nonlinear Dirac field". Comm. Math. Phys. 105 (1): 35–47. Bibcode:1986CMaPh.105...35C. doi:10.1007/BF01212340. S2CID 121018463.
  7. ^ J. Cuevas-Maraver; P.G. Kevrekidis; A. Saxena; A. Comech & R. Lan (2016). "Stability of solitary waves and vortices in a 2D nonlinear Dirac model". Phys. Rev. Lett. 116 (21): 214101. arXiv:1512.03973. Bibcode:2016PhRvL.116u4101C. doi:10.1103/PhysRevLett.116.214101. PMID 27284659. S2CID 15719805.