Soler model information

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

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

where $g$ is the coupling constant, $\partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}$ in the Feynman slash notations, ${\overline {\psi }}=\psi ^{*}\gamma ^{0}$ . Here $\gamma ^{\mu }$ , $0\leq \mu \leq 3$ , are Dirac gamma matrices.

The corresponding equation can be written as

i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi

,

where $\alpha ^{j}$ , $1\leq j\leq 3$ , and $\beta$ are the Dirac matrices. In one dimension, this model is known as the massive Gross–Neveu model.^[3]^[4]

^ Dmitri Ivanenko (1938). "Notes to the theory of interaction via particles" (PDF). Zh. Eksp. Teor. Fiz. 8: 260–266.
^ 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.
^ 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)
^ 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.

[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.

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