Soler model

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 and re-introduced and investigated in 1970 by Mario Soler as a toy model of self-interacting electron.

This model is described by the Lagrangian density

ℒ = \overline{ψ} (i \partial / - m) ψ + \frac{g}{2} {(\overline{ψ} ψ)}^{2}

where $g$ is the coupling constant, $\partial / = \sum_{μ = 0}^{3} γ^{μ} \frac{\partial}{\partial x^{μ}}$ in the Feynman slash notations, $\overline{ψ} = ψ^{*} γ^{0}$ . Here $γ^{μ}$ , $0 \leq μ \leq 3$ , are Dirac gamma matrices.

The corresponding equation can be written as

i \frac{\partial}{\partial t} ψ = - i \sum_{j = 1}^{3} α^{j} \frac{\partial}{\partial x^{j}} ψ + m β ψ - g (\overline{ψ} ψ) β ψ

,

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

Generalizations

A commonly considered generalization is

ℒ = \overline{ψ} (i \partial / - m) ψ + g \frac{{(\overline{ψ} ψ)}^{k + 1}}{k + 1}

with $k > 0$ , or even

ℒ = \overline{ψ} (i \partial / - m) ψ + F (\overline{ψ} ψ)

,

where $F$ is a smooth function.

Features

Internal symmetry

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

Renormalizability

The Soler model is renormalizable by the power counting for $k = 1$ and in one dimension only, and non-renormalizable for higher values of $k$ and in higher dimensions.

Solitary wave solutions

The Soler model admits solitary wave solutions of the form $ϕ (x) e^{- i ω t},$ where $ϕ$ is localized (becomes small when $x$ is large) and $ω$ is a real number.

Reduction to the massive Thirring model

In spatial dimension 2, the Soler model coincides with the massive Thirring model, due to the relation $(\bar{ψ} ψ)^{2} = J_{μ} J^{μ}$ , with $\bar{ψ} ψ = ψ^{*} σ_{3} ψ$ the relativistic scalar and $J^{μ} = (ψ^{*} ψ, ψ^{*} σ_{1} ψ, ψ^{*} σ_{2} ψ)$ the charge-current density. The relation follows from the identity $(ψ^{*} σ_{1} ψ)^{2} + (ψ^{*} σ_{2} ψ)^{2} + (ψ^{*} σ_{3} ψ)^{2} = (ψ^{*} ψ)^{2}$ , for any $ψ \in ℂ^{2}$ .

References

↑ 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.
↑ Galindo, A. (1977). "A remarkable invariance of classical Dirac Lagrangians". Lettere al Nuovo Cimento. 20 (6): 210–212. doi:10.1007/BF02785129. S2CID 121750127.
↑ 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.
↑ 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.

[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.