Supriya Ghosh (Editor)

Soler model

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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 1970 by Mario Soler as a toy model of self-interacting electron.

Contents

This model is described by the Lagrangian density

L = ψ ¯ ( i ∂ / − m ) ψ + g 2 ( ψ ¯ ψ ) 2

where g is the coupling constant, ∂ / = ∑ μ = 0 3 γ μ ∂ ∂ x μ in the Feynman slash notations, ψ ¯ = ψ ∗ γ 0 . Here γ μ , 0 ≤ μ ≤ 3 , are Dirac gamma matrices.

The corresponding equation can be written as

i ∂ ∂ t ψ = − i ∑ j = 1 3 α j ∂ ∂ x j ψ + m β ψ − g ( ψ ¯ ψ ) β ψ ,

where α j , 1 ≤ j ≤ 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 = ψ ¯ ( i ∂ / − m ) ψ + g ( ψ ¯ ψ ) k + 1 k + 1

with k > 0 , or even

L = ψ ¯ ( i ∂ / − m ) ψ + F ( ψ ¯ ψ ) ,

where F is a smooth function.

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.

References

Soler model Wikipedia
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