Solitons and Spontaneous Symmetry Breaking in 2 and 4 Dimensions

We show that mass generation in 1+1 and 3+1 dimensions may occur together with spontaneous symmetry breaking. 1. Thirring model in the chirally broken phase Skyrme’s conjectured [1] that the solitons of the sine-Gordon model have the properties of fermions and couple by an interaction of the Thirring model type. The sine-Gordon model is a model of a bosonic field θ(x) in 1+1–dimensional space–time with a Lagrangean which in Coleman’s [2] notation reads L (x) = 1 2 ∂μθ(x)∂θ(x)+ α β2 (cosβθ(x)−1). (1) The Thirring model [3] describes a self-coupled Dirac-field ψ(x) in 1+1 dimension L (x) = ψ̄(x)(iγ∂μ −m)ψ(x)− 1 2 g ψ̄(x)γψ(x)ψ̄(x)γμψ(x), (2) where m is the mass of the fermion field and g is a dimensionless coupling constant. The field ψ(x) is a spinor field with two components ψ1(x) and ψ2(x). The Presented by Manfried Faber at the NATO Advanced Research Workshop “Confinement, Topology, and other Non-Perturbative Aspects of QCD”, January 21–27, 2002, Stará Lesná (Slovakia). Supported in part by Fonds zur Förderung der Wissenschaftlichen Forschung P13997-TPH. 2 γ–matrices are defined in terms of the well–known 2×2 Pauli matrices, γ 0 = σ1, γ1 = −iσ2, γ5 = γ0γ1 = −iσ1σ2 = σ3. Coleman suggested a perturbative approach to the understanding of the equivalence between the sine–Gordon and the Thirring model. He developed a perturbation theory with respect to α and m in order to compare the n–point Green functions in the sine-Gordon and the massive Thirring model in coordinate representation. Under the assumption of the existence of these two theories, Coleman concluded that they should be equivalent if the coupling constants β and g obey the relation [2] 4π β2 = 1+ g π (3) and the operators ψ(x) and θ(x) satisfy the Abelian bosonisation rules Z m ψ̄(x) (