12.7. Equivalence of the Sine–Gordon and Thirring models Consider the Sine–Gordon model of a scalar bosonic field ϕ, whose Lagrangian is L = 1 2

(∂ϕ)2 + m2

β2 (cosβϕ −1).

Use the bosonization formulae, prove that this Lagrangian can be transformed in the Lagrangian of the Thirring model L = i ¯ γμ ∂μ −M ¯ − 1 2

g ( ¯ γμ)( ¯ γμ)

where  is a complex fermionic field, with the coupling constants related as

β2 4π

= 1 1+ g

π

.

Note that β2 = 4π is equivalent to g = 0, i.e. a free fermionic model!