Affine lie algebras in massive field theory and form factors from vertex operators

We present a new application of affine Lie algebras to massive quantum field theory in 2 dimensions by investigating theq→1 limit of theq-deformed affine $$\widehat{sl\left( 2 \right)}$$ symmetry of the sine-Gordon theory, this limit occurring at the free fermion point. Working in radial quantization leads to a quasi-chiral factorization of the space of fields. The conserved charges which generate the affine Lie algebrasplit into two independent affine algebras on this factorized space, each with level 1 in the anti-periodic sector, and level 0 in the periodic sector. The space of fields in the anti-periodic sector can be organized using level-1 highest weight representations if one supplements the $$\widehat{sl\left( 2 \right)}$$ algebra with the usual local integrals of motion. Introducing a particle-field duality leads to a new way of computing form-factors in radial quantization. Using the integrals of motion, a momentum space bosonization involving vertex operators is formulated. Form-factors are computed as vacuum expectation values of vertex operators in momentum space.