Existence and renormalization of the Sine-Gordon field theory in all dimensions

The Euclidean field theory of the Sine-Gordon model (SG) is investigated by using methods of statistical physics. The SG model is shown to be the continuum limit of the ferromagnetic Harmonic Rotator model (HR) which itself is equivalent to a classical Coulomb plasma of unit charges on a lattice. Using our recent results for the latter models in all Euclidean dimensionsD we determine the existence of the SG field theory in terms of the HR parameters, temperaturet and magnetic fieldB. The following results are derived: 1. InD=1 the SG exists neart=B=0; the quantum theory in zero space dimensions is just the Mathieu equation as Schrödinger equation. 2. ForD>2 the SG field theory exists for allt nearB=0. The theory is actually constructed and is equivalent to a free massive scalar theory. 3. In the most interesting case ofD=2 the SG field theory exists for allt<t=8 nearB=0, it does not exist fortt. All necessary renormalizations are performed and all necessary subtractions are obtained in closed form which proves that the theory is superrenormalizable when it exists. We also discuss the relations between the structure of the particle spectrum of the SG, the phase transitions of the HR, and the binding properties of the classical Coulomb charges.