Correlation Functions in the Schwinger Model at Zero and Finite Temperature

The general correlations between massless fermions are calculated in the Schwinger model at arbitrary temperature. The zero temperature calculations on the plane are reviewed and clarified. Then the finite temperature fermionic Green's function is computed and the results on the torus are compared to those on the plane. It is concluded that a simpler way to calculate the finite temperature results is to associate certain terms in the zero temperature structure with their finite temperature counterparts. A testimony to the complexity of QCD is in the many outstanding problems of the theory that are still not solved. For example, it is not fundamentally understood how quarks are confined or chiral symmetry is broken, though nature seems to adhere to these properties. Many have studied the Schwinger model (QED in two Euclidean dimensions with a massless fermion) [2, 3, 4] in the hopes of gaining intuition for tackling various such problems. Schwinger [1] proposed this model to show that gauge invariance of a vector field need not imply the existence of a massless particle. The gauge field phase shifts the right and left handed massless fermions with respect to one another causing a breakdown of chiral symmetry. This leads to an anomaly which takes form as the generation of a photon mass (m = g/ √ π). The vacuum has a non-trivial topology analogous to the case in QCD. This theory also has charge confinement because of Gauss' law in one space dimension. Since the Schwinger model is exactly solvable, having a form for the general n-point correlation function would be useful in many calculations. The result has been approached on the plane [5], on the sphere [6], and on the torus [7]. Compact surfaces are preferred in calculations because the infinities are kept under control. Fermions on the sphere are difficult to define and for finite temperature considerations the boundary conditions of the torus are more natural. Therefore, work on the torus is desirable. Interest in finite temperature calculations in field theories have led to the evaluation of two-and four-point correlation functions [9]. In this paper, we will provide explicit calculations for a six-point correlation function at finite temperature. When combined with the previous results, the present result yields the generic form for the n-point correlation function for the Schwinger model at finite temperature. Exact finite temperature calculations of correlation function in field theory are of paramount importance for the …