Thermal field theory: Algebraic aspects and applications to confined systems

Abstract

A resume of recent trends in thermal field theory is presented with emphasis on algebraic aspects. In this sense, some representations of Lie symmetries provide, in particular, a unified axiomatization, via the so-called thermofield dynamics (TFD) approach, of different methods treating thermal systems. First, a connection between imaginary and real time formalism is presented, with emphasis on physical paradigms of thermal physics. The study of Poincare Lie algebra leads us to a derivation of Liouville-like equations for the scalar and Dirac field, and as an application the Juttiner distribution for bosons is obtained. Exploring the fact that a finite temperature prescription results to be equivalent to a path-integral calculated on R D−1 × S 1, where S 1 is a circle of circumference β = 1/T, a generalization of the thermal quantum field theory is presented in order to take into account the space confinement of fields. In other words, we consider the TFD and the Matsubara mechanism on a \( R^{D - N} \times S^{1_1 } \times S^{1_2 } ... \times S^{1_N } \) topology, describing time (temperature) and space confinement. The resulting geometrical approach is then applied to analyse the 3 — D N — component Gross-Neveu model compactified in a square of side L, at a temperature T. The main result is a closed expression for the large-N effective coupling constant, g(L, T). For large values of the fixed coupling constant, we obtain simultaneously asymptotic freedom, spacial confinement and a decoupling transition at a temperature T d. Taking the Gross-Neveu model as describing the effective interaction between quarks, the confining length and the deconfining temperature obtained are of the order of the expected values for hadrons.