In the first part of the work, we consider problems of the field theory approach to the quantum Hall effect (QHE) and the anyon superconductivity. In the remaining part, we study the convexity property of the effective potential and some features of the high temperature behaviour of field theory models with two scalar fields. Describing the QHE, we adapt the Chern--Simons gauge theory in the holomorphic gauge which seems to incorporate Laughlin's picture of the incompressible quantum fluid and Jain's composite fermion approach. Further, the 2+1 dimensional relativistic field theory is used to describe the Meissner effect in the planar superconducting matter. Using Bogolubov's concept of quasiaverages and analyzing the boundary conditions in the path integral representation of the Green functions generating funcitonal, a self-consistent calculation of the effective potential is proposed. Studying the effective potentials for the two Higgs models, it is demonstrated that the symmetry can be spontaneously broken in some high temperature interval, being exact at zero and very high temperatures.
Mathematics Subject Classification: 81T99.
Key words and phrases: Chern-Simons theory, quantum Hall effect, similarity transformation, complex gauge potentials and holomorphic quantization, anyon superconductivity, structure functions, effective potential, convexity, ground state, symmetry breaking.