**Merab Eliashvili**

## On Quantum Fields and Systems with Ordered Ground State

**abstract:**

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.