Coleman-Weinberg potential
Encyclopedia
The Coleman–Weinberg model represents quantum electrodynamics
of a scalar field in four-dimensions. The Lagrangian
for the model is
where the scalar field is complex,
is the electromagnetic field tensor, and 
the covariant derivative containing the electric charge 
of the electromagnetic field. The model illustrates the generation of mass by fluctuations of the vector field. Equivalently one may say that the model possesses a first-order phase transition
as a function of
. The model is the four-dimensional analog of the three-dimensional Ginzburg–Landau theory used to explain the properties of superconductors
near the phase transition
. Interestingly, the three-dimensional version of the Coleman–Weinberg model has both a first and a second-order phase transition depending on the ratio of the Ginzburg–Landau parameter
, with a tricritical point
near
which separates type I
from type II superconductivity
.
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
of a scalar field in four-dimensions. The Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
for the model is
where the scalar field is complex,
is the electromagnetic field tensor, and the covariant derivative containing the electric charge of the electromagnetic field. The model illustrates the generation of mass by fluctuations of the vector field. Equivalently one may say that the model possesses a first-order phase transitionPhase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
as a function of
. The model is the four-dimensional analog of the three-dimensional Ginzburg–Landau theory used to explain the properties of superconductorsSuperconductivity
Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...
near the phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
. Interestingly, the three-dimensional version of the Coleman–Weinberg model has both a first and a second-order phase transition depending on the ratio of the Ginzburg–Landau parameter
, with a tricritical pointTricritical point
In condensed matter physics, dealing with the macroscopic physical properties of matter, a tricritical point is a point in the phase diagram of a system at whichthree-phase coexistence terminates...
near
which separates type IType I superconductor
Superconductors cannot be penetrated by magnetic flux lines . This Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs...
from type II superconductivity
Superconductivity
Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...
.