Statistical field theory
Encyclopedia
A statistical field theory is any model in statistical mechanics
where the degrees of freedom
comprise a field
or fields. In other words, the microstates
of the system are expressed through field configurations. It is closely related to quantum field theory
, which describes the quantum mechanics
of fields, and shares with it many phenomena, such as renormalization
.
If the system involves polymers, it is also known as polymer field theory
.
In fact, by performing a Wick rotation
from Minkowski space
to Euclidean space
, many results of statistical field theory can be applied directly to its quantum equivalent. The correlation function
s of a statistical field theory are called Schwinger function
s, and their properties are described by the Osterwalder–Schrader axioms.
Statistical field theories are widely used to describe systems in polymer physics
or biophysics
, such as polymer
films, nanostructured block copolymers or polyelectrolyte
s.
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
where the degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
comprise a field
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
or fields. In other words, the microstates
Microstate (statistical mechanics)
In statistical mechanics, a microstate is a specific microscopic configuration of a thermodynamic system that the system may occupy with a certain probability in the course of its thermal fluctuations...
of the system are expressed through field configurations. It is closely related to quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, which describes the quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
of fields, and shares with it many phenomena, such as renormalization
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
.
If the system involves polymers, it is also known as polymer field theory
Polymer field theory
A polymer field theory is a statistical field theory describing the statistical behavior of a neutral or charged polymer system. It can be derived by transforming the partition function from its standard many-dimensional integral representation over the particle degrees of freedom in a functional...
.
In fact, by performing a Wick rotation
Wick rotation
In physics, Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable...
from Minkowski space
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
to Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, many results of statistical field theory can be applied directly to its quantum equivalent. The correlation function
Correlation function (quantum field theory)
In quantum field theory, the matrix element computed by inserting a product of operators between two states, usually the vacuum states, is called a correlation function....
s of a statistical field theory are called Schwinger function
Schwinger function
In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to the ordered set of points in Euclidean space with no coinciding points...
s, and their properties are described by the Osterwalder–Schrader axioms.
Statistical field theories are widely used to describe systems in polymer physics
Polymer physics
Polymer physics is the field of physics that studies polymers, their fluctuations, mechanical properties, as well as the kinetics of reactions involving degradation and polymerisation of polymers and monomers respectively....
or biophysics
Biophysics
Biophysics is an interdisciplinary science that uses the methods of physical science to study biological systems. Studies included under the branches of biophysics span all levels of biological organization, from the molecular scale to whole organisms and ecosystems...
, such as polymer
Polymer
A polymer is a large molecule composed of repeating structural units. These subunits are typically connected by covalent chemical bonds...
films, nanostructured block copolymers or polyelectrolyte
Polyelectrolyte
Polyelectrolytes are polymers whose repeating units bear an electrolyte group. These groups will dissociate in aqueous solutions , making the polymers charged. Polyelectrolyte properties are thus similar to both electrolytes and polymers , and are sometimes called polysalts. Like salts, their...
s.