Partition function (quantum field theory)
Encyclopedia
In quantum field theory
, we have a generating functional, Z[J] of correlation functions
and this value, called the partition function is usually expressed by something like the following functional integral:
where S is the action
functional
.
The partition function in quantum field theory is a special case of the partition function in mathematics
, and is related to the partition function in statistical mechanics
. The primary difference is that the countable collection of random variable
s seen in the definition of such simpler partition functions has been replaced by an uncountable set, thus necessitating the use of functional integrals over a field
.
is the Green's function
, propagator
or correlation function
for the field
between points 
and 
in space.
s; this is not so. Rather, the i has to do with the fact that the fields
are to be interpreted as quantum-mechanical probability amplitude
s, taking on values in the complex projective space
(complex Hilbert space
, but the emphasis is placed on the word projective, because the probability amplitudes are still normalized to one). By contrast, more traditional partition functions involve random variables that are real-valued, and range over a simplex
--a simplex, being the geometric way of saying that the total of probabilities sum to one. The factor of i can be understood to arise as the Jacobian of the natural measure of volume in complex projective space. For the (highly unusual) situation where the complex-valued probability amplitude is to be replaced by some other field taking on values in some other mathematical space
, the i would be replaced by the appropriate geometric factor (that is, the Jacobian) for that space.
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...
, we have a generating functional, Z[J] of correlation functions
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....
and this value, called the partition function is usually expressed by something like the following functional integral:
where S is the action
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
functional
Functional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
.
The partition function in quantum field theory is a special case of the partition function in mathematics
Partition function (mathematics)
The partition function or configuration integral, as used in probability theory, information science and dynamical systems, is an abstraction of the definition of a partition function in statistical mechanics. It is a special case of a normalizing constant in probability theory, for the Boltzmann...
, and is related to the partition function in statistical mechanics
Partition function (statistical mechanics)
Partition functions describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas...
. The primary difference is that the countable collection of random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s seen in the definition of such simpler partition functions has been replaced by an uncountable set, thus necessitating the use of functional integrals over a field
.Uses
The prototypical use of the partition function is to obtain Feynman amplitudes by differentiating with respect to the auxiliary function (sometimes called the current) J. Thus, for example:is the Green's function
Green's function
In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
, propagator
Propagator
In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. Propagators are used to represent the contribution of virtual particles on the internal...
or correlation function
Correlation function
A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...
for the field
between points and in space.Complex-valued action
Unlike the partition function in statistical mechanics, that in quantum field theory contains an extra factor of i in front of the action, making the integrand complex, not real. It is sometimes mistakenly implied that this has something to do with Wick rotationWick 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...
s; this is not so. Rather, the i has to do with the fact that the fields
are to be interpreted as quantum-mechanical probability amplitudeProbability amplitude
In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...
s, taking on values in the complex projective space
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
(complex Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, but the emphasis is placed on the word projective, because the probability amplitudes are still normalized to one). By contrast, more traditional partition functions involve random variables that are real-valued, and range over a simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
--a simplex, being the geometric way of saying that the total of probabilities sum to one. The factor of i can be understood to arise as the Jacobian of the natural measure of volume in complex projective space. For the (highly unusual) situation where the complex-valued probability amplitude is to be replaced by some other field taking on values in some other mathematical space
Mathematical space
In mathematics, a space is a set with some added structure.Mathematical spaces often form a hierarchy, i.e., one space may inherit all the characteristics of a parent space...
, the i would be replaced by the appropriate geometric factor (that is, the Jacobian) for that space.
Books
- Kleinert, HagenHagen KleinertHagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany , at theWest University of Timişoara, at thein Bishkek. He is also of the...
, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition, World Scientific (Singapore, 2004); Paperback ISBN 981-238-107-4 (also available online: PDF-files)