Partition function (quantum field theory)

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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:

$Z[J] = \int \mathcal{D}\phi e^{i(S[\phi]+\int d^dx J(x)\phi(x))}$

where S is the action functional.

The partition function in quantum field theory is a special case of the mathematical partition function, and is related to the statistical partition function in statistical mechanics. The primary difference is that the countable collection of random variables seen in the definition of such simpler partition functions has been replaced by an uncountable set, thus necessitating the use of functional integrals over a field $\phi$.

Uses

The prototypical use of the partition function is to obtain probability amplitudes in Feynman path integral by differentiating with respect to the auxiliary function (sometimes called the current) J. Thus, for example:

$\langle G(x_1,x_2)\rangle = \left. -\frac{\delta}{\delta J(x_1)} \frac{\delta}{\delta J(x_2)} \log Z[J] \right|_{J=0}$

is the Green's function, propagator or correlation function for the field $\phi$ between points $x_1$ and $x_2$ in space.

Complex-valued action

Unlike the partition function in statistical mechanics, the partition function 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 rotations; this is not so. Rather, the i has to do with the fact that the fields $\phi$ are to be interpreted as quantum-mechanical probability amplitudes, 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 a compact geometric domain admitting a cumulative sum of 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.

Books

• Kleinert, Hagen, 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)