Partition function (quantum field theory)

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In quantum field theory, the partition function Z[J] is the generating functional of correlation functions. It 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is that the countable collection of random variables seen in the definition of such simpler partition functions has been replaced by an uncountable set, thus necessitating the use of functional integrals over a field \phi.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[edit]

The n-point correlation functions G_n can be expressed using the path integral formalism as


G_n(x_1,...,x_n)
\equiv \langle \Omega | T \{ \phi(x_1) \cdots \phi(x_n) \} | \Omega \rangle
= \frac{\int \mathcal{D} \phi \, \phi(x_1) \cdots \phi(x_n) \exp ( iS[\phi]/\hbar )}
{\int \mathcal{D} \phi \, \exp ( iS[\phi]/\hbar )}

where the left-hand side is the time-ordered product used to calculate S-matrix elements. The \mathcal{D} \phi on the right-hand side means integrate over all possible classical field configurations \phi(x) with a phase given by the classical action S[\phi] evaluated in that field configuration.[1] The generating functional Z[J] can be used to calculate the above path integrals using an auxiliary function J (called current in this context). From the definition (in a 4D context)

Z[J] = \int \mathcal{D}\phi \exp \left\{ \frac{i}{\hbar} \left[ S[\phi]+\int d^4 x J(x)\phi(x)) \right] \right\}

it can be seen using functional derivatives that the n-point correlation functions G_n(x_1,...,,x_n) are given by


G_n(x_1,...,x_n) = (-i \hbar)^n \frac{1}{Z[0]} \left. \frac{\partial^n Z}{ \partial J(x_1) \cdots \partial J(x_n)} \right|_{J=0}

Connection with statical mechanics[edit]

The generating functional is the quantum field theory analog of the partition function in statistical mechanics: it tells us everything we could possibly want to know about a system. The generating functional is the holy grail of any particular field theory: if you have an exact closed-form expression for Z[J] for a particular theory, you have solved it completely.[2]

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[according to whom?] 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[citation needed], 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[edit]

  • 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)

References[edit]