- Partition function (quantum field theory)
In

quantum field theory , we have agenerating functional , Z [J] of correlation functions and this value, called the**partition function**is usually expressed by something like the followingfunctional 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 partition function in mathematics, and is related to the partition function in statistical mechanics. The primary difference is that the

countable collection ofrandom variable s seen in the definition of such simpler partition functions has been replaced by an uncountable set, thus necessitating the use offunctional integral s over a field $phi$.**Uses**The prototypical use of the partition function is to obtain

Feynman amplitude s by differentiating with respect to the auxiliary function (sometimes called the "current") "J". Thus, for example::$langle\; G(x\_1,x\_2)\; angle\; =\; left.frac\{delta\}\{delta\; J(x\_1)\}\; frac\{delta\}\{delta\; J(x\_2)\}\; log\; Z\; [J]\; ight|\_\{J=0\}$

is the

Green's function ,propagator orcorrelation function for the field $phi$ between points $x\_1$ and $x\_2$ in space.**Complex-valued action**Unlike the 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 rotation s; this is not so. Rather, the "i" has to do with the fact that the fields $phi$ are to be interpreted as quantum-mechanicalprobability amplitude s, taking on values in thecomplex projective space (complexHilbert 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 asimplex --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 theJacobian 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 othermathematical 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: [*http://www.physik.fu-berlin.de/~kleinert/b5 PDF-files*] )"

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