Why is the partition function

$$Z[J]=\int\ \mathcal{D}\phi\ e^{iS[\phi]+i\int\ d^{4}x\ \phi(x)J(x)}$$

also called the generating function?

Is the partition function a q-number or a c-number?

Does it make sense to talk of a partition function in classical field theory, or can we define partition functions only in quantum field theories?

Is the source $J$ a q-number or a c-number?

  • $\begingroup$ are you just posting around homework problems? $\endgroup$ – gradStudent Oct 7 '16 at 21:12
  • 2
    $\begingroup$ These are not homework problems. And I am not a graduate student. Sorry! $\endgroup$ – nightmarish Oct 7 '16 at 21:16

It is called a generating function, because one can use it to generate $n$-point functions with the aid of functional derivatives with respect to the source $J$. For instance, one can compute the two point function as follows: $$ \langle\phi(x_1)\phi(x_2)\rangle = \frac{\delta}{\delta J(x_1)} \frac{\delta}{\delta J(x_2)} Z[J]_{J=0} . $$ The result would be a $c$-number. Hence, the generating function itself is also a $c$-number.

Formally one can treat classical field theories also with the aid of such generating functions, provided that, if you set $J=0$, you recover the original theory.

These generating functions are based on the path integral approach in which fields can be interpeted as $c$-numbers as apposed to the $q$-numbered operator-valued fields used in the second quantization approach. As a result, the source $J$ is also interpeted as a $c$-number.

  • $\begingroup$ -How would you define the partition function in classical field theory? The is no path-integral in classical field theory. Right? $\endgroup$ – SRS Dec 26 '16 at 13:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.