A Partition function in quantum field theory

[spaghetti3451]

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?

 

[Lucas SV]

Why is the partition function

Z[J]=∫ Dϕ eiS[ϕ]+i∫ d4x ϕ(x)J(x)Z[J]=∫ Dϕ eiS[ϕ]+i∫ d4x ϕ(x)J(x)Z[J]=\int\ \mathcal{D}\phi\ e^{iS[\phi]+i\int\ d^{4}x\ \phi(x)J(x)}

also called the generating function?

Because it is the generator of green's functions (also called n-point correlation functions). See the first section of https://arxiv.org/abs/0712.0689 for definitions.

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

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?

Yes, it does make sense to talk about the partition function of a classical field theory.

Is the source JJJ a q-number or a c-number?

c-number.

 

[spaghetti3451]

How can you have operators on one side of the equation and a c-number on the other side?

 

[Lucas SV]

How can you have operators on one side of the equation and a c-number on the other side?

There are no operators. $\phi$ is a real valued function, called the field configuration, and the path integral is taken over all such configurations. Operators fields are only introduced in the canonical formulation. $\phi$ and $\hat{\phi}$ are related, but not the same thing.

 

[spaghetti3451]

But you mentioned that $Z$ and $J$ are c-numbers.

Does it make sense to talk about c-numbers if we are in classical field theory anyway?

After all, the whole point of c-number and q-number is to distinguish between operators and complex numbers, right?

 

[vanhees71]

In the entire expression only c-numbers are involved. The path-integral formalism is a formalism using only c-numbers (and Grassmann numbers for fermions).

 

[Fightfish]

I found John Cardy's lecture notes (http://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf) quite useful in understanding the entire family of generating functionals in QFT (of course this should ideally be complemented by a textbook). It might be helpful to recognise the analogy between the "partition function" here and the same partition function which arises in standard statistical mechanics (that they have the same name is no accident). In the same way that you derive correlation functions between quantities by differentiating the partition function in stat mech, we also obtain the Green's functions (which are also correlation functions) by functional differentiation of $Z[J]$.

As it turns out, this correspondence is more than just a lucky accident - the imaginary time treatment of the temperature parameter in statistical mechanics leads to a path integral formulation.