Vacuum Energy from Correlation Functions

[masteralien]

TL;DR Summary

In QFT the n point functions contain all the information about a QFT so how would one compute the vacuum energy just from the n point functions

In QFT the objects of interest are the n point Correlation functions which contain all the information about the theory and can be used to compute any expectation value in principle. However I cant figure out how to compute the vacuum energy from the correlation functions alone and cant find any sources or articles which discuss this computation.

Is there a way to do it with the n point functions only. I know how to compute the Vacuum Energy in QFT with the field operators but want to know how to compute it with the n point functions alone without the field operators.

 

[Demystifier]

The Hamiltonian density is something like
$${\cal H}=\frac{1}{2}\dot{\phi}(x) \dot{\phi}(x)+\ldots$$
Hence the vacuum energy density with the operators is
$$\frac{1}{2}\langle 0|\dot{\phi}(x) \dot{\phi}(x) |0\rangle +\ldots$$
But $\langle 0|\dot{\phi}(x) \dot{\phi}(x) |0\rangle$ is just the 2-point function
$$\langle \dot{\phi}(x) \dot{\phi}(x)\rangle = \lim_{x'\to x} \partial_0\partial'_0\langle \phi(x) \phi(x')\rangle$$
which tells you how, in principle, to compute the vacuum energy density in terms of $n$-point functions.

 

[masteralien]

I see also is it true in principle any expectation value can be computed with the n point functions

 

[Demystifier]

I would say any expectation value in the vacuum can be computed with the n-point functions.

 

[masteralien]

I would say any expectation value in the vacuum can be computed with the n-point functions.

What about expectation values for states which aren’t the vacuum like particle states can all those be computed with the n point functions

 

[Demystifier]

Not directly.

 

[masteralien]

Also one more question can the Propagator G_2(x1-x2) be thought of as the Probability amplitude for a Particle to be found at spacetime point x1 created by a delta function source at spacetime point x2

 

[Demystifier]

Also one more question can the Propagator G_2(x1-x2) be thought of as the Probability amplitude for a Particle to be found at spacetime point x1 created by a delta function source at spacetime point x2

Yes, at least in nonrelativistic QM.