How Uncertainty Principle, Vacuum fluctuations and Energy Conservation coexist in QFT? Recently I had a debate about the uncertainty principle in QFT that made me even more confused..
Because we use Fourier transforms in QFT, we should have an analogue to the usual Heisenberg uncertainty principle in QFT between the 4-vector of space-time coordinates and the conjugate momentum, but I found no reference for that, so is my guess is wrong?
We know that there is no universal Hermitian operator for time in QM, even so there is an uncertainty principle for time and energy, well, in QFT time is just a parameter, the same as the spatial coordinates, so is there an uncertainty principle for energy in QFT?
The last question made me confused regarding the energy conservation law in QFT: we use this law in QFT during calculations of propagators only (as I remember), it means we are using it with "bare" particles, while we suppose that these particles don't "interact" with vacuum fluctuations, so does that mean energy conservation law is a statistical law?
This brings to my mind the vacuum expectation value, that we suppose is zero for any observer, but it is zero statistically. At the same time we usually use Noether's theorem to deduce that energy is conserved (locally at least, and not statistically).
I believe I'm missing something here, can you please advise me?
 A: 
...uncertainty principle in QFT between the 4-vector of space-time coordinates and the conjugate momentum...

Conjugate momentum is "conjugate" to a particular generalized coordinate. Which are field values in case of QFT.  Space-time coordinates  (as you've noted yourself) are just parameters. So, I'm afraid, you are mixing two different things here.

... is there uncertainty principle for energy in QFT?

Yes. In QM and QFT observables are hermitian operators. If some pair of those operators do not commute -- you get an uncertainty principle. Time-energy  uncertainty is a little bit tricky, and nicely explained, say, in this question.

... we use energy conservation law in QFT during calculations of propagators only

That is a strange claim. In the beginning of most QFT textbooks you can find a derivation of an energy-momentum conservation by means of Noether's theorem.

... energy conservation law is a statistical law? 

Here is the list of statistical laws. As you can see there is not even a hint on energy conservation law. Therefore, either:


*

*The answer is "no" and you should be satisfied with it.

*You are for some reason (winning a debate?) inventing your own terminology. Which is a bad idea anyway.
A: There is an uncertainty relation in QFT for averaged field and the corresponding momentum operators. For a detailed discussion see here. (e-Print: arXiv:1208.3647 [hep-th])
Similarly to normal QM there is no "energy-time" uncertainty relation which would have the same meaning. 
A: I am only familiar with QM. Pauli (or maybe Dirac) wrote that there is a symmetry: energy-time is perfectly analogous to momentum-position, and one can think of energy as the momentum a thing has as it travels thru time.
Einstein tells us that one man's space is another man's time, so one man's momentum IS another man's energy.
How can energy be conserved AND uncertain? Remember basic vector space theory: every state can be expressed as a linear combination of other states (which form a basis). So a state which is not a state of definite energy can be expressed as a linear combination of states which ARE of (different) definite energies. In each of those states, energy is conserved; the answer to your question follows.
