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?