Recently I had a debate about the uncertainty principle in QFT that made me even more confused..
Because we use Furrier 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 uncertainty principle for time and energy, well, in QFT time is just a parameter same as spatial coordinates, so is there uncertainty principle for energy in QFT?
The last question made me confused regarding energy conservation law in QFT: we use this law in QFT during calculations of propagators only (as I remember), it means we was using it with "bare" particles, while we supposing that this particles don't "interact" with vacuum fluctuations, so dose that mean energy conservation law is a statistical law? this brings to my mind vacuum expectation value, that we suppose it as zero for any observer, but it is zero statistically, in the same time we used to use Noeather theorem to deduce that energy is conserved (locally at least, and not statistically).
I believe I'm missing something here, can you please advice me?