# How Uncertainty Principle, Vacumm 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 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).

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I similar question I found here, but unfortunately it gives no sufficient answer. –  TMS Mar 5 at 8:50

...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.

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The uncertainty principle derivation you pointed is for QM not QFT, and it uses Schrodinger equation directly, that is not what I'm looking for, also I mentioned the "use" of conversation law in real calculations, Noether theorem has no direct use there, also I'm looking for to analyze the the question here, providing list of statistical laws adds nothing to my understandings, please check my own answer on the question, may be you will correct me there more clearly. –  TMS Mar 8 at 21:11
@TMS There is nothing to derive. The uncertainty principle carries over exactly from QM to QFT, you just have to be clear what the operators are. The basic operators are the fields and their conjugate momenta, which obey commutation relations as discussed in any QFT book. Write any operator you like, including the Hamiltonian, in terms of the basic operators and you can work out the commutation relations. Then apply the usual uncertainty principle to that. That's all there is to it. There are no new conceptual difficulties in QFT (at least as far as this is concerned). –  Michael Brown Mar 9 at 0:22
@TMS And what do you want to use for the time-energy uncertainty relation except for the Schrodinger equation? The time evolution generated by the Schrodinger equation is what the time-energy uncertainty relation is all about. –  Michael Brown Mar 9 at 0:24
@MichaelBrown: I'm trying to derive and understand it in QFT theory, here we have no Schrodinger equation! in other wards, I can't see how relativity, vacuum fluctuations doesn't affect this uncertainty. –  TMS Mar 9 at 7:51
@TMS You can do any "picture" - Schrodinger, Heisenberg (this is the usual one in QFT), or interaction. Vaccum fluctuations don't modify energy conservation for the same reason that they don't in ordinary QM: the theory has a time translation symmetry. This means that there is a time-independent Hamiltonian which defines the energy eigenstates. In QFT these are states which contain superpositions of many field configurations (this is what we mean by "vacuum fluctuations"), just like the energy eigenstates state of a harmonic oscillator is smeared in position space. –  Michael Brown Mar 9 at 11:57

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.

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As much as nobody answered my question, I will post what I get into, please correct/support me:

Strictly speaking, Energy Conversation in QFT is a Statistical Law, BUT (a big one!) it is statistical up to uncertainty principle (explanations follows), in other words, we never can measure this statistical behavior of the law, thus we can stick with it's precise formulation.

When we deal with bare particles, there is no field interactions with vacuum, then we always using precise energy conservation law, also when we calculating S-matrix, it is stationary treatment of the theory, averaged over infinite time, thus there we can also use energy conservation law strictly.

Vacuum interaction is basically affect on the probabilities of the output (state), but the energy is always conserved, anyway this happens only if we are dealing out of uncertainty condition, this happens because average expectation value of vacuum is always zero, again is zero up to uncertainty principle, and we never can explain what happened below uncertainty principles, and more important, what happens there will not affect the energy conservation (will only change probability of the output), and for that reason we still capable to say that all fields are Lorentz invariant (and thus energy conserved, not statistically), without specifying the scale we are talking about (within or not uncertainty principle).

P.S

The only left thing is to find/derive uncertainty principle (especially for Energy) within QFT framework,I will be glad if anybody can point for a reference.

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Actually I asked clearly to support/correct me, not to down vote, please specify your point of view. –  TMS Mar 8 at 21:04