# Interactions in QFT and consistent field energy...

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## Main Question or Discussion Point

I have a not-very-well formulated question about the interaction picture of QFT.

I understand that, in an interaction picture, particle numbers are not well defined (except as t goes to infinity and you're back at free fields). However, at the very least, in an interaction picture, a field carries a certain amount of energy. Does this energy remain during interaction? Or is it transferred to other fields - or is there no answer?

An example would help to clarify what I mean above (which is probably not clear). Let's take a simple interaction between two electrons (at reasonable energy). I formulate a scattering experiment: two electrons go in, two electrons come out...

Now, one may not be able say that a specific number of electrons are present during the interaction - but can one at least say the electron field remains excited throughout? Or is it possible that, all energy is somehow transferred to the photon field and back? (or some other odd situation) I ask in part because this is a question of ontology (what actually exists...). The two electron case is just a simple example, but I'm wondering more generally what can be said.

Please forgive the naive nature of the question - I don't have sufficient background in QFT to determine what can be said about field excitations during interactions.

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PeterDonis
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at the very least, in an interaction picture, a field carries a certain amount of energy
No, it doesn't. During the interaction, there is no well-defined value for anything like "the energy carried by a particular field".

I ask in part because this is a question of ontology (what actually exists...)
I don't see how. A quantum field that is in its vacuum state (i.e., with no excitations) still exists.

Thanks @PeterDonis - so you're effectively saying that nothing can be said about the excitation of an individual field? That is, I may have two electrons that go into my scattering experiment, but during the the interaction, nothing at all can be said about the field states (of not just the electron field, but the photon field, etc.)?

So one cannot say that the electron field still carries excitations (the original electrons) - but is there no mathematical description of of the interaction?

I mention ontology in the sense of what can be said about the interactions at all - what field excitations remain and how they change - not in the sense of the underlying fields themselves (which clearly exist).... it seems that during the interaction, one can't even specify the state (excitation) of any of the fields at all?

Thanks for your patience with these questions.

PeterDonis
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so you're effectively saying that nothing can be said about the excitation of an individual field?
Not during an interaction, since the whole point is that during an interaction you're not measuring anything (you only measure the "in" state and the "out" state, not what happens in between), and you can't assign definite values to things you aren't measuring.

during the the interaction, nothing at all can be said about the field states (of not just the electron field, but the photon field, etc.)?
No, because you're not measuring them. See above.

You seem to be wanting to think about an interaction as some smooth change of "field states" from a set of "in" states to a set of "out" states. You can't think about interactions in QFT that way; that kind of thinking is a holdover from classical physics and needs to be abandoned if you want to understand quantum physics.

But surely there is a mathematical description of what's happening?

What you're saying above, if I read correctly, is that you can't say anything about anything really - what even exists during the interaction (except presumably the fields themselves). If I start with asymptotic free states with definite energies, I certainly shouldn't have more energy during an interaction?? (as an example)

Further, low energy interactions happen all the time (chemical reactions) - can we really say nothing about these cases also?

Nugatory
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But surely there is a mathematical description of what's happening?
There is. But it doesn't do what you want it to do.
The mathematical description allows you to calculate the probability, given the "in" state, of any given "out" state when and if you get around to measuring the "out" state. It says absolutely nothing about what happens in between.
Further, low energy interactions happen all the time (chemical reactions) - can we really say nothing about these cases also?
That's right, even if you don't dig deep into the quantum electrodynamics of how electrons interact to form chemical bonds. Look at the high-school simple reaction $2H_2+O_2\rightarrow 2H_2O$ - it has an "in" state and an "out" state and says nothing about what happens in between.

PeterDonis
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What you're saying above, if I read correctly, is that you can't say anything about anything really
Obviously not. You can say there's a definite "in" state, and once you measure the outcome, you can say there's a definite "out" state. You can also say plenty about the probabilities of various "out" states for a given "in" state. You just can't describe the interaction in terms of smoothly changing states between the "in" and "out" states.

If I start with asymptotic free states with definite energies, I certainly shouldn't have more energy during an interaction??
If you're not measuring the energy during the interaction, you can't say anything about its value.

Again, you seem to be stuck in a mindset that everything ought to have definite values during the interaction, instead of just before and after. That is a holdover from classical physics and you need to abandon that mindset if you want to understand quantum physics. Please read this paragraph again and again until it sinks in.

@PeterDonis - not definite values... just even a mathematical description - are there no probability distributions over intermediate states / conditions?

I'm just wondering if it's truly the case that two electrons (field excitations) go in - and then, who knows ... maybe during the interaction the electron field ends up in the vacuum state (electrons gone) and the photon field is excited, or some other field is excited (in some probabilistic sense, I don't know?). Maybe the two electrons fly off to the sun, do a little dance, and come back....

So I return to my original question about existence ... my body is made up of lots of protons and neutrons and electrons, and even when not observed, I have a hard time believing that nothing at all can be said about the system that is 'me'. When I'm not looking at the moon, at least there's a wavefunction to think about...

@Nugatory - thanks ... I had always assumed there was more behind the scenes regarding the formation of water ...

If two electron field excitations pass each other at a distance (say 100 meters), essentially what's being said above (corrections welcome) is that all bets are off regarding whether you can even talk about excitations in the electron field existing at all (as they go by... ) - you have your in state, your out state, but who knows if those field excitations persist at all... or something much more bizarre happens?

Nugatory
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my body is made up of lots of protons and neutrons and electrons, and even when not observed, I have a hard time believing that nothing at all can be said about the system that is 'me'.
No one is asking you to believe that. Your body is composed of an enormous number of randomly interacting particles, and we can say a great deal about the behavior of such a system without knowing anything about what's going on "inside" of the innumerable random interactions between these particles.

At the risk of pushing the hydrogen/oxygen analogy past the breaking point.... if we have a mix of $O_2$ and $H_2$ in a given volume, we can make exquisitely accurate calculations of what will happen. We can say how rapidly the reactions to $H_2O$ will proceed, how much energy will be released, the volume, temperature, and pressure of the resulting cloud of water vapor; for all practical purposes, the behavior of mixtures of hydrogen/oxygen mixtures is a solved problem. And we can do all of this just with our understanding of the "in" state and the "out" state when hydrogen and oxygen molecules interact, and then considering the effects of a large number of these interactions; what happens "inside" any one interaction is irrelevant.

strangerep
- not definite values... just even a mathematical description - are there no probability distributions over intermediate states / conditions?
The only mathematical description we have is perturbative QFT, to evaluate scattering probabilities between initial and final asymptotic states. The various terms in the perturbation series are usually represented as Feynman diagrams. But these diagrams do not give a physical picture of "what's really going on" that you're looking for, any more than the terms in a Taylor series for $\sin(x)$.

Any decent textbook on introductory QFT will explain this in great detail.

PeterDonis
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are there no probability distributions over intermediate states / conditions?
No. QFT (or QM in general) gives probability distributions for possible values of observables. There are no observables corresponding to "intermediate states / conditions" in interactions, only corresponding to initial and final states. That's the definition of an interaction--you only observe the initial and final states, not anything in between.

Demystifier
There are no observables corresponding to "intermediate states / conditions" in interactions, only corresponding to initial and final states. That's the definition of an interaction--you only observe the initial and final states, not anything in between.
I think this statement may be misleading. In the intermediate states there are no observations, but we can still compute observables at intermediate states, that is time-dependent Hermitian operators. For instance, if the total Hamiltonian is
$$H=H_1+H_2+H_{\rm int}$$
one can compute the operator $H_1(t)$ at any intermediate time $t$. But of course, such a computation doesn't really answer the OP's question what is there at intermediate times, because the operator is a bunch of matrix elements, not a single value.

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Demystifier
I ask in part because this is a question of ontology (what actually exists...)
The standard textbook formulation of quantum theory says nothing about ontology. It only talks about results of measurements, in cases in which measurements are actually performed. If you are interested in ontology, then you must go beyond standard quantum theory. You must deal with interpretations of quantum theory, on which there is no consensus among physicists. Some interpretations deny the existence of ontology, but that's probably not the kind of interpretations you would be satisfied with. The best known ontological interpretation is the Bohmian interpretation, an introduction to which can be seen e.g. in
https://arxiv.org/abs/quant-ph/0611032
The Bohmian interpretation of QFT is a more difficult subject, for one approach to this see e.g. my
https://arxiv.org/abs/0904.2287

Demystifier
But surely there is a mathematical description of what's happening?
In standard formulation of quantum theory - no! In alternative formulations such as Bohmian mechanics - yes.