Which Things Exist in Quantum Field Theory

If you ever think metaphysics is easy, learn a little quantum field theory.

Someone asked me recently about virtual particles. When talking to the public, physicists sometimes explain the behavior of quantum fields with what they call “virtual particles”. They’ll describe forces coming from virtual particles going back and forth, or a bubbling sea of virtual particles and anti-particles popping out of empty space.

The thing is, this is a metaphor. What’s more, it’s a metaphor for an approximation. As physicists, when we draw diagrams with more and more virtual particles, we’re trying to use something we know how to calculate with (particles) to understand something tougher to handle (interacting quantum fields). Virtual particles, at least as you’re probably picturing them, don’t really exist.

I don’t really blame physicists for talking like that, though. Virtual particles are a metaphor, sure, a way to talk about a particular calculation. But so is basically anything we can say about quantum field theory. In quantum field theory, it’s pretty tough to say which things “really exist”.

I’ll start with an example, neutrino oscillation.

You might have heard that there are three types of neutrinos, corresponding to the three “generations” of the Standard Model: electron-neutrinos, muon-neutrinos, and tau-neutrinos. Each is produced in particular kinds of reactions: electron-neutrinos, for example, get produced by beta-plus decay, when a proton turns into a neutron, an anti-electron, and an electron-neutrino.

Leave these neutrinos alone though, and something strange happens. Detect what you expect to be an electron-neutrino, and it might have changed into a muon-neutrino or a tau-neutrino. The neutrino oscillated.

Why does this happen?

One way to explain it is to say that electron-neutrinos, muon-neutrinos, and tau-neutrinos don’t “really exist”. Instead, what really exists are neutrinos with specific masses. These don’t have catchy names, so let’s just call them neutrino-one, neutrino-two, and neutrino-three. What we think of as electron-neutrinos, muon-neutrinos, and tau-neutrinos are each some mix (a quantum superposition) of these “really existing” neutrinos, specifically the mixes that interact nicely with electrons, muons, and tau leptons respectively. When you let them travel, it’s these neutrinos that do the traveling, and due to quantum effects that I’m not explaining here you end up with a different mix than you started with.

This probably seems like a perfectly reasonable explanation. But it shouldn’t. Because if you take one of these mass-neutrinos, and interact with an electron, or a muon, or a tau, then suddenly it behaves like a mix of the old electron-neutrinos, muon-neutrinos, and tau-neutrinos.

That’s because both explanations are trying to chop the world up in a way that can’t be done consistently. There aren’t electron-neutrinos, muon-neutrinos, and tau-neutrinos, and there aren’t neutrino-ones, neutrino-twos, and neutrino-threes. There’s a mathematical object (a vector space) that can look like either.

Whether you’re comfortable with that depends on whether you think of mathematical objects as “things that exist”. If you aren’t, you’re going to have trouble thinking about the quantum world. Maybe you want to take a step back, and say that at least “fields” should exist. But that still won’t do: we can redefine fields, add them together or even use more complicated functions, and still get the same physics. The kinds of things that exist can’t be like this. Instead you end up invoking another kind of mathematical object, equivalence classes.

If you want to be totally rigorous, you have to go a step further. You end up thinking of physics in a very bare-bones way, as the set of all observations you could perform. Instead of describing the world in terms of “these things” or “those things”, the world is a black box, and all you’re doing is finding patterns in that black box.

Is there a way around this? Maybe. But it requires thought, and serious philosophy. It’s not intuitive, it’s not easy, and it doesn’t lend itself well to 3d animations in documentaries. So in practice, whenever anyone tells you about something in physics, you can be pretty sure it’s a metaphor. Nice describable, non-mathematical things typically don’t exist.

12 thoughts on “Which Things Exist in Quantum Field Theory

  1. itaibn

    I don’t agree with the commonly spoken claim that virtual aren’t real, but are “merely” entities that appear in calculation schemes physicists use for making approximate predictions. I agree they do appear in such calculation schemes, I just don’t agree with the “mere”. After all, the scheme in question is the calculation of an amplitude by partitioning into different pathways and adding the amplitudes in each pathway. (To be clear, I’m not saying that each field configuration has to correspond “come from” a single Feynmann diagram; I know that’s not true.) This is no different from what quantum theory says happens with any physical object: It has no fixed state but an amplitude can be computed by adding the amplitudes for all possible trajectories it can take. So virtual particles are as real as atoms and baseball bats.

    Like

    Reply
    1. 4gravitons Post author

      I think that if you characterize virtual particles as “particles” in any sense, you’re assuming some kind of perturbative picture. And we know that’s not the full story for many theories, we know that if you simply add up every Feynman diagram you don’t get the right results for QCD, even if you could add up an infinite number, you need nonperturbative effects as well. Now, whether or not that means virtual particles are “real”…eh. “Real” is a tricky word to use for anything in QFT, and doesn’t tend to be all that useful.

      Like

      Reply
      1. itaibn

        I agree with you that “real” is a tricky word in QFT — that was the main point of your post and it is correct. Part of the reason it’s tricky is that is already very tricky in QM. Yet people do care about understanding with QM in what sense atoms and baseball bats are real. What I am arguing is that in whatever sense atoms and baseball bats are real, virtual particles have roughly same kind of existence. Although Feynmann diagrams are not always applicable as a calculation tool, whenever they are applicable the role virtual particles take in them is the same as the role an atom or baseball bat takes in calculating a QM amplitude. Although baseball bats exhibit decoherence to a much greater extent than atoms or virtual particles, and most of the time only virtual particles appear in interference patterns where they only exist in some of the branches, they seem to me to all be essentially the same kind of stuff.

        Like

        Reply
        1. 4gravitons Post author

          I think there’s a difference between “this is true in a certain approximation” and “this is true at a certain level of abstraction”. Baseball bats and atoms are “real” in the latter sense, they’re useful ways to clump together certain kinds of phenomena. Feynman diagrams are an approximation though. They can be a very good approximation, but except in certain very special situations (read: planar N=4 SYM) they won’t give the whole result, even to all orders.

          I think that makes asserting them to be real a different kind of claim. If you approximate the world on a grid, it would be really weird to claim that the grid is real: you added the grid yourself!

          Like

          Reply
  2. Martin

    Thanks a lot for this nice post. I can only emphasize your last two paragraphs: the only things we certainly now to exist are measured observables. Period. Theories (whatever “real-sounding” terms they might use) can never be made real, but the only observables they might (or might not) predict.

    However, I think the term “virtual particles” is actually not completely misleading once we use a proper definition of what we mean by a “particle”. In your examples of neutrino oscillations it sound misleading but in different experiments, eg. a 2->2 scattering, the virtual particles might become on-shell and manifestate themselves as a peak with a certain height and width in an observable energy spectrum. The properties of this peak could be called “the particle” which we might associate with an excitation of a field. At least this is what I always think of a proper definition of a “particle”, i.e. the whole chain of predicting and measuring peaks in energy spectra.

    So yes, virtual particles are not in the in- and out-going states and therefore do not exist in terms of the experiment which prepares and measures only the properties of the in- and out-states.
    However, If we would not include them in our theory, the predictions to be compared with the experiment would simply be wrong. Think of the W/Z/H-Bosons: they actually never have been detected in an W/Z/H-Detector but only there decay products. The same even applies onto QED: we don’t have electron- or photon-detectors but only observe their energy-deposition patterns.. So a philosopher might even doubt the existence of all particles (not only the virtuals) which imho is only because of that language barrier and doesn’t necessarily lead to anything interesting.

    Like

    Reply
    1. 4gravitons Post author

      That’s a fair point, sure. Part of the issue here is that “virtual particles” are more of a pop science term than a physics term, so their meaning isn’t very specific. I think if by “virtual particles” you’re talking about thresholds and resonances and Landau singularities and so on, then sure, there’s something meaningful there, an on-shell skeleton that seems to lurk within quantum field theory.

      Like

      Reply
      1. Martin

        Yep, that is the direction I was pointing to.

        Could you then please elaborate what you mean by “Is there a way around this? Maybe. But it requires thought, and serious philosophy.”? This was the sentence that actually triggered me. What kind of philosophy? Since – as I was trying to explain – I don’t think that philosophers can give more contributions to the question of the existence of something but actually only the experimentaltheoretical physics workflow.

        Like

        Reply
        1. 4gravitons Post author

          Two things here:

          One, I think you’ve probably got a bit of an inaccurate stereotype of philosophers, as just sort of going around doubting stuff like Descartes. The impression I get is that modern philosophers tend to be more careful and less trivial: the idea is to sort out a kind of precise meaning of “what things exist” that they can prove nontrivial things with, in the same way a mathematician finds a precise meaning for, say, “continuous”. I think that that kind of thing can be genuinely valuable, if it’s properly informed. (Though I have seen some modern contributions to the subject that raised my eyebrow, like a philosopher arguing that holography was incoherent because the number of dimensions can’t be ambiguous.)

          The other thing is that “things should be called whatever is most practical for getting work done” is a real philosophical position with its own adherents, one I’m broadly sympathetic to based on the little I’ve heard about it. So I suspect I’m largely on your side anyway, but I don’t discount the ability of metaphysics philosophers to come up with something valuable and interesting.

          Like

          Reply
  3. Madeleine Birchfield

    Doesn’t Hilbert’s sixth problem and the developments resulting from it make theoretical physics just a branch of mathematics? In which case the philosophical question of whether elementary particles or fields actually exist belongs to the the same category of questions as whether the natural numbers or the real numbers actually exist.

    Like

    Reply
    1. 4gravitons Post author

      Hilbert’s sixth problem (and “the developments resulting from it”, which I assume means axiomatic QFT?) doesn’t mean anything of the sort. In general, it’s good to pursue physics using the language and tools of mathematics, for the same reason it’s good to pursue chemistry or economics using those tools. That doesn’t mean that economics and chemistry are branches of mathematics.

      In this case, regardless, the question is being asked is “do these things exist physically” not “do these things exist as mathematical objects”. Unless you’re extremely Tegmark-y in outlook those are different questions.

      Like

      Reply
      1. Madeleine Birchfield

        The problem isn’t really rooted in quantum field theory though; the same could be said about classical field theory as well. Do gravitational and electromagnetic fields actually exist, or are they just mathematical abstractions/metaphors to explain otherwise inexplicable observations in our black box physical world?

        Like

        Reply

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s