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In the framework of QM, we have known that particle, like electron, cannot be a wavepacket, because if it is a wavepacket then it will become "fatter" due to dispersion and it's impossible.

However in the framework of QFT, a real scalar particle is defined as an excitation or a perturbation from the vaccum of a real scalar field. If it is a free theory, there is no problem to consider a particle as a wavepacket of a real scalar field, because the wavepacket in free theory is stable. However, if it is an interacting scalar field, for example $\phi^4$ theory, we cannot consider a particle as a wavepacket, since if a particle is a wavepacket then it must be soliton otherwise it cannot be stable. However Derrick's No-Go theorem says that in $3+1$-dim there is no stable soliton in real scalar field.

Therefore my question is what is a particle's classical counterpart in a field theory? If it is a wavepacket, then why is my argument wrong?

PS: It's too difficult to talk about standard model. Let's assume that only the toy model, massive $\phi^4$ theory, is taken into consideration. Then obviously we can have a stable state that has only one static $\phi$ particle. This state must be stable because this particle cannot decay into other particles and it has a energy gap from the vaccum state. So physically this state must exist and I want to know what's the classical field configuration corresponds to this state.

346699
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  • A local excitation of a quantum field is a quantum, not a particle. The particles don't come into sight until you let a plane wave solution of qft interact weakly with a matter background. Unfortunately, they probably don't mention this in most books and classes on the topic. – CuriousOne May 08 '16 at 22:30
  • Maybe this piece of research can help too http://journals.aps.org/pr/abstract/10.1103/PhysRev.132.2353 – gatsu May 09 '16 at 12:11

2 Answers2

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A particle is not a wavepacket. And there are no particle states for interacting theories.

We define particle states in QFT by expanding the free field into its Fourier modes and using these modes as creation/annihilation operators for particle states - the mode of momentum $p$ creates the particle state $\lvert p\rangle$ with momentum $p$. The Hilbert space of free theories is the Fock space built by using these operators.

The Hilbert space of interacting theories is, in general, unknown, but it is not the space of particles of the free theory. This is Haag's theorem. Whenever you hear people talking about "particles", they mean state of the theory in the asymptotic future/past where the interaction is turned off and we have a notion of particle states. But for the interacting theory, we have no formal notion of a particle state.

ACuriousMind
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  • Well, one does draw particle lines in a feynman diagram, and the invention of QFD was so that one could calculate elementary particle interactions using them and compare them with measurements, you know : "electrons", "neutrinos", "photons" as defined in our measurements. The creation and annihilation operators create and annihilate "particles" after all, quanta of the field – anna v May 08 '16 at 17:23
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    @annav the external lines in feynman diagrams are asymptotic states (i.e., "free"), and internal lines are not particles – AccidentalFourierTransform May 08 '16 at 17:37
  • They have names, they are measured and tabulated as electron proton crossections, Higgs production, etc. It is extremely confusing "we have no formal notion of a particle state" and what are those "free " states if not a quantum of the field i.e. a particle? – anna v May 08 '16 at 17:47
  • @AccidentalFourierTransform: right...so what about a hydrogen atom? Does it mean anything to say that is made of an electron and a proton with the electron being in a certain state (if we imagine the proton to be infinitely more massive than the electron that is)? – gatsu May 08 '16 at 18:24
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    @gatsu: See this question for a discussion of bound states/resonances. – ACuriousMind May 08 '16 at 18:32
  • @ACuriousMind: Thanks for the link. Note that I don't dispute using the term "resonances" instead of excited bound state of a hydrogen atom. I dispute or rather question the claim that these resonances are not, in some sense, associated to actual particle states (here in particular electron states). The fact that they are short lived is irrelevant to my question I believe. And even if for some reason resonances never do the job, then does the ground state at least qualify to talk about the quantum state of an electron interacting with a proton? – gatsu May 08 '16 at 18:47
  • @gatsu: The phenomenology of particle detection is quite clear about what it takes to have particles: a weak background interaction that breaks the symmetries of the vacuum. I don't know why people are in the dark about this... it's daily bread and butter for anybody who has ever worked in an accelerator facility of who had to deal with cosmic rays or sufficiently high energy radiation of any kind. – CuriousOne May 08 '16 at 22:33
  • Let's assume that only the toy model, massive $\phi^4$ theory, is taken into consideration. Then obviously we can have a stable state that has only one static $\phi$ particle. This state must be stable because this particle cannot decay into other particles and it has a energy gap from the vaccum state. So physically this state must exist and I want to know what's the classical field configuration corresponds to this state. – 346699 May 09 '16 at 04:54
  • @CuriousOne: QFT is not the realm of only high energy physics with only free fields to grant a particle ontology, it can also be used quite effectively to comprehend low energy physics. The formal map between a model in QM (chemistry-like Schrodiner equation) and a model in QFT can be done without the standard model of particle physics by using the notion of Schrodinger field. I notice more and more, and I hope I am wrong, that particle physicists are so focused on LHC-type experiments that they believe only those experiments make sense. – gatsu May 09 '16 at 06:41
  • @gatsu: Nobody has ever seen particles in low energy phenomena in which weak measurement is not the dominant interaction. They only exist in cases where we aren't getting anywhere close to the localization limit given by the uncertainty relation in individual measurements. There is an awful lot of confusion about this, even among physicists who should know the phenomenology, but seem to be having a hard time putting one and one together, i.e. in which regime individual quantum effects lie. Quanta are everywhere, particles are not. – CuriousOne May 09 '16 at 07:34
  • @user34669: 1. Well, I don't know about "physically", but rigorously, no one knows the Hilbert spaces of interacting theories, not even of $\phi^4$ in 4D. Some bits are known in two dimensions from the work of Glimm and Jaffe (and others). Haag-Ruelle theory defines "surrogates" of the free particle states in the interacting space, but in the end, all it gives you is how to turn free in states into free out states, and nothing useful about the interacting states. 2. Why do you think there is such a thing as a "classical field configuration" associated to a QFT state? – ACuriousMind May 09 '16 at 09:17
  • @CuriousOne: There are two points in your comment: 1) Do you seriously use names only for things that can be measured in the weak interaction limit of QFT? and 2) There is a misunderstanding. I am only saying that if there is a QFT of bound states, then one can talk about some states that qualify as quantum states of a single particle in this bound system. At that stage, I don't care whether you can detect it or not; it is just "ontology matching" where the single electron states of the atoms correspond to a higher level of ontology than that of the QFT states. – gatsu May 09 '16 at 09:25
  • @gatsu: 1) I seriously use names for phenomena that are real (localized objects) and that people keep mistaking for non-localized objects (quantum fields). 2) Single particles are a kiddie model of the universe and that's what causes the all the heartbreak for the early learners who are mistaking single particle quantum mechanics for an actual self-consistent model of reality, which it is not and can not be because one can't even perform measurements in a single particle approximation. We need to stop teaching inconsistent models for more than they are. – CuriousOne May 09 '16 at 10:05
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My 2 cents on it is that in QM (be it "standard" QM or QFT) one describes only the state of a particle. Having said that, the most general state for a single particle is indeed a wave packet.

Now, if you localise certainly a particle at some point in time, then later on it will be associated with a spreading wave packet because of Heisenberg indeterminacy principle exactly as you say.

The point is then to figure out how long it will take for the wave packet to spread when compared to the time scale of interest.

2D wave packet in a box

In the gif above what you can see is that although we start from a quite localised packet the packet eventually spreads because of the momentum dispersion. However, we see it actually "bounces on the walls" like a classical particle would for quite some time after which it spreads entirely in the box.

Even when taking quite ridiculous constraints on the precision, you will find that a well localised atom for example will spreads over a distance of the order of meter in few microseconds while a pebble will spread over a distance of one millimetre over a period of time much bigger than the age of the universe.

EDIT 1: In reaction to ACuriousMind's answer that focuses on the description of particles from the point of view of the quantisation of free fields (and therefore as making sense only as asymptotic states in any interacting system), I would tentatively claim that choosing asymptotically free states is one limiting case that leads to a non ambiguous formal description of a single particle state. I would however argue that a perfectly bound state (like the ground state of a hydrogen atom or even a particle in a box) would, in principle, be equally valid to talk about a particle state. I would say that qualifying those states of an interacting QFT model that can be matched onto states of an equivalent single particle in QM (with the same intrinsic parameters) in a potential as being single particle states for instance seems to be an equally legitimate choice to talk about single particle state. Of course, not all possible states would necessary qualify for such a terminology and one may prefer the term "resonances" to talk about these borderline cases, as discussed in one of ACuriousMind's comment.

EDIT 2: I just remarked that the OP's question is not so much about "classicality" for there are classical variants of a problem with no particle ontology. This is the case of the $\phi^4$ model and also of electromagnetism that does not have any photon classically. I therefore conclude that the original question refers to the existence of local "beables" (to use Bell's terminology) which are localised particle states. From a theoretical and experimental point of view, I think it is worth looking at what has been done in quantum optics where single photon wave packets play an important role (in optical cavities for example. This reference seems to be quite related to the matter https://www.weizmann.ac.il/chemphys/dayan/notes/Lecture3.pdf .

gatsu
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    I'm not sure how this is supposed to answer the question of what a particle state in an interacting QFT is. – ACuriousMind May 08 '16 at 17:43
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    @ACuriousMind: It provides a clarification to the OP's first statement about the wave packet becoming fatter in standard QM. And it also relates quite explicitly with the idea of classicality or "beability" of a particle that is mentioned in the question. – gatsu May 08 '16 at 18:10
  • +1 for the lovely animation. I hadn't seen this one before, but I find it quite cool. A gaussian state is, by the way, not what we talk about when we talk about particles in high energy physics. Those are really plane waves under the influence of weak measurement. – CuriousOne May 08 '16 at 22:27
  • @CuriousOne: It might be that the OP meant "high energy physics" but as far as the first version of his question is concerned, only QM, QFT and classicality are the keywords that attracted my attention. It seems to me that everybody else seems to be ignoring the "classical" tone of the OP's question. The way I interpreted his question had more to do with explaining classicality from the best theory we have so far which is QFT. Since this is quite beyond me, I went back to QM to discuss some aspects of his question. – gatsu May 09 '16 at 06:31
  • @gatsu: The OP simply doesn't seem to understand what quantum mechanics is and it totally doesn't matter where his question is going when he is already starting out with the wrong mental model about what is going on. QM is not about particles and never has been. It has always been about waves. Non-relativistic QM is simply a completely linear non-interacting theory where the effective potential defined by the environment is being introduced in an ad-hoc way. In QFT there is no such thing, at all, and the effective field is generated by the self-interaction. There are no particles in either. – CuriousOne May 09 '16 at 07:38
  • @CuriousOne: In my EDIT 2, I have given an example of a localised single photon wave packet in the context of quantum optics that is essentially being detected "as is" in various experiments. I don't really care whether this can stand as an absolute, ever-stable, definition of a particle (or beable) but it does the job in the sense of my answer to the question...maybe not in the seemingly restrictive sense you seem to use it, I admit it. – gatsu May 09 '16 at 09:06
  • @gatsu: I don't see anything in your example that moves the ontology from quanta to particles. – CuriousOne May 09 '16 at 10:07
  • @CuriousOne: you could not see anything related to localised single photon states (particle) in a link to lecture notes on localised single photon states? – gatsu May 09 '16 at 12:20
  • @gatsu: No, I can't. The em field only localizes when there is matter present, in which case it's not the em field any longer. The free field retreats at the speed of light and, as I keep pointing out to everyone (who does or does not want to hear this trivial fact), one can't catch light. – CuriousOne May 09 '16 at 17:41
  • @CuriousOne: I absolutely don't get your logic...so you never speak about classical EM in vacuum because we can't catch light either (or you only look at plane waves)? If you scan through these notes you will actually see that this localised single photon wave packet formalism, no matter how impossible you may think it is, is actually and practically used in photonics; to model things like laser pulses or photons in cavities. In this context, it is absurd to say that these things don't exist. – gatsu May 09 '16 at 18:12
  • Let us continue this discussion in chat. – gatsu May 09 '16 at 18:17
  • @gatsu: You can use whatever you want in photonics, that still doesn't change the ontology of quantum mechanics for LOCAL measurements. You are simply conflating different physical scenarios. – CuriousOne May 09 '16 at 18:17