# From the viewpoint of field theory and Derrick's theorem, what's the classical field configuration corresponding to particle? Is it a wavepacket?

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.

• 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 8 '16 at 22:30
• Maybe this piece of research can help too journals.aps.org/pr/abstract/10.1103/PhysRev.132.2353 – gatsu May 9 '16 at 12:11

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.

• 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 8 '16 at 17:23
• @annav the external lines in feynman diagrams are asymptotic states (i.e., "free"), and internal lines are not particles – AccidentalFourierTransform May 8 '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 8 '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 8 '16 at 18:24
• @gatsu: See this question for a discussion of bound states/resonances. – ACuriousMind May 8 '16 at 18:32

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.

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 .

• I'm not sure how this is supposed to answer the question of what a particle state in an interacting QFT is. – ACuriousMind May 8 '16 at 17:43
• @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 8 '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 8 '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 9 '16 at 6:31
• @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 9 '16 at 7:38