# Why should there be one-particle states in an interacting quantum field theory?

I'm a mathematician trying to learn quantum field theory. This question has two parts: first, I want to double check that I'm thinking about the surrounding issues correctly, after that I'll ask my main question.

It's taken me a while to understand how I should think about particle states in an interacting quantum field theory and how they relate to the field operators. Here's what I think is true; please tell me if I'm wrong!

1. When we pass from a free to an interacting theory, no one has a description of the state space that's as nice as the Fock space picture. Instead, we simply posit that the combined spectrum of the momentum operators $$P^\mu$$ contains some isolated hyperboloids $$p^2=m^2$$. (This is the famous "mass gap".) We decide to call the corresponding eigenstates "one-particle states with mass $$m$$". This name is somewhat sensible since, in particular, these states will be eigenstates of the Hamiltonian $$H=P^0$$, so they are stable and have the sorts of momenta we would expect from a particle of mass $$m$$. These states have very little to do with any one-particle states for any free theory.

2. For a field operator $$\phi$$, look at the Källén-Lehmann representation of the corresponding two-point function. If $$|p\rangle$$ is a one-particle state for which $$\langle p|\phi(0)|\Omega\rangle\ne 0$$, it turns out there will be corresponding pole in the two-point function and vice versa. (When this condition is met, $$\phi$$ is an "interpolating field" for $$|p\rangle$$.) We can use this to construct a one-particle state out of an appropriately smeared version of $$\phi$$ applied to the vacuum; this cruicially requires the isolatedness of the eigenvalue mentioned above.

3. If the interpolating field is one of the "basic" fields appearing in the Lagrangian, the state we build in this way is an "elementary" particle, and if it's some more complicated combination of products of basic fields it's a "composite" particle. For a given one-particle state, there are in general lots of different fields you could use for this purpose, but we always assume there is at least one.

4. We can build $$n$$-particle states by doing a similar smearing trick with well-separated smearing functions and letting $$t\to\pm\infty$$; in particular, unlike the one-particle states, these multiple-particle states are only really meaningful in this asymptotic limit. But once we have these, we can define the S-matrix in terms of their overlaps, and we can write down the LSZ formula and a bunch of Feynman diagrams and we're off to the races.

My main question (aside from "is everything I just said correct?") is why should we expect these assumptions to hold? In particular, the parts that are the least obvious to me are (a) why should the spectrum of the $$P^\mu$$'s contain these isolated hyperboloids, and (b) why should I assume that the corresponding states all have interpolating fields as in (2) above?

I'm aware that there are some problems extracting a completely rigorous version of this whole story and that massless particles, gauge symmetries, and so on introduce complications; that's not what I'm asking about. I'm asking for intuition: right now these assumptions --- the mass gap and the existence of interpolating fields --- feel arbitrary to me, and I'm wondering if anyone has a way of thinking about the logical structure that makes it feel more natural.

• You might want to have a look at Section 3 of www-m5.ma.tum.de/foswiki/pub/M5/Allgemeines/WojciechDybalski/… – Abdelmalek Abdesselam Feb 23 '19 at 23:49
• Thanks for the reference! On a first reading, though, it seems like it doesn't quite address the same question. I think I'm convinced that a sensible scattering theory can be derived from (among others) the mass gap assumption; I'm asking why I should expect that assumption to hold in the first place. – Nicolas Ford Feb 24 '19 at 0:21
• We don’t expect the properties to hold in general, just in some models. As you pointed out, it is an open question whether Yang-Mills is one of them. But if you read through the millennium prize problem description, you’ll notice that mass gap is just part 2 of the problem, part 1 being to define a theory to begin with... We are at a point where we don’t even know for sure if QFT exists, and there are no physical reasons why it must (QFT is just an approximation to quantum gravity). All we can really do is run lattice simulations, and there’s numerical evidence that your properties do hold. – Prof. Legolasov Feb 24 '19 at 2:53
• Just curious, what reference have you mainly been following? – InertialObserver Feb 24 '19 at 5:46
• @InertialObserver: I took a class in grad school that was taught out of Srednicki, which left me pretty confused about questions of this form. I've sort of cobbled together the picture I currently have from a bunch of places, most recently Duncan's "Conceptual Framework of Quantum Field Theory". This is part of what makes me worried that I'm understanding something wrong! – Nicolas Ford Feb 24 '19 at 7:14