Relativistic many-particle dynamics as a field theory subsector In more than 2 dimensions, the so-called "no-interaction theorem" of Leutwyler (see this article for a proof) states that naive attempts at constructing $n$-particle classical relativistic dynamics face certain problems. A certain set of requirements, which include Poincare invariance, non-trivial interaction and a "covariant" behaviour of particle trajectories (by which time-evolved canonical coordinates of the particles are meant), turn out to be incompatible with each other.
There is, however, a known example of such system in 2 dimensions by Ruijsenaars and Schneider (see, e.g, article "A New Class of Integrable Systems and Its Relation to Solitons"). They constructed a Poincare invariant classical Hamiltonian which describes the $n$-soliton solutions of sine-Gordon equation. The exact construction of soliton trajectories from this theory is, however, somewhat obscure: it is not like we just time-evolve canonical coordinates with the given hamiltonian to get them. In that sense, they get around the last assumption of the theorem.
I am looking for some intuition concerning this construction. Are there any other known interesting (by which I mean interacting) examples of classical Hamiltonian theories realizing the $n$-particle sectors of field theories?
 A: I can't directly address the case you bring up, by Ruijsenaars and Schneider, and I notice that it's in 2D, while I don't know if there are lower-dimensional analogues to Leutwyler or not. But, I can shed some light on exactly what the Leutwyler Theorem (as well as what may be regarded as its quantum version, the Haag Theorem) are actually saying.
The best way to understand the question of many body interaction dynamics in relativistic theories is to step back a bit and lay it out side-by-side with the non-relativistic case, where there are non-trivial many-body dynamics.
The key players in both settings - relativistic and non-relativistic - are the momentum 3-vector $$, and angular momentum 3-vector $$. For non-relativistic many-body dynamics, they are assumed to be additive quantities. So, it seems natural to make a similar assumption in relativistic many-body dynamics -- and that's the setup for the Leutwyler Theorem, as well as for the construction of many-body state spaces (i.e. Fock spaces) in quantum theory.
Counting the components separately, this adds up to 6 quantities. The linear and angular momentum correspond, respectively, to spatial translations and spatial rotations. Under relativistic kinematics, 10 quantities in all are at play. The others are the total energy $E$, which corresponds to time translations, and the 3 components of a quantity $$ that isn't normally named at all, and corresponds to boosts (i.e. changes made to the velocity of the reference frame). It represents the mass moment of a system, taken with respect to a reference time.
In non-relativistic theory $E$ has no analogue. Instead, there are two other quantities which do, which have well-defined non-relativistic limits: the relativistic mass $M = E/c^2$ and the kinetic energy $H = E - E_0$, where $E_0$ is the energy in the rest-frame of the system (where $ = $). In relativity, the rest-frame value of $M$ is the rest-mass $m$, and is related to the rest energy as $E_0 = mc^2$. In non-relativistic theory $M = m$ in all frames. It is assumed to be additive, while $H$ is not. For many-body dynamics, the system total for $H$ consists of the kinetic energies of its components, and an additional term $U$ - the potential - where the interaction is to be found.
For relativistic many-body dynamics, a similar assumption is made for $E$; which brings us, last, to $$. In non-relativistic dynamics it is additive, for relativistic dynamics, it is largely determined by the condition that it - taken in conjunction with $$, $$ and $E$ - comprise the generators for the symmetry group for relativistic kinematics: the Poincaré group. And that's where everything falls apart and crashes: you're forced into a situation where $U$ has to be trivial.
A similar crash does not occur for non-relativistic dynamics, nor for its underlying kinematics. That's a strong indication that we have the wrong setup, and took the wrong inverse to the non-relativistic limit. But, which kinematics are we talking about here, when we say ‟non-relativistic limit”? You begin to see where part of the problem lies when looking at this in greater depth; because, if you haven't already noticed, there are not 10 generators at all, but 11. An inverse to the non-relativistic limit also has to have 11, not 10. Both $H$ and $M$ have to be treated independently.
The symmetry group for non-relativistic kinematics is called the Bargmann group. It is a one-parameter extension of the Galilei group which adds in a linear invariant - $M$ itself. The Galilei group corresponds to the case $M = 0$. The additional quantity corresponds to a central charge, and the group, itself, is called a central extension of the Galilei group. To invert the ‟non-relativistic limit”, you should actually be starting from the Bargmann group, not the Galilei group, which means you land in a symmetry group that would be the relativistic version of Bargmann: a direct product of Poincaré and $E(1)_μ$, where the invariant $μ = M - H/c^2$ plays the role analogous to the central charge in non-relativistic theory. For relativistic systems, it is just the rest mass $m$ of a body. For many-body systems, if we exploit the analogy to the non-relativistic case and take $U$ as the rest value of $H$, as the limiting value of the total $H$ when all the component bodies are brought to rest, then that plays the role of potential energy, and we have $μ = m - U/c^2$. You have a place to stick in the potential energy.
This unravels part of the issue, but does not - by itself - get to the root of the matter. The non-relativistic setup for many-body dynamics has additivity applying to three sets of quantities: $$, $$, but also $$. You can't do this in relativistic dynamics. But what, precisely is it that you can't do here? The answer is: you can't have both $$ and $$ be additive. If you make one additive, then the need to make the quantities conform to the Poincaré group will force a situation where either $$ or $$ will have to have additional contributions from $U$. If you make $$ additive, then $$ has to get those contributions, and that's where the crash occurs: $U$ is forced into being trivial. Not even the first fix, I've just described, will get you out of that conundrum.
The theorem is telling you, in the clearest possible way, that it is $$ that you have additive, not $$.
The root cause of the problem spotlighted by Leutwyler is the inconsistent treatment you're giving to the 4-vector $(E, )$. You made part of it additive, but not all of it. You made part of it non-additive, but not all of it. You have to pick a side: either it's all additive (i.e. no interaction) or it's all non-additive (i.e. both $E$ and $$ have potentials, not just $E$).
A lot of people have tried to grapple with the relativistic many-body dynamics issue - which also falls under the header ‟relativistic action at a distance dynamics”: Dirac, Feynman, Wheeler in the 1940's, Bakamjian and Thomas in the 1950's, Leutwyler, Wigner, Jordan and a lot of others in the 1960's. This is discussed in greater depth in the Edward Kerner's compilation ‟The theory of action-at-a-distance in relativistic particle dynamics.” (https://www.biblio.com/book/theory-action-distance-relativistic-particle-dynamics/d/1004253756). Leutwyler's no-interaction paper is re-printed in it, by the way.
The paper I consider to be the foundation of the whole enterprise is Dirac's 1949 paper “Forms of Relativistic Dynamics” (https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.21.392), in which he lays out 3 different setups.
One of the setups The Instant Form is the one used by most people trying to do many-body dynamics in both the classical setting and in quantum theory (the Fock Space), and is the one that Leutwyler starts with. It's the one which has $$ and $$ being additive.
Another setup, The Front Form, takes a null triad out of $(,E)$ and four components out of $(,)$ as being additive. I don't know of anyone who has explored these cases in depth, and don't know if there is any kind of no-interaction theorem blocking them. I suspect there is, because it also treats the 4-vector and 6-vector unevenly, as does the Instant Form, and that's where the problems arose with the latter.
The third setup, The Point Form, is the one I just described: where 6-vector $(,)$ is taken to be additive, while the 4-vector $(,E)$ is the one having the interaction terms in it - or with the fix I just laid out - where the 5-vector $(,H,M)$ is endowed with the interaction terms, possibly with the additional assumption that $μ$ be made additive (in which case $m$ will acquire $U$-dependent terms, in what might be regarded as a classical analogue to mass-renormalization).
I don't know of any no-interaction theorem for the Point Form. The question of whether a non-trivial relativistic many-body dynamics exists with it is, as far as I'm aware, still open. Or ... it might be trivial to resolve in the affirmative: consider the dynamics for an electrical charge $e$, $d( + e)/dt = -∇V$, $d(H + eφ)/dt = ∂V/∂t$, for a body with charge $e$, momentum $ = M$, relativistic mass $M$, kinetic energy $H$ and velocity $$ under the action of an electromagnetic field, with vector potential $$, and scalar potential $φ$, where $V(,t) = e (φ(,t) - ·(,t))$. The example seems to point the way to non-trivial interacting many-body systems in the Point Form.
I suggest that you take a closer look at the construction you mentioned, by Ruijsenaars and Schneider, and see which (lower-dimensional analogue) setup it actually conforms to; and whether it can be scaled up to 3+1 dimensions or not (or 4+1 dimensions, with the ‟relativistic Bargmann” extension I described above). If it's making the 2D version of $(,)$ additive, then the scaled up 3+1 dimensional version should be set up with the Point Form, not the Instant Form.
Edit : I've listed a few extra references, including some I just came across. Leutwyler did his proof in position-momentum space, but assumed the total energy's Hessian with respect to the one-body momenta components is non-singular ... so that actually reduces it to position-velocity space, with a non-singular Lagrangian. Marmo, Mukunda and Sudarshan, or MMS (‟Relativistic particle dynamics—Lagrangian proof of the no-interaction theorem”, Phys. Rev. D 30, 2110 – Published 15 November 1984, https://journals.aps.org/prd/abstract/10.1103/PhysRevD.30.2110) start in position-velocity space and allow singular Lagrangians, but assume $(,)$-additivity.
The starting assumption for Leutwyler is the transformation law on the position coordinates. I'm not sure that would be applicable for the Point Form, since an interaction term for $$ will produce an interaction-dependent term for the translations of the coordinate operators. So, it looks like that's where they slipped in an effective Instant Form assumption, and why additivity was so easy to derive from it (just a simple canonical transform was needed).
The starting assumpion in MMS was the ‟World Line Condition” (WLC) - which (once $(,)$-additivity is assumed) is equivalent to posing the corresponding transformation law on the one-body coordinates as an ansatz - one that's similar to Leutwyler's.
Ciaglia, Di Cosmo, Ibort, Marmo over on ArXiv (‟Descriptions of Relativistic Dynamics with World Line Condition” https://arxiv.org/pdf/1910.06447.pdf) is a latter-day follow-up of MMS which has key phrases in it: ‟Dirac”, ‟Instant Form” and (surprise) ‟eleventh generator formalism”. That can only be something equivalent to what I already described, since there isn't room anywhere else to put the extra generator.
They cite the 1981 Mukunda and Sudarshan ‟Form of relativistic dynamics with world lines” (https://www.osti.gov/biblio/6404311-form-relativistic-dynamics-world-lines) in their bibliography as the ground reference for the formalism and also summarize it in some detail. The extra generator has to come in their $Γ$ (which is the same as what I called $M$) in their equation (25). The explicit form for $M$ would be $M = μ + H/c^2 = m + (H-U)/c^2$, which reduces to $M = μ = m$ in the non-relativistic limit.
There is a natural 5-D geometry for all of this, obtained by adding an extra coordinate $u$ as conjugate to the mass invariant $μ$, such that the 1-form $·d - H dt + μ du$ transforms as an invariant. The corresponding line element & constraint ${|d|}^2 + 2 dt du = dx^2 + dy^2 + dz^2 + 2 dt du = 0$ yields a 4-D geometry for non-relativistic theory, its relativistic form $dx^2 + dy^2 + dz^2 + 2 dt du + 1/c^2 du^2 = 0$ giving you Minkowski geometry with the proper time $s$ given by $s = t + u/c^2$. (The light-speed adjusted time dilation $u = c^2 (s-t)$ actually has a meaningful non-relativistic limit!) In both cases they're subspaces in an underlying (4+1)-D geometry.
It would be an interesting exercise to clean up and rehabilitate section 4 and on, of the Ciaglia et al. paper, by recasting it geometrically in terms of the underlying (4+1)-D geometry.
I also found something on Point Form dynamics from 2002 here by Klink ‟Point Form Relativistic Quantum Mechanics and an Algebraic Formulation of Electron Scattering” (https://www.slac.stanford.edu/econf/C0107094/papers/Klink569-576.pdf). So, it looks like there are people trying to work with this setup, too.
