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In QFT, elementary particles correspond to irreps of the universal cover of the Poincaré group, and the full quantum field is then the direct sum of fields living in different irreps. So the unitary $U_{\Delta t} $ representing a time translation $ t\mapsto t+\Delta t $ can be written as a direct sum of unitaries acting on each irrep individually. But this unitary just gives us the time evolution of the system. Therefore it seems that the component in a given irrep will evolve independently of the other components - in other words, the fields don't interact.

Clearly I have misunderstood something, since fields certainly do interact. I'd be grateful for any clarity on this issue.

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There are several confusions here. First, you seem to be mixing up particles (described by infinite-dimensional unitary irreps of the Poincare group) with fields (described by finite-dimensional, generally nonunitary irreps of the Lorentz group). I'll just focus on the particles, but for more on this point see this question and the linked questions there.

Now, there's a simpler objection that is equivalent to yours. The Schrodinger equation says that for a time-independent Hamiltonian, each energy eigenstate just rotates its phase as $e^{- i E t / \hbar}$. Since these phases just rotate independently and uniformly, how can anything nontrivial ever happen?

The reason is that the energy eigenstates for any nontrivial system are extremely complex. For example, an unstable particle can decay, but that's because the particle itself is not in an energy eigenstate. The true energy eigenstates of the system are exceedingly complicated superpositions of the particle and its decay products, which nobody can compute. Thus, conversely, starting with an initial condition of just the particle present actually corresponds to taking an exceedingly complicated superposition of energy eigenstates, with their phases aligned just right. As the phases start to rotate, their relation to each other changes in time, causing something nontrivial to happen.

In principle it's similar to, e.g. normal modes in classical mechanics. Many linear nondissipative system in classical mechanics can be written as a bunch of independently oscillating normal modes, but that can give rise to complex time evolutions.

In the case of the Poincare group, the situation is just the same, since the heart of your complaint is just about the factorization of the time evolution operator. But you might be wondering, don't textbooks seem to imply that the time evolution is more complex than this? Yes, and the reason is that in general, they break the Hilbert space into Poincare irreps under the free time evolution, thereby defining "in" and "out" states. (Here, "free" is defined as any Hamiltonian that is simple enough so that you can actually perform this procedure.) By definition, these states don't interact with each other, but when they get close, the full time evolution takes over. The effect of this time evolution on the free irreps is, of course, described by the $S$-matrix.

In summary, the "true" Poincare irreps indeed have "trivial" time evolution, but we see nontrivial time evolution in practice because these irreps are related to the free Poincare irreps in an exceedingly complicated way. Textbooks always construct the free Poincare irreps, because it's not feasible to say anything about the true ones.

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  • $\begingroup$ Thanks for the response. You're right that I was getting my groups mixed up. However, I'm confused as to the distinction between "free" and "true" Poincare irreps - the Poincare group is the same thing regardless of the situation, so shouldn't its irreps always be the same? $\endgroup$ May 17, 2020 at 0:10
  • $\begingroup$ @Jacob Drori Nope, recall that one of the Poincaré generators is the Hamiltonian. The “free” irreps correspond to taking the free Hamiltonian. $\endgroup$
    – knzhou
    May 17, 2020 at 0:44
  • $\begingroup$ Ah ok - my confusion was due to "irreps of Poincare" and "irreps of a given rep of Poincare" meaning different things. I see now why the different fields can interact, and that it's only the full field (i.e. direct sum over the irreps) which must evolve unitarily. However I'm still unclear about the relation between the field and the particle pictures in terms of representations: is it correct to say that states with a fixed number of particles live in irreps of Poincare with the free Hamiltonian? Feel free to point me to someone else's answer if this is getting off topic. $\endgroup$ May 17, 2020 at 10:56
  • $\begingroup$ @JacobDrori I think that comment is still mixing up particles and fields, and hence Poincare and Lorentz representations. I would suggest you search up past questions along these lines (e.g. see here), and if you're not satisfied, ask a new one! $\endgroup$
    – knzhou
    May 17, 2020 at 18:47
  • $\begingroup$ ok - thanks for the help $\endgroup$ May 18, 2020 at 13:06
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The point you correctly discovered - as far as I understand the question - is that they don't interact naturally. One has to assemble them into a Lagrangian that provides 'couplings' of the different fields.

Criteria like "gauge invariance", "L has to be scalar", and renormalizability (iiekk) reduce the possible choices of couplings, otherwise the mess would be immense.

I don't now what you mean exactly by "the full quantum field". I don't think this notion exists in general QFT. But if you are like me, I would prefer a single 'full' fundamental field, where interactions are naturally built in. In fact, I'm working on such a thing (no references yet)

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