# Poincare representations for interacting field theory

I was going through Rudolf Haag's memoir http://link.springer.com/article/10.1140%2Fepjh%2Fe2010-10032-4 and came across these lines:

'..in quantum ﬁeld theory (or for any system of interacting particles) the equivalence class of the representation of the Poincare group is independent of the interaction. It depends only on the types of stable particles described and is explicitly known. This result was at ﬁrst sight rather counterintuitive since the Hamiltonian –which is one of the generators of the group – contains a term characterizing the interaction. But this is due to the choice of variables in terms of which the Hamiltonian is written. It is not a purely group theoretical feature.'

It will be helpful for me if anyone can explain what exactly he is talking about or can give any references to where this result is proved/ discussed.Haag refers to his notes from 1954 - 'Lecture Notes Copenhagen CERNT/RH1 53/54'. But this does not seem available online.

To me the first statement seems to be saying that representations of Poincare group are labelled by mass and spin, so if I know the particle content of an interacting theory including bound states, I basically know the representation. Is that correct? It is the second part which I can't make much sense of . why is there an apparent contradiction? Different interaction Hamiltonians do contain the information about what bound states can exist and thereby determine the representation, no? What is the role of 'the choice of variables in terms of which the Hamiltonian is written'?

It sounds to me like he is saying the appearance of the interaction term in the Hamiltonian is due to our choice of fields with which to express the Hamiltonian. If we choose fields which annihilate the bound states instead, clearly the 1 particle sector of the theory looks just like a free theory.

It might seem there is a problem with doing this with states describing many interacting particles, but in terms of the Poincare group these must just be described as reducible representations. So we can describe the interacting theory with assymptotic in and out states of many particles, which transform just like a free theory under the Poincare group.

The dynamics is still there but it's now it's hidden in the transformation from in to out states (the S matrix) instead of the Hamiltonian.

Take away, for a moment, any prejudgment about the answer to "what is a particle?"; QM provides us a precise answer. We know that physics’ laws should be invariant under the Poincare group. Since this group defines how Poincare transformations act on classic objects (tensors), in order to find out how they act on quantum objects (vectors in Hilbert space), we have to implement them as unitary representations, acting on some Hilbert space. The Lie algebra structure of this group admits two Casimir invariants, whose values indexing the irreducible representations of the group. Further analysis of the physical meaning of the Casimirs yields that one of them is the (square of) rest-mass operator, and the other is some kind of intrinsic angular momentum, unrelated to the particle’s motion; let’s call it spin (or helicity, for massles particles).

Having in mind that the group of Poincare transformations leaves invariant its irreducible representation’s vector space, now we are in position to answer what a particle is. Taking as self-evident that Poincare transformations should leave invariant the identity of a particle, from the above argument we deduce that the set of admissible one-particle Hilbert spaces shall be, exactly, the irreducible representations of the Poincare group. In any such representation, the rest mass is invariant, as it should, as well as this spin (or helicity) thing.

You see, now, how by imposing the Poincare invariance principle (the essence of Special relativity theory) into QM, we get the answer of what is a single particle and which are its properties. So, the identity of the particle is defined as a pair of the possible values of the invariant. Therefore, yes, if you know the particle content you know the representation, by definition.

Now, as stated above, it is the structure of Poincare algebra that determines (the Casimirs and therefore) the set of admissible particles and their properties. This structure is determined by the composition law of Poincare transformations. The Hamiltonian operator is included in Poincare algebra, as the generator of temporal translations. Therefore, its role in the structure of the Poincare algebra does not depend on how you will implement the Hamiltonian, by defining it explicitly as a formal function of some variables, since the structure is determined by the properties of space-time (and particularly on how Poincare transformations are composed).