# Excitations and representation categories

Apologies if this question is a little vague — I'm fuzzy on the details, which is why I ask.

In quantum mechanics, we commonly encounter the idea that types of particle-like charges or excitations correspond to irreducible representations of some symmetry group. For example, in non-relativistic QM, the types of spin correspond to representations of $$\text{Spin}(3)$$. In relativistic quantum field theory, the types of one particle states correspond to representations of the Poincare group. In the quantum Ising chain in the symmetric phase, there is single type of excitation — a spin flip — and I believe this corresponds to the single non-trivial representation of the symmetry group $$\mathbf{Z}_2$$.

My first question is: to what extent can we make this correspondence more precise? Specifically, is the following "rule of thumb" true?

\begin{align} \text{types of charges/particle-like excitations above a symmetric vacuum}\, &\simeq\\ \text{objects of the category of representations of the symmetry group, } &\text{Rep}(G) \end{align}

From some of the condensed matter literature, I am pretty sure this (or something close to it) is true at least in lattice systems with finite symmetry groups, and maybe more generally. One thing I am confused about is with, e.g., the Poincare group, not all of the irreducible representations correspond to physical particle states — there are tachyonic representations, for example. There is also the issue of the representations needing to be unitary.

My second question is, if the above rule of thumb is true, what is the physical meaning of the morphisms in the representation category? It seems that one of the lessons of category theory is that the morphisms in a category are at least as important as the objects, so I think they should have some significance. If the rule of thumb isn't true, I still think the representation category of the symmetry group has some physical meaning, so I suspect there should be some physical interpretation of the morphisms anyway.

(Note that I'm specifically asking about the 1-categorical case. In the higher case, I think there is an interpretation of $$k$$-morphisms as membrane-like excitations of appropriate dimension. It may be that an answer to my question(s) involves moving to the higher case.)