What are unitary representations used for in physics?

Question

Specifically in quantum mechanics I have seen unitary representations crop up a few times. I understand what they are and how they work mathematically, I'm unsure as to what use they have in physics?

Because my understanding is that they are linear representations which are also unitary operators, which are applied to Hilbert spaces and preserve inner product. But why not just use unitary operators instead of unitary reperenstations, is there something about the groups being represented that makes them important to the Hilbert space too.

Answer 1

Observables are (in the simplest cases) hermitian operators, not unitary operators. Exponentiation of hermitian operators give unitary operators, v.g. the time evolution operator $U(t)=e^{-it\hat H}$ when the Hamiltonian is time independent.

Unitary operations often encapsulate fundamental physical symmetries of the system at the global level, without affecting the norm of the states, i.e. guaranteeing probability is not lost through symmetry operations. For instance, by rotational invariance, one can always make a (unitary) rotation of the system and choose the quantization axis for the angular momentum to be $\hat z$.

Of course a unitary transformation will also take you from a basis where $\hat L_z$ is diagonal to a basis where (say) $\hat L_x$ is diagonal, and this gives you insight into the possible outcomes of measuring $\hat L_x$: since the physics does not depend on the orientation of axes, it must be that the possible outcomes of measuring $\hat L_x$ are identical to those of $\hat L_z$. (The probabilities, which depend on the basis, can be different for a given state.)

Beyond angular momentum one can also think of various relations between cross-sections in theories where operators are connected by symmetries, which must in turn be implemented by unitary transformations.

There are some applications - for instance in quantum optics - where the actual group representations are very useful, as this paper on SU(2) and SU(1,1) interferometers shows. There are generalizations of this to more modes.

They are also useful in constructing coherent states and can thus be used as starting points of phase space methods (v.g. $P$-functions, $Q$-functions or Wigner functions). The wave-functions of rigid rotors are properly symmetrized functions of group representations.

There are other applications of course but the ones above are directly applicable to SU(2), for which the representations are well-known.

Finally, there is some work done on non-unitary representations of states. This was done in the context of particle decay, for instance as was done here by Barut and Raczka, but this never really "caught on".