Group Theory and Energy Eigenstates in Quantum Mechanics

[praharmitra]

I have only recently begun to study group theory from Lie Algebra for Particle Physicists by Georgi. I am slightly confused about the language used by physicists.

What does it mean when the following is stated

"The energy eigenstates transform like irreducible representations of the group G"

(G is a transformation group that is a symmetry of a quantum mechanical system.

Does it mean that if u take irreducible representations of the group G (which are linear operators) and act them on the energy eigenstates, you get new eigenstates with the same energy?

 

[fresh_42]

The language physicists use in the area is quite a bit different from how mathematicians would say it.

The eigenstates are vectors, say of ##V##. Then the group ##G## as well as its Lie algebra ##\mathfrak{g}## act on ##\mathfrak{gl}(V)## e.g. via the adjoint representations if ##V=\mathfrak{g}##. I assume that we have the latter here, i.e. ##\varphi\, : \,\mathfrak{g} \longrightarrow \mathfrak{gl}(V)##. So if we have any representation ##(V,\varphi)## of ##\mathfrak{g}##, then, since ##\mathfrak{g}## is semisimple, ##V=V^{(1)}\oplus \ldots V^{(m)}## splits into irreducible subspaces ##V^{(k)}##, i.e. ##\{\,0\,\}## and ##V^{(k)}## are the only ##\varphi-##invariant subspaces. Nevertheless, the ##V^{(k)}## split into a direct sum ##V^{(k)}=V^{(k)}_1\oplus \ldots \oplus V^{(k)}_{n_k}## of subspaces which are invariant under the CSA of ##\mathfrak{g}## and the root spaces of ##\mathfrak{g}## shift vectors from on ##V^{(k)}_i## to another ##V^{(k)}_j##. The energy eigenstates are represented by those irreducible components ##V^{(k)}##.