Is there a connection between Lie Groups and observable quantities in physics? Good evening everybody.
I have some questions about the relation between Lie groups and observables in physics. Indeed, taking the example of spin formalism of Quantum mechanics I know that Pauli's matrices $\{\sigma^i\}$ correspond to an observable quantity ($\frac{1}{2}$ eigenvalues of spin projection) since these latter are hermitian operators.
However, by a recent study on group theory I understood that $\frac{1}{2}$ spin eigenstates of classical QM can span the vector space which transform under a representation of the Lie group $SU(2)$.
It is always the case that an hermitian basis of a Lie algebra represent an observable quantity in QM or it is only the case for $SU(2)$ group? What is the general connection between group theory and observable quantities of QM and QFT?
 A: Let $A$ be the C*-algebra of a quantum mechanical system, and suppose that $G$ acts on $A$ by symmetries through the group homomorphism $\alpha:G\to\text{Aut}(A)$. Let us further assume that $G$ is a simply connected Lie group with trivial $H^2(\mathfrak g,\mathbb R)$, $\mathfrak g$ being the Lie algebra of $G$. If $\pi$ is an $\alpha$-regular representation, meaning that the transition probability $P_{\omega\to\omega\circ\alpha_g}$ is continuous in $g\in G$ for every state $\omega$, then $G$ can be unitarily represented on the Hilbert space of $\pi$ in such a way that
$$u_g\pi(a)u_g^* = \pi(\alpha_g(a)),\qquad\forall a\in A,$$
by Bargmann's theorem. This expresses the fact that the pair $(\pi,u)$ is a covariant representation of the $G$-algebra $A$. Since $u$ is a representation of $G$, it induces a representation of $\mathfrak g$ on the Hilbert space of $\pi$ as well through self-adjoint operators. Hence if you allow the generators to live in the larger C*-algebra generated by $\pi(A)$ and all the $u_g$ for any $g\in G$ (the crossed product $A\rtimes_{(\pi,u)}G$), you can interpret them as observables. In most cases, these operators can be given a physical meaning (e.g. the generators of translations are the momentum operators, etc...).
