I am trying to understand the Quantum theory of non-relativistic angular momentum in terms of representation theory with full precision. In particular, I would like to deduce that angular momentum is composed of orbital angular momentum as well as spin using mathematics, experimental results, and extremely basic physical assumptions.
What I mean by this is that I want to deduce that given an arbitrary Hilbert space $\mathcal{H}$, there exists a projective representation of $SO(3)$ that we identify with the “spin of the system” and a distinct other projective representation of $SO(3)$ which we identify with the orbital angular momentum of the system (i.e. should allow only integer values of orbital angular momentum) on $\mathcal{H}$.
To begin, I define angular momentum as the quantity that is conserved when the system is invariant under rotations. In non-relativistic Quantum Mechanics, I assume that the space modeling our reality is Euclidean three dimensional space, denoted $\mathbb{R}^3$. I also assume that the abstract group $SO(3)$ properly captures 3D spatial rotations.
I am content with explanations found online about the necessarily projective representations of $SO(3)$ on finite dimensional Hilbert spaces. Instead of considering projective representations of $SO(3)$, we consider the equivalent genuine representations of its double cover. We actually take another step and consider representations of $SU(2)$ which is isomorphic to the double cover of $SO(3)$. Justified by a couple of theorems, we can then characterize the irreducible representations of $SU(2)$. In this finite dimensional case, we identify each irreducible representation with a spin.
However, the above derivation of course requires that the Hilbert space we are furnishing a projective representation of $SO(3)$ on is finite dimensional. In general, Hilbert spaces are infinite dimensional. Here is where many online resources assume that we define an arbitrary Hilbert space as $\mathcal{H} = \mathcal{H}_s \otimes \mathcal{H}_l$ where the first tensor factor is finite dimensional (encodes finite dimensional degrees of freedom) and the second tensor factor is infinite dimensional. It can be shown (cf. Hall's book of mathematical Quantum Mechanics) that only odd dimension irreducible representations of $SO(3)$ on $\mathcal{H}_l$ exist. Hence, no half integer orbital angular momentum.
My questions:
- The partitioning of a Hilbert space into finite and infinite degrees of freedom seems either i) assuming that spin exists in the first place, or ii) absolutely arbitrary. There are a large number of ways you can partition an infinite dimensional space into a finite dimensional space tensor an infinite dimensional space.
- Even if we accept the partitioning of Hilbert space, resources like Hall's book begin at the outset by looking for genuine representations of $SO(3)$ (as opposed to projective) on $\mathcal{H}_l$. I don't understand why we wouldn't be motivated to look for projective representations here as the phase gauge of Quantum Mechanics is a property of the theory, not the dimension of Hilbert space you are working in.
Best Answer
It is impossible to do so. You cannot start with an abstract Hilbert space, search for projective representations of $SO(3)$ and then pop out something corresponding to what we would physically associate with spins and orbital angular momentums.