# Spin via Change of Phase

Thinking of spin as arising from a change in the phase of a wave function:

The angular momentum is defined by the change of the phase of the wave function under rotations, which may come from the dependence of the wave function on space, but also from the transformations of the components of the wave function among each other, which is possible even if everything is localized at a point. So even point-like objects may carry an angular momentum in quantum mechanics, the spin.

Is it possible to see the existence of spin using the quasi-classical wave function $\phi (\vec{r}) = e^{i \theta} = e^{iS/\hbar}$? If the action being invariant under a rotation gives angular momentum then $\phi$ should remain as $\phi$ yet the quote above seems to be saying that $\phi (\vec{r}) = e^{i(S+\lambda \hbar)/\hbar} = Ae^{iS/\hbar}$ can happen, furthermore it can happen in two ways (which I do not see).

If there is a nice way to see this, perhaps one can also somehow understand spin, when thought of as arising from Lorentz invariance, if you think of the Lorentz group as being generated by unitary operators $T = \Pi_{\mu} e^{s_{\mu}K_{\mu}} = e^{\sum _{\mu} s_{\mu}K_{\mu}}$ and somehow see this as like the phase of the wave: $T \phi = \Pi_{\mu} e^{s_{\mu}K_{\mu}} \phi = Ae^{iS/\hbar} = \phi (\vec{r})$, if that makes sense?

• You might be interested by this previous answer to a quite similar question. – Trimok Sep 30 '14 at 9:29
• Yeah that's a very cool argument about discreteness of $J$ but unfortunately doesn't tackle this mixing up of components business or the infinitesimal generator issue via the quasi-classical wave function. – bolbteppa Sep 30 '14 at 11:58
• Are you familiar with this stuff -- en.wikipedia.org/wiki/… ? Your question is very hard to understand. Why do you say "ϕ should remain as ϕ"? What can "happen in two ways"? – Steve Byrnes Oct 2 '14 at 15:40
• Yes I'm familiar with the material in the link, it doesn't address my question. The two ways angular momentum (defined by the change of the phase of the wave function under rotations) can be changed are a) the dependence of the wave function on space, b) via the transformations of the components of the wave function among each other. If spin is just a change of the phase due to transformations of the components, can this be seen via the quasi-classical wave function? Can it be seen in the way I mentioned in my post? If so, how does all this relate to the relativistic interpretation of spin? – bolbteppa Oct 2 '14 at 21:25