/sci/ - Science & Math

>>11056713
>>11056713
For particles with spin, the [math](2j+1)[/math]-multiplets of a spin-[math]j[/math] particle are given as representations of some [math]SU[/math] group (for example for spin-[math]1/2[/math] we have [math]SU(2)[/math]). I'm fairly confused as to if indeed these are representations of the group or its lie algebra. In the spin-1/2 case, the groups [math]SU(2)[/math] and [math]SO(3)[/math] have the same lie algrbra:
[math] [J_a, J_b] = i \epsilon _{abc} J_c [/math]

so elements of [math]SO(3)[/math] might as well satisfy it. How do i know if i'm representing [math]SU(2)[/math] or [math]SO(3)[/math]?

Sorry if i'm being a bit naive, just starting to grasp how reps fit into the picture (not pic related, thats unrelated).

>>10621037
Okay. Then the choice of initial and final state is arbitrary because whatever i choose is the same: "anything can happen between the two states"?
And choosing the GS just makes it easier since we'll probably have it?

I asked my professor today what the transition amplitude from the vacuum to itself means and he told me that it doesn't have a meaning and that its trivial. That perplexed things a bit.