Proofs of Group Theory for Theoretical Physicists - Daniel

[dextercioby]

It's always annoying when one finds in books (written by (theoretical) physicists for (theoretical) physics students) statements like those below without a mere cross-reference for a mathematically-rigurous proof. And that's what I'm searching for right now: either point me to some books, or post the proofs right here (that would be perfect). So here's what's been bugging me lately:

1. A noncompact Lie group has no nontrivial finite dimensional unitary linear irreps.
2. The linear representations of a simply connected Lie group are fully reducible.
3. The unitary linear irreps of a simply connected Lie group can be put in bijective correspondence with (essentially) selfadjoint irreps of the corresponding Lie algebra.
4. The Theorem of Nelson. The only reference for a proof that i found is the original article by Nelson, but, unfortunately, it's not within my reach.

The relevance of this thread: these mathematical results are fundamental in understanding the concept of implementing space-time symmetries in the Hilbert space language of QM through the so-called "Wigner-Weyl method".

Daniel.

 

[matt grime]

The underlying idea is the following:

any lie group homomorphism is determined uniquely by the induced lie algebra map.

See Fulton and Harris, Representation Theory (A First Course) Springer GTM 129 (1991) Lecture 8 onwards.

I don't know that it'll answer all your questions, but it seems like it'll be a start in the right direction. It is quite a cheap and readily available book. You library should have a copy.