sl(2,C) and su(2)xsu(2)
john baez
In article <5562ba$i...@lyra.csx.cam.ac.uk>,
Kitty <bje...@cus.cam.ac.uk> wrote:
>I have a problem that's been bothering me for a few days now. Perhaps
>someone can give me a fresh perspective on it [....]
One answer to your question about exponentiation
might be that 1) any finite-dimensional Lie algebra is the Lie
algebra of a simply connected Lie group, unique up to isomorphism,
2) every finite-dimensional representation of the Lie algebra
then gives rise to a finite-dimensional representation of this Lie
group and vice versa, and 3) the representation of the Lie algebra
is Hermitian if and only if the representation of the Lie group
is unitary. Things work very nicely.
I think what's bugging you is that sl(2,C) is not isomorphic
to su(2) x su(2). One way to see this is to use 1) and note
that su(2) x su(2) is the Lie algebra of a compact simply
connected group, namely SU(2) x SU(2), while sl(2,C) is the
Lie algebra of a noncompact simply connected group, namely SL(2,C).
Another way is to note that sl(2,C) has no finite-dimensional
Hermitian representations, while su(2) hence su(2) x su(2) has
lots --- for example, su(2) has an obvious Hermitian
representation on C^2.
It might also be enlightening to write down the isomorphism
you're thinking of, and see that it's not really an isomorphism.
For example, while su(2) is a Lie subalgebra of sl(2,C) in an obvious
way, this does not extend to a Lie algebra isomorphism between
su(2) x su(2) and all of sl(2,C) as you seem to think. This is
related to how rotations form a subgroup of the Lorentz group, while
Lorentz boosts do not.
By the way, there are noncompact groups with finite-dimensional
unitary representations, e.g. the real line.
Matt McIrvin
In article <5566na$4...@charity.ucr.edu>, ba...@math.ucr.edu (john baez) wrote:
> It might also be enlightening to write down the isomorphism
> you're thinking of, and see that it's not really an isomorphism.
> For example, while su(2) is a Lie subalgebra of sl(2,C) in an obvious
> way, this does not extend to a Lie algebra isomorphism between
> su(2) x su(2) and all of sl(2,C) as you seem to think. This is
> related to how rotations form a subgroup of the Lorentz group, while
> Lorentz boosts do not.
One thing that might be contributing to the confusion is that among
physicists classifying finite-dimensional representations of SL(2,c),
sl(2,c) is often treated as if it *were* su(2) x su(2), with one factor for
right-chiral and one for left-chiral particles. However, the "generators"
in these "su(2)" factors are linear combinations *involving factors of i*
of the sl(2,c) generators; whereas the generators of the actual group
transformations *cannot* involve those extra factors of i.
So, although the finite-dimensional representations have the same numbers
of dimensions as su(2) x su(2) representations (providing an
extraordinarily convienient way to enumerate and classify them), the
infinitesimal transformations on them are not really infinitesimal
rotations, but combinations of infinitesimal rotations and infinitesimal
Lorentz boosts.
The spinor of a massive spin 1/2 particle at rest (which transforms
infinitesimally under a representation of su(2)) and the left-handed Weyl
spinor of a massless neutrino (which transforms infinitesimally under a
representation of one of those so-called "su(2)"'s in sl(2,c)) may look
similar, but they aren't the same thing-- and the reason is that the Lie
algebras are not really the same thing either.
--
Matt McIrvin <http://world.std.com/~mmcirvin/>