meaning of SO(4) -- SU(2)xSU(2)

[IRobot]

Hi,

I was doing an exercise in my QFT book asking me to show that the Lorentz Group SO(4) is isomorphic to SU(2)*SU(2) but not explaining why. I was wondering, and asking myself that maybe it has some "deep" meaning, about the relation between the spin and the relativity. Am I totally wrong?

[tom.stoer]

I was wondering, and asking myself that maybe it has some "deep" meaning, about the relation between the spin and the relativity. Am I totally wrong?You are right, the origin of spin is essentially due to the Spin(N) groups which are related to SO(1, N-1). This works for arbitrary N, whereas the factorization is special for N=4.

I don't want to post too man formulas here, so will try to give you a brief summary and find a good reference.

The idea is to take the six generators of the Lorentz group Ja for the rotations w.r.t. the a-axis (= angular momentum) and Ka for the boosts along the a-axis. The J's generate the usual su(2) = so(3) algebra, whereas the K's don't as their commutator is a J again.

Now one defines two new sets of generators, namely

L+a = Ja + iKa
L-a = Ja - iKa

One can check that both sets generate the usual su(2) = so(3) algebra and that one set commutes with the other. So one has two copies of the SU(2), one generated by the L+, one by the L-

[haael]

Does this "L" have any physical meaning, i.e. is it related to some property of a particle?

[IRobot]

thank for responding so fast, I did the calculation using the commutators of K

[IRobot]

thank for responding so fast, I did the calculation using the commutators of K and found my answer

[tom.stoer]

Does this "L" have any physical meaning, i.e. is it related to some property of a particle?
No. It's a complexification i.e. a linear combination with an "i" of a rotation and a boost - I don't think it has some interpretation.

[arivero]

Question, is there some other case where so(n) is isomorphic to other product g*g of Lie algebras?

[dextercioby]

Using a metric mostly minus, the restricted Lorentz group is not SO(4), neither SO(1,3), but \mbox{SO(1,3)}_{\uparrow} . One can show that this is homomorphic (NOT isomorphic!!) to the direct product of 2 SU(2)'s (proof based on the polar decomposition theorem and the existence of a homomorphism between SO(3,R) and SU(2)). At the level of Lie algebras

\mbox{so(1,3)_{\mathbb{C}}} \simeq \mbox{su(2)}\oplus \mbox{su(2)}

Note that the Lie algebras are directly summed, there's no product.

[tom.stoer]

Question, is there some other case where so(n) is isomorphic to other product g*g of Lie algebras?
1) its not * but +
2) so(4) is the only non-semi-simple so(n), that means for all higher n there's no such decomposition

[arivero]

1) its not * but +
2) so(4) is the only non-semi-simple so(n), that means for all higher n there's no such decomposition

So, I was wondering, is there some group theoretic meaning in heterotic string T-duality? In this case, SO(32) reveals itself as having a hidden "E8xE8". Is there some paralell to the SO(4) - SU(2)xSU(2)? Is there some general concept containing both "dualities"?

[fzero]

So, I was wondering, is there some group theoretic meaning in heterotic string T-duality? In this case, SO(32) reveals itself as having a hidden "E8xE8". Is there some paralell to the SO(4) - SU(2)xSU(2)? Is there some general concept containing both "dualities"?

The relationship is a little bit more obscure than an isomorphism between groups and is explained in a paper by Ginsparg (preprint available at http://www-lib.kek.jp/cgi-bin/kiss_prepri.v8?KN=&TI=&AU=&AF=&CL=&RP=HUTP-86%2FA053&YR=).

In 10D, the string states in the heterotic string are parameterized by the vectors in either the SO(32) root lattice \Gamma_{16} or in two copies of the E_8 lattice, \Gamma_8. If we further compactify either of these on a circle, we obtain additional states living in the even 2d Lorentzian lattice U. In either case, the states live in an even self-dual Lorentzian lattice \Pi_{17,1}. But all such lattices are unique up to SO(17,1) transformations. Therefore states in \Gamma_{16}\oplus U and those in \Gamma_8\oplus \Gamma_8 \oplus U are related by an SO(17,1) transformation. This is a T-duality.

[tom.stoer]

In this case, SO(32) reveals itself as having a hidden "E8xE8".
As said by fzero the relationship is more complicated. E(8)*E(8) and SO(32) have the same dimension, but SO(32) does not factorize like SO(4).

[qsa]

http://www.cosmolearning.com/video-lectures/lecture-33-groups-compact-and-non-compact-groups-universal-covering-group-9755/