Question on the 2-dim representation of the Lorentz group

[Kontilera]

Hello! I'm currently reading some QFT and have passed the concept of Weyl spinors 2-4 times but this time it didn't make that much sense..
We can identify the Lorentz algebra as two su(2)'s. Hence from QM I'm convinced that the representation of the Lorentz algebra can be of dimension (2s_1 + 1)(2s_2 + 1).
The Weyl spinor is two dimensional so it's either a (s_1, s_2) = (1/2, 0) or a (0,1/2) representation (i.e. left or right handed).

But it then seems (since one representation of the su(2)s is the trivial) as if I only need to specify three parameters when lorentz transforming my Weyl spinors.. What happened to my choice of three rotations and three boosts?


Thanks! :)

 

[Bill_K]

But it then seems (since one representation of the su(2)s is the trivial) as if I only need to specify three parameters when lorentz transforming my Weyl spinors.. What happened to my choice of three rotations and three boosts?

But they are three complex parameters. The SU(2) factorization of the Lorentz group involves the complex generators J ± iK. A Weyl spinor remains invariant under one of these subgroups.

Specifically, the infinitesimal transformation of a left-handed spinor is (Peskin & Schroeder, p44)

ψ_L → (1 - iθ·σ/2 - β·σ/2) ψ_L

where θ is a rotation and β is a Lorentz boost. Individually they change ψ_L, and any real combination of them changes ψ_L, but a complex combination of them leaves ψ_L invariant.

 

[Avodyne]

Another way to say the same thing is to let $\vec A$ be the generators for one of the SU(2)'s and $\vec B$ be the generators for the other SU(2). Then the angular momentum generators are $\vec J=\vec A +\vec B$ and the boost generators are $\vec K=i(\vec A-\vec B)$. For the (1/2,0) rep, $\vec A=\vec\sigma/2$ and $\vec B=0$. Then $\vec J =\vec\sigma/2$ and $\vec K=i\vec\sigma/2$.

 

[Kontilera]

Yeah, I with you. The su(3) has three generators... but the relation above gives me six generators and that is what I can work with. :)
Or equivalently, I work with three generators but they now span a complex vectorspace, hence my coefficients amount to six degrees of freedom.

Thanks.

 

[Kontilera]

Does it exists a Loretnz transformation such as (1, 1/2) ? In that case, have do we make A and B to same dimension?

 

[dextercioby]

Does it exists a Loretnz transformation such as (1, 1/2) ? In that case, have do we make A and B to same dimension?

Of course there is. It's 'half' of a Rarita-Schwinger spinor. The dimension of the spinor space is (2x1+1)(2x1/2+1) = 6.

 

[Kontilera]

Of course there is. It's 'half' of a Rarita-Schwinger spinor. The dimension of the spinor space is (2x1+1)(2x1/2+1) = 6.

Haha. But what about the dimension? I guess we use a reducible rep for the lower s_i? Does it matter which one we choose?