About the Lie algebra of our Lorentz group

[Kontilera]

Hello!

I'm currently reading Ryder - Quantum Field Theory and am a bit confused about his discussion on the correpsondence between Lorentz transformations and SL(2,C) transformations on 2-spinor.
He writes that the Lie algebra of Lorentz transformations can be satisfied by setting
$$\vec{K} =\pm \frac{i \vec{\sigma}}{2}.$$
Here it seems as if the dimensions are mixed up. The Pauli matrices are 2 times 2 while the Loretnz generators are 4 times 4.

Secondly he argues that the Lorentzgroup can be ''factorized'' into $$SU(2) \times SU(2)$$ but how come this goes along with the fact that the Loretnz group is non-compact.
It seems as if we take the product group of two compact group the resulting group is compact?
Am I wrong about this?

 

[Bill_K]

Here it seems as if the dimensions are mixed up. The Pauli matrices are 2 times 2 while the Loretnz generators are 4 times 4.

He means K and σ as abstract operators which satisfy a certain set of commutation relations, not a particular matrix representation of those operators.

Secondly he argues that the Lorentzgroup can be ''factorized'' into SU(2)×SU(2) but how come this goes along with the fact that the Loretnz group is non-compact.

To go from one set of generators to the other you have to consider the complexified Lie algebra, A = (J + i K)/2 and B = (J − i K)/2. This does not preserve group compactness.

 

[Kontilera]

Ah cool!
Maybe I'm a bit under the level that this book is written on.. I come back with future confusions. :)
Thanks Bill!