A How is the invariant speed of light enocded in SL(2,C)?

[jakob1111]

In quantum field theory, we use the universal cover of the Lorentz group SL(2,C) instead of SO(3,1). (The reason for this is, of course, that representations of SO(3,1) aren't able to describe spin 1/2 particles.)

How is the invariant speed of light enocded in SL(2,C)?

This curious fact of nature, is encoded in SO(3,1), because this is exactly the group that leaves the Minkowski metric invariant. In contrast, SL(2,C) is just the group of complex 2x2 matrices with unit determinant.

 

[PeterDonis]

representations of SO(3,1) aren't able to describe spin 1/2 particles

They aren't? SO(3,1) has spinor representations, doesn't it? SU(2) certainly does, and SO(3,1) is isomorphic to SU(2) x SU(2).

 

[jakob1111]

They aren't? SO(3,1) has spinor representations, doesn't it? SU(2) certainly does, and SO(3,1) is isomorphic to SU(2) x SU(2).

No, SO(3,1) has no spinor representations. It is the complexified Lie algebra of the Lorentz group so(3,1)_C which is isomorphic to the Lie algebra su(2) x su(2). This process off complexification is a Lie algebra deformation and changes something fundamental. Other names for this complexification are Weyl's unitary trick or Wick rotation. We map the boost generators K_i to iK_i and this way we get the Lie algebra so(4), which is isomorphic to su(2) x su(2).

When we then do representation theory of the complexified Lie algebra so(3,1)_C or equivalently of su(2) x su(2) and use the exponential map to get the corresponding group representations, we do not only get representations of SO(3,1), but instead the representations of SL(2,C). Some of these representations are also representations of SO(3,1), but we get more than that. For example,the scalar and vector representation are also representations of SO(3,1), but the spinor represntations aren't.

 

[jakob1111]

My question has been answered at stackexchange: http://physics.stackexchange.com/qu...the-invariant-speed-of-light-encoded-in-sl2-c

 

[vanhees71]

Indeed the answer is given in the stackexchange posting.

Just to clarify the thing with spin 1/2. If you consider only rotations, the SO(3), the covering group is SU(2), and the fundamental representation of SU(2) describes spin 1/2. To extend this to representations of the, proper orthochronous Lorentz group, $\mathrm{SO}(1,3)^{\uparrow}$ you are led to its covering group $\mathrm{SL}(2,\mathbb{C})$. However, now there are two two-dimensional non-equivalent representations corresponding to two sorts of Weyl spinors.

To also be able to describe space reflections (parity) you have to add these two representations, which leads to the four-dimensional Dirac-spinor representations, and the two Weyl spinors these are made of are precisely the states of left and right-handed chirality. For details, see Appendix B of

http://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf