# Representation of SL(2,C)

• paweld
In summary, there are only two inequivalent two-dimensional representations of the SL(2,C) group, (1/2,1/2) and (1/2,0)+(0,1/2). They are responsible for the Lorentz transformation of left and right Weyl spinors, with the difference being the spin of the particle. The photon transforms irreducibly while the electron does not. The matrices of general representation (m,n) are (2m+1)(2n+1) dimensional and related to traditional matrices of Lorentz transformation. The representation of SL(2,C) is derived from the representation of SU(2) and the pair (m,n) represents a tensor product of

#### paweld

Is it true that there are only two inequivalent two-dimensional representation of
SL(2,C) group and they are responsible for Lorentz transformation of left and right
Weyl spinor.

What's the difference between representation (1/2,1/2) and (1/2,0)+(0,1/2) of
SL(2,C)?

It's simple, one describes an electron (positron), the other a photon. It's the spin which makes a difference. Electron 1/2, photon 1.

One can also say that the photon transforms irreducibly wrt the SL(2,C) group, while the electron not.

Thanks. Do you happen to know how the matricies of general representation (m,n)
look like. As far I know the representation (m,n) is (2m+1)(2n+1) dimensional so
these matricies should be also (2m+1)(2n+1) dimensional. In case of (1/2,1/2)
it gives 4x4 matricies which are probably somehow related to traditional matricies of
lorentz transformation of spacetime points. But on the other hand I heard that
the representation of SL(2,C) are derived from representation of SU(2) and the pair
(m,n) says probably that this representation of SL(2,C) is a tensor product (?) of
(2m+1)-dimensional and (2n+1)-dimnsional representation of SU(2). Does anyone know
the details?

paweld said:
Thanks. Do you happen to know how the matricies of general representation (m,n)
look like.

I don't know, but I can point you to the vast literature on this issue. Try one of Moshe Carmeli's books on group theory and General Relativity. The introduction treats SL(2,C) extensively.

As far I know the representation (m,n) is (2m+1)(2n+1) dimensional so
these matricies should be also (2m+1)(2n+1) dimensional. In case of (1/2,1/2)
it gives 4x4 matricies which are probably somehow related to traditional matricies of
lorentz transformation of spacetime points.

Absolutely correct.

But on the other hand I heard that
the representation of SL(2,C) are derived from representation of SU(2) and the pair
(m,n) says probably that this representation of SL(2,C) is a tensor product (?) of
(2m+1)-dimensional and (2n+1)-dimnsional representation of SU(2). Does anyone know
the details?

Willard Miller's book on group theory deals with the connection between SO(3), restricted Lorentz, SU(2) and SL(2,C) and the way the finite dim. of these Lie groups are related.