Can we simply comment that the Tr[TarTbr]≡C(r)δab depends on the representation. For the case of SU(2) and SO(3), we can relate this to the spin-S representation. By the manner that for SU(2) group is in a spin 1/2 representation and SO(3) group is in a spin 1 representation. One can write down the relation of spin operators as:
S2x+S2y+S2z=S(S+1)I2s+1.
(
ℏ=1).
And
∑a(Sa)2=S2x+S2y+S2z=∑a=x,y,z(Ta)2=3(Tb)2
here
b can be
x,y,z.
So, combine the two relations above:
12Tr[TarTbr]=12S(S+1)3Tr[I2s+1]=S(S+1)(2S+1)6
For SU(2), spin-1/2 representation, we have:
Tr[TarTbr]=2S(S+1)(2S+1)6|S=1/2=1/2
For SO(3), spin-1 representation, we have:
Tr[TarTbr]=2S(S+1)(2S+1)6|S=1=2
.
For spin-3/2 representation, we have:
Tr[TarTbr]=2S(S+1)(2S+1)6|S=3/2=5,
etc.
Shall we say the level k quantization of SU(2) C-S and SO(3) C-S are thus related by a factor of:
(1/2)/2=1/4.
And this quantization value presumably is a measurable quantized value for the spin-Hall conductance. See for example, the discussion in this paper: Symmetry-protected topological phases with charge and spin symmetries: response theory and dynamical gauge theory in 2D, 3D and the surface of 3D: arXiv-1306.3695v2, in Eq(26) and its p.7 right column and in p.8 left column. See also this Phys Rev B paper.
This way of interpretation simplifies Dijkgraaf-Witten or Moore-Seiberg's mathematical argument to a very physical level of the spin S property. Would you agree?
Any further thoughts/comments?
This post imported from StackExchange Physics at 2014-04-04 16:26 (UCT), posted by SE-user Idear