Here a short summary
In a gauge theory with algebra generators satisfying [tα,tβ]=iCγαβtγ
it can be checked that the field strength tensor
Fβμν transforms as follows
δFβμν=iϵαCβγαFγμν
We want to construct Lagrangians. A free-particle kinetic term must be a quadratic combination of
Fβμν and Lorentz invariance and parity conservation restrict its form to
L=−14gαβFαμνFβμν
where
gαβ may be taken symmetric and must be taken real for the Lagrange density to be real as well. The Lagrangian above must be gauge-invariant thus it must satisfy
δL=ϵδgαβFαμνCβγδFγμν=0
for all
ϵδ. In order not to impose any functional restrictions for the field strengths
F the matrix
gαβ must satisfy the following condition
gαβCβγδ=−gγβCβαδ
In short, the product
gαβCβγδ is anti-symmetric in
α and
γ.
Furthermor the rules of canonical quantization and the positivity properties of the quantum mechanical scalar product require that the matrix
gαβ must be positive-definite. Finally one can prove that the following statements are equivalent
- There exists a real symmetric positive-definite matrix gαβ that satisfies the invariance condition above.
- There is a basis for the Lie algebra for which the structure constants Cαβγ are anti-symmetric not only in the lower indices β and γ but in all three indices α, β and γ.
- The Lie algebra is the direct sum of commuting compact simple and U(1) subalgebras.
The proof for the equivalence of these statements as well as a more in-detail presentation of the material can be found in the aforementioned book by S. Weinberg.
A proof for the equivalence for gαβ=δαβ (actually the most common form) was given by M. Gell-Mann and S. L. Glashow in Ann. Phys. (N.Y.) 15, 437 (1961)
This post imported from StackExchange Physics at 2015-01-19 14:11 (UTC), posted by SE-user Stan