Yang-Mills Lagrangian: Is ##F^{\mu \nu}F_{\mu \nu}## a Number?

[AndrewGRQTF]

I am sorry for asking this stupid question, but in the Yang-Mills lagrangian, there is a term $Tr(F^{\mu \nu}F_{\mu \nu})$. Isn't $F^{\mu \nu}F_{\mu \nu}$ a number?

 

[Orodruin]

It is a Lorentz scalar, but generally the fields at each point are elements of the Lie algebra of the gauge group. In other words, the Fs are matrices.

 

[AndrewGRQTF]

It is a Lorentz scalar, but generally the fields at each point are elements of the Lie algebra of the gauge group. In other words, the Fs are matrices.

So is this true: $F^2 = (F^{\mu\nu}F_{\mu\nu}) \Sigma_a (T^aT^a)$ where the a goes from a=1,...,n and the n is the number of generators of the group? This the the $F^2$ that we put inside the trace, right?

 

[Orodruin]

No. Again, the $F_{\mu\nu}$ are matrices. If you want to express them in terms of the Lie algebra generators you must write $F_{\mu\nu}= F_{\mu\nu}^a T^a$.

 

[AndrewGRQTF]

No. Again, the $F_{\mu\nu}$ are matrices. If you want to express them in terms of the Lie algebra generators you must write $F_{\mu\nu}= F_{\mu\nu}^a T^a$.

On the right hand side, since we wrote out the T which are matrices, is the $F_{\mu \nu} ^a$ a number? For example is $F^1_{22}$ a number?

 

[Orodruin]

Yes.

 

[DarMM]

On the right hand side, since we wrote out the T which are matrices, is the $F_{\mu \nu} ^a$ a number? For example is $F^1_{22}$ a number?

At a single point, although as @Orodruin mentioned above it varies from point to point so really it is a function.