The discussion revolves around the nature of the term ##F^{\mu \nu}F_{\mu \nu}## in the context of the Yang-Mills Lagrangian, specifically whether it can be considered a number or if it has a more complex structure due to the properties of the fields involved.
Participants generally agree that ##F^{\mu \nu}F_{\mu \nu}## is a Lorentz scalar, but there is disagreement regarding its interpretation as a number due to the matrix nature of the fields involved. The discussion remains unresolved regarding the implications of this scalar being expressed in terms of matrices.
The discussion highlights the dependence on the definitions of the fields and their representations, as well as the varying interpretations of scalar quantities in the context of gauge theories.
Orodruin said: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.
Orodruin said: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##.
At a single point, although as @Orodruin mentioned above it varies from point to point so really it is a function.AndrewGRQTF said: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?