The discussion centers on the interpretation of the term ##Tr(F^{\mu \nu}F_{\mu \nu})## in the Yang-Mills Lagrangian. It is established that while ##F^{\mu \nu}F_{\mu \nu}## is a Lorentz scalar, the fields ##F_{\mu\nu}## are matrices representing elements of the Lie algebra of the gauge group. The expression for ##F^2## is clarified as ##F^2 = (F^{\mu\nu}F_{\mu\nu}) \Sigma_a (T^aT^a)##, where ##F_{\mu\nu}= F_{\mu\nu}^a T^a##. The components ##F_{\mu \nu}^a## are indeed numbers at specific points, but they vary as functions across different points in space.
PREREQUISITESThe discussion is beneficial for theoretical physicists, particularly those specializing in quantum field theory, gauge theories, and mathematical physics. It is also relevant for graduate students studying advanced topics in particle physics.
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?