A New Pedagogical Way of Finding Out the Gauge Field Strength Tensor in Abelian and Non-Abelian Local Gauge Field Theories

Abstract

The gauge field strength tensor $F_{\mu \nu}$ in Abelian and non-Abelian local gauge field theories is a key object in the construction of the Lagrangian since it provides the kinetic term(s) of the gauge field(s) $A_\mu$. When introducing this object, most of textbooks employ as a tool the commutator of the gauge covariant derivatives $D_\mu \psi$ of a fermion field $\psi$: $F_{\mu \nu} \psi = (i/g)[D_\mu,D_\nu]\psi$. I argue that such a construction, although completely correct and valid, is not pedagogical enough for many students that approach the gauge field theories for the first time. Another construction, based on the object $D_\mu A_\nu$: $F_{\mu \nu} = D_{[\mu} A_{\nu]}$, which avoids the introduction of additional and, for the case in consideration, spurious degrees of freedom such as the fermion one, simpler, more pedagogical in many cases, and suitable for first-time students, is presented.