The discussion centers on the kinetic terms for vierbein fields and spin connection fields in the context of gauge theory of gravity. It is established that the kinetic term for spin connections is included in the local Lorentz invariance, while the vierbein terms are included under Poincaré local invariance. The commutator of covariant derivatives is not the kinetic term for either field when considered independently. This clarification resolves the initial uncertainty regarding the relationship between these fields and their kinetic terms.
PREREQUISITESThe discussion is beneficial for theoretical physicists, particularly those focusing on gauge theories, general relativity, and the mathematical foundations of gravity. It is also relevant for students and researchers interested in the interplay between vierbeins and spin connections.
luxxio said:when we try to build the gauge theory of gravity we needs the vierbeins fields and spin connections fields. In the case we take them independent, which are the kinetic terms for two fields? or in other words, is the commutator of covariant derivatives the kinetic term for vierbeins and for spin connections?