Abstract:  In gauge field theory, gauge invariant quantities are endowed with physical meaning, while gauge dependent quantities are regarded as redundant mathematical objects. In that case, why are gauges still widely used in modern physics? Based on the analysis of some specific gauge fields, Rovelli gives an explanation that the coupling between gauge fields needs to be described by gauge dependent quantities. Electromagnetic field is the simplest gauge one, and the Lagrangian of free field can be described by gauge invariant relative generalized coordinates. The interaction between the electromagnetic fields of two ideal media at the interface can be expressed by the relative generalized coordinate difference of the two fields. In order to preserve the gauge invariance of this action, the gauge functions of the two fields must be time independent and their gradients are equal． 

Key words:  gauge transformation, 2+1 dimensional electromagnetic field, Lagrangian