Question

What is the meaning of the gauge-dependent term?

Answer 1

1. Whether or not observable quantities must be gauge-independent?

2. If the observable quantities are gauge-dependent. What is the meaning of them?

For the first question, I find some quantities are gauge dependent and these quantities are measured by experiments; for example, the charge densities on the different sublattice of graphene. The symmetry of pseudo-spin is not a good symmetries of Lagrangian, which leads to the fact that there is not a gauge-covariant fields to couple with pseudo-spin. As a results, the 0-components of pseudo-spin currents are gauge dependent and these quantity can be measured by STM. Thus, the question is to explain the physical significance of gauge-dependent terms.

Answer 2

A gauge is something you yourself can choose when doing computations. Your personal choices when doing computations cannot influence experimental observations except if you have magical powers. If do not understand much of your explanations, but it seems to me you simply do not have a gauge symmetry.