Basis change is not generally coordinate change

Even an allowed finite relabelling requires more to define a smooth, time-oriented coordinate transformation. If an actual coordinate change $T^{\prime }=f(T)$, with $f^{\prime }>0$, is supplied, tensor transformation gives in the zero-shift case

No such transformation law follows from applying $U_C$ alone. A simultaneous passive transformation of states and observable matrices is instead a change of representation.