Gauge Invariance, Gauge Fixing, and Gauge Independence*

1 Why a fixed law of dynamics for each system? $O$ne of the implicit but standard preconceptions in physical sciences seems to be that “a physical theory should describe a specific physical system with a fixed law of dynamics” Sticking to this belief causes the following difficulties: i $) $ difficulty caused by “singular constrained dynamics ” with a de-generate symplectic form (typically in a gauge theory to be quantized), and ii) (mathematical) difficulty in treating “explicitly broken symmetris” (e.g., broken scale invariance and approximate flavour symmetry of hadrons). If we are free from the above prejudice, we can easily relativize the situ-ation concerning overlap vs. separation between responses of a physical system against two sorts of actions on its internal ( $\Rightarrow $ symmetries) and ex-ternal ( $\Rightarrow $ dynamics) degrees of freedom. A geometric analogy: resolution of singularity Between $i$) & ii) with no inevitable relations seen at first, some aspect of mutual duality starts to show up. To understand it, the analogy to geometric configurations of two diagrams (like an arc and a surface) with or without intersections will be helpful; by shifting the diagrams along the axis orthogonal to the intersections, we can freely change their contact relations with or without intersections, which is a simple-minded picture of the blowing-up method for singularity resolutions. Or, even if we do not touch on given two separated diagrams, they can be viewed as intersecting by choosing suitable angles of axis of our sight. The former is in the active version of deformations, while the latter in the passive one