*[Other parts in this series: 0,1,2,3,***4**,5,6.]

In an ideal world, the formalism that you use to describe a physical system is in a one-to-one correspondence with the physically distinct configurations of the system. But sometimes it can be useful to introduce additional descriptions, in which case it is very important to understand the unphysical over-counting (e.g., gauge freedom). A scalar potential is a very convenient way of representing the vector force field, , but any constant shift in the potential, , yields forces and dynamics that are indistinguishable, and hence the value of the potential on an absolute scale is unphysical.

One often hears that a quantum experiment measures an *observable*, but this is wrong, or very misleading, because it vastly over-counts the physically distinct sorts of measurements that are possible. It is much more precise to say that a given apparatus, with a given setting, simultaneously measures all observables with the same eigenvectors. More compactly, **an apparatus measures an ***orthogonal basis* – not an observable.We can also allow for the measured observable to be degenerate, in which case the apparatus simultaneously measures all observables with the same degenerate eigenspaces.… [continue reading]