Please note that I just read about 20 forum discussions, none of which answered my question. This question is related to my earlier question Is spacetime symmetry a gauge symmetry?.
I am looking for a definition of "Gauge Theory" in purely formal terms. What is a gauge theory in the context of classical field theory?
Suppose you are given a number of fields $\{\phi_i\},$ $i=1,\dots,n$ on $\mathbb{R^4}$, which are defined to transform in some way (and perhaps mix) under Poincaré Transformations, a Lagrangian density $\mathcal{L}$ (which depends on the values of the fields and their derivatives) and an action (integral of $\mathcal{L}$ evaluated at fields evaluated at points). Further suppose that for all fields $\phi=(\phi_i)$ and Poincaré transformations $P$ $$S[\phi]=S[P\cdot \phi].$$ I think most people will agree that this data specifies a Classical Special Relativistic field theory. Is it a meaningfull question to ask: "What are the gauge symmetries of this theory?" ? In other words: Is this a well-defined notion with only the data given above? Given some transformation $\phi\mapsto \phi'$, is it meaningfull to ask "Is this a gauge transformation?"?
There seems to be some controversy "whether GR is a gauge theory" and "in what sense GR is a gauge theory", so I assume the answer to above question is no, but I wanted to be sure. I am aware of an article by Terence Tao also called What is a gauge? where in his opinion a choice of coordinates is a special case of gauge fixing.
To get back to the above question and make it more interesting than asking for "No" as an answer, consider the following: As far as I know, free klein Gordon theory of a real scalar field in $3+1$ dimensions has no gauge symmetries (whatever that means). How would you construct a classical field theory that is "physically equivalent" to klein Gordon theory after removing gauge degrees of freedom? I'm asking for an example if this is reasonably simple.
Another version of the very first question is the following: Can one have the same Lagrangian with two different gauge groups making two different theories? Supposedly the fact that the canonical momentum associated to $A^0$ in Maxwell theory is non-dynamical (but rather constrained) somehow implies a gauge symmetry being present. However, if one cannot determine the gauge transformations by any well-defined procedure, could one construct a self-consistent (perhaps physically meaningless) theory, where all components of $A$ are considered to be "observable"? I assume not, since the time evolution of such field configurations would turn out to not be unique.
On the other hand, in the theory of a complex scalar field, all momenta are dynamical and the equation of motion fixes the entire field for all time given reasonable initial conditions. Is it possible in this case to consider something else as "the gauge symmetry" of the theory, perhaps nothing, rather than $U(1)$?