*1) Why is it called a symmetry if it is not a symmetry? what about Noether theorem in this case? and the gauge groups U(1)...etc?*

Gauge symmetry is a local symmetry in CLASSICAL field theory. This may be why
people call gauge symmetry a local symmetry. But we know that our world is quantum.
In quantum systems, gauge symmetry is not a symmetry, in the sense that the gauge transformation does not change any quantum state and is a do-nothing transformation.
Noether's theorem is a notion of classical theory. Quantum gauge theory (when described by the physical Hilbert space and Hamiltonian) has no
Noether's theorem. Since the gauge symmetry is not a symmetry, the gauge group
does not mean too much, in the sense that two different gauge groups can sometimes
describe the same gauge theory. For example, the $Z_2$ gauge theory
is equivalent to the following $U(1)\times U(1)$ Chern-Simons gauge theory:

$$\frac{K_{IJ}}{4\pi}a_{I,\mu} \partial_\nu a_{J,\lambda} \epsilon^{\mu\nu\lambda}$$ with $$K= \left(\begin{array}[cc]\\ 0& 2\\ 2& 0\\ \end{array}\right)$$ in (2+1)D.

Since the gauge transformation is a do-nothing transformation and the gauge group is unphysical, it is better to describe gauge theory without using
gauge group and the related gauge transformation. This has been achieved by string-net theory. Although the string-net theory is developed to describe topological order, it can also be viewed as a description of gauge theory without using gauge group.

The study of topological order (or long-range entanglements) shows that
if a bosonic model has a long-range entangled ground state, then the low energy effective theory must be some kind of gauge theory. So the low energy effective
gauge theory is actually a reflection of the long-range entanglements in the ground state.

So in condensed matter physics, gauge theory is not related to geometry or curvature. The gauge theory is directly related to and is a consequence of the long-range entanglements in the ground state. So maybe the gauge theory in our vacuum is also a direct reflection of the long-range entanglements in the vacuum.

*2) Does that mean, in principle, that one can gauge any theory (just by introducing the proper fake degrees of freedom)?*

Yes, one can rewrite any theory as a gauge theory of any gauge group.
However, such a gauge theory is usually in the confined phase and the effective theory at low energy is not a gauge theory.

Also see a related discussion:
Impossibility of breaking gauge-symmetry in lattice gauge theories

This post imported from StackExchange Physics at 2014-04-04 15:40 (UCT), posted by SE-user Xiao-Gang Wen