Demystifier said:
The way I see it, you deeply disagree with Schwartz. He seems to be saying that gauge invariance is just a convenient mathematical trick which makes some calculations easier, while you seem to saying that gauge invariance is a deep truth without which it is absolutely impossible to get correct physical results.
Well, it may well depend on a reader's opinion which meaning he reads into the words of an author.
In my understanding what Schwartz does here is to underline the important difference between global symmetries, which indeed have observational consequences in terms of conservation laws a la Noether, while a local gauge symmetry indicates a redundancy in the description of the dynamics of a physical model.
Let's stick with electrodynamics and Abelian gauge symmetry. On the quite general level we are discussing here there's not so much difference for the more general non-Abelian case.
In classical electrodynamics the Maxwell equations summarize about 200 years of empirical knowledge about electromagnetic phenomena in terms of fields, at the time a brand-new concept discovered by Faraday to solve the old enigma of actions at a distance disturbing the physicists since Newton's gravitational froce law (including Newton himself). The Maxwell equations are written in terms of the observable fields ##\vec{E}## and ##\vec{B}## (the electric and magnetic field or taken together the electromagnetic field) and the charge-current densities ##\rho## and ##\vec{j}##.
Using the homogeneous Maxwell equations you get the potentials, i.e., there exists a scalar field ##\Phi## and a vector field ##\vec{A}## such that (working with natural Heaviside-Lorentz units with ##c=1##)
$$\vec{B}=\vec{\nabla} \times \vec{A}, \quad \vec{E}=-\vec{\nabla} \Phi - \partial_t \vec{A}.$$
Then it's immediately clear that the physical situation, given by ##\vec{E}## and ##\vec{B}##, is not in one-to-one correspondence with the potentials, because for given ##(\vec{E},\vec{B})## any other potentials connected to one solution by a gauge transformation
$$\vec{A}'=\vec{A}-\vec{\nabla} \chi, \quad \Phi'=\Phi+\partial_t \chi$$
leads to the same fields, because
$$\vec{\nabla} \times \vec{A}'=\vec{\nabla} \times \vec{A}=\vec{B}, \quad -\vec{\nabla} \Phi'-\partial_t \vec{A}' = -\vec{\nabla} \Phi-\partial_t \vec{A} + \partial_t \vec{\nabla} \chi -\vec{\nabla} \partial_t \chi=-\vec{\nabla} \Phi-\partial_t \vec{A}=\vec{E}.$$
So the physical fields ##\vec{E}## and ##\vec{B}## determine the potentials only modulo a gauge transformation with an arbitrary scalar field ##\chi## and thus the physics is not in the potentials but only in the potentials modulo a gauge transformation. A gauge transformation is not a symmetry, because it just expresses the redundancy of the description of the theory in terms of the potentials, i.e., the physical situation is described not uniquely by the potentials but only modulo a gauge transformation. In other words there are more field degrees of freedom used than are physical dynamical fields.
This becomes clear using the action principle. It turns out that the canonically conjugate field momentum of ##\Phi## is identically 0, i.e., ##\Phi## cannot be a true dynamical degree of freedom. This leads to characteristic trouble when trying to use canonical quantization and you either have to fix a gauge and then quantize canonically or you use the general formalism for Hamiltonian systems with constraints a la Dirac.
The upshot is that indeed a local gauge symmetry is not a symmetry as a global one. While the global symmetry before introducing the em. field of the free matter fields (usually Klein-Gordon or Dirac), i.e., the symmetry under multiplication with spacetime independent phase factors, yields the conservation law of a charge-like quantity in the "gauged version", i.e., after introduction of the electromagnetic field charge conservation follows from the Bianchi identity and not as an independent conservation law.
Another important conclusion, not correctly discussed in almost all textbooks on QFT (the one exception coming to my mind is Duncan, The conceptual framework of QFT), including my favorite Weinberg, is that local gauge symmetries cannot be spontaneously broken. Indeed, if you try to spontaneously break a local gauge theory you do not end up with a degenerate vacuum/ground state but with an equivalence class of different representations of the ground state connected by gauge transformations, i.e., the ground state is not degenerate as you might expect from the analogous case of a spontaneously broken global symmetry. In the latter case you get massless Goldstone modes as physical degrees of freedom, while for a local gauge symmetry indeed there are no such massless Goldstone modes, because the "would-be Goldstone fields" are just absorbed via a gauge transformation into the gauge fields, providing for the additional 3rd degree of freedom of a massive vector field and so providing a mass to the gauge fields (or some of the gauge fields depending on the pattern of the would-be symmetry breaking of the gauge group, as in the electroweak part of the standard model where you have four gauge fields based on the gauge group ##\mathrm{SU}(2) \times \mathrm{U}(1)##, which is "pseudo-broken" to ##\mathrm{U}(1)##, making three of the four gauge fields massive by eating up 3 would-be Goldstone modes into three of the gauge fields, which then get the W and Z boson fields and keeping one massless, describing the photon).
Another argument comes from the representation theory of the Poincare group. If you want to construct a microcausal theory of massless spin-1 fields you realize that it must be a gauge theory if you don't want continuous polarization degrees of freedom, which indeed is not observed in Nature.
What I understand Schwartz means with a "mere convenience" is that in perturbation theory you like to work with manifestly Dyson renormalizable models to calculate (a priori not physical!) proper vertex functions as building blocks to get S-matrix elements (physical!). It depends on a gauge whether the theory on the level of the proper vertex functions is manifestly renormalizable or not. E.g., if you try to work with physical field degrees of freedom for a Higgsed gauge theory, i.e., with the would-be Goldstone's absorbed and only having the corresponding massive gauge bosons the usual power counting doesn't work out anymore, because the gauge-field propgator is not of counting order ##-2## but of order ##0## (because of the contribution ##\propto k_{\mu} k_{\nu}/(m^2 k^2)## for a massive spin-1 field). However, you can choose another gauge ('t Hoofts ##R_{\xi}## gauge, where the gauge-field propagator indeed has the "right" power counting)). Then you have the would-be Goldstone fields left in the formalism, but they together with the usual Faddeev-Popov ghosts conspire to eliminate order by order in the loop/##\hbar## expansion of perturbation theory the also unphysical gauge-field degrees of freedom. The S-matrix is gauge invariant and the unitary gauge just is a limit of the ##R_{\xi}## gauges. This shows that the S-matrix depends only on physical degrees of freedom and is unitary and no causality/locality constraints are violated.
