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There is a question and corresponding downvoting of my answer, so I decided to ask this question.

There is my answer on it:

"...The most theories of free fields are invariant under global gauge transformations. It can be interpreted as instantaneous "rotation" of all space-time relatively gauge space, which contradicts the causality (of gauge transformations) in relativity case, because the speed of interactions is limited by $c$ according to it. So we introduce local gauge transformations, where "rotation" depends on space-time point. The gauge states is indistinguishable in a case of free field. But when our global gauge invariance field interacts with other fields, degeneracy is lifted, and this leads to instant changes in interaction. Thus we satisfy causality principle for gauge transformations.

Also I can add some example. Let's have global gauge isotopic invariance of QFT. So we identify proton and neutron as the state with some value of isotopic spin, and then if we choose that what to call a proton at one space-time point, we also must choose what to analogically call a proton at other points, so it little contradict the principle of local field theory. If the nucleon interacts with EM field, we may instantly change it state by using global gauge transformations, which also instantly changes the interaction between nucleon and EM field in all points of the space-time. It leads to non-local theory and contradict the causality.

Corresponding arguments were used by Yang and Mills in 1954...."

Where did I make the mistake (I really do not see the error)?

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I don't know what some of the statements in what you wrote mean... For example what is a "global gauge transformation?" If you mean a global symmetry transformation, then the statement that global symmetries are instanteous doesn't parse... It's a mathematical property of the structure of the theory that has consequences, such as conserved currents by Noether's theorem. But in classical field theory it's not that we ever imagine the field actually doing a global transformation at some point during its classical evolution. Field theories are local, there aren't instantaneous interactions. – Andrew Oct 15 '13 at 17:52
@Andrew . "...For example what is a "global gauge transformation?.." For example, for $U(1)$ symmetry it is $$ \psi \to \psi {'} = e^{i \alpha}\psi, \quad \alpha \neq \alpha (x). $$ "...Field theories are local, there aren't instantaneous interactions...", so we need to make the local gauge invariance theory, of course. And global gauge invariance theory isn't suitable in case of interacting fields. – user8817 Oct 15 '13 at 17:58
Surely a rotation is just using a different coordinate system? That isn't a gauge transformation. – John Rennie Oct 15 '13 at 18:06
@JohnRennie . I wrote about a rotation relative to the gauge space. In the space-time it looks as instantaneous change of one-particle states independently on the point of space-time. – user8817 Oct 15 '13 at 18:09
OK, one more try but then I am really done. Neither lepton nor baryon number are treated as local gauge symmetries in the standard model, even though the standard model has interactions. That's exactly the point I'm trying to make. If you are still confused or disagree about this point, try these notes by Willenbrock:, in particular look at the text above equation 30. I think you are confused by Yang/Mills original motivation. They wanted to make isospin local, but we now know isospin is not gauged; isospin is an approximate global symmetry in the SM. – Andrew Oct 15 '13 at 21:45

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