In most physics textbooks, local gauge invariance is simply postulated---you start with a global symmetry, e.g. the global phase, then allow it to depend on the spacetime point, make the necessary adjustments to the derivative (i.e. introduce the gauge field) in order to preserve the invariance, and out fall, say, Maxwell's equation. If this procedure is justified at all, it's mostly due to its successes (which are certainly quite impressive).

However, occasionally one meets arguments that seem to allege that such a procedure is, in some sense, necessary due to the requirement of locality in field theory. The probably earliest such argument is in the original paper by Yang and Mills, where they write:

This means that ... the orientation of the isotopic spin is of no physical significance. The differentiation between a neutron and a proton is then a purely arbitrary process. As usually conceived, however, this arbitrariness is subject to the following limitation: once one chooses what to call a proton, what a neutron, at one space-time point, one is then not free to make any choices at other space-time points. It seems that this is not consistent with the localized field concept that underlies the usual physical theories.

Similar arguments are found elsewhere, e.g. David Gross in Conceptual Foundations of Quantum Field Theory, p. 58:

In the standard model, non-Abelian gauge symmetry dictates the electro weak and strong forces. Today we believe that global symmetries are unnatural. They smell of action at a distance. We now suspect that all fundamental symmetries are local gauge symmetries. Global symmetries are either broken, or approximate, or they are the remnants of spontaneously broken local symmetries.

Or Sunny Auyang, in How is Quantum Field Theory Possible?, p.55:

If we have chosen to designate a certain state as proton at one spatio-temporal point, we are not free to designate another state as proton elsewhere. The global convention requires that all field operators share a common state space. It violates the spirit of local field theories, in which descriptions concentrate on a point and its infinitesimal vicinity. The relaxation of the global requirement is the starting point of gauge field theories.

However, these are obviously somewhat heuristic formulations. My question now is, is there a way to make them more precise? I.e. is there actually some rigorous sense in which global symmetries are inconsistent with local field theory? Do we get some 'action at a distance', or some other conflict with the principles of spatiotemporal locality?

Or is this just to build some intuition, to provide a kinda-sorta justification for letting the symmetry transformation depend on the spacetime point?


1 Answer 1


No, global symmetries are not at odds with local field theory, since the global transformations are just gauge transformations that are constant in space and time, and are thus naturally a subset of the gauge transformations. Gauge symmetries thus include a global symmetry.

All these quotes intend to say is to provide a heuristic for why the relaxation from a global symmetry to a local (gauge) symmetry seems natural from the field theoretic viewpoint.

  • $\begingroup$ I'm certain OP is aware of the first point and is just asking if a global-only field theories must be ruled out for describing physics if locality principles are assumed. $\endgroup$
    – Nikolaj-K
    Oct 1, 2015 at 12:09
  • $\begingroup$ @NikolajK: Not sure if that's the question, because there's the obvious counterexample: All of the usual special relativistic QFTs, which have a global, but not local, Lorentz symmetry, but are perfectly consistent local theories. $\endgroup$
    – ACuriousMind
    Oct 1, 2015 at 12:17
  • $\begingroup$ Well, in a sense the question exactly is if the global Lorentz symmetry is somehow in tension with purely global symmetries, e.g. phase transformations. This, to me, seems to be what the quotes want to imply, but I don't immediately see why that should be the case. $\endgroup$
    – Jochen
    Oct 1, 2015 at 12:29
  • $\begingroup$ @ACuriousMind: Maybe that's the answer for him, then. $\endgroup$
    – Nikolaj-K
    Oct 1, 2015 at 17:40

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