In this question, I'll define a gauge symmetry to be a symmetry that is merely a redundancy of our description of the system. Typically, gauge symmetries are also local symmetries, and a global symmetry can be made into a local gauge symmetry by introducing a covariant derivative and a gauge field.

But what about the case of turning a global symmetry into a global gauge symmetry? For example, the Schrodinger equation has a global $U(1)$ symmetry by multiplying the wavefunction by phases. This global symmetry is typically gauged, i.e. we regard vectors in the Hilbert space as physically equivalent if they differ by a phase.

However, I've never heard of other cases of this. Are there situations where such a global gauge symmetry would be useful, or is there some reason this isn't useful?

  • $\begingroup$ No time for an answer, but there are discrete gauge symmetries - i.e. redundancies which are globla in the snese that there is no locally varying transformation parameter. $\endgroup$ – Toffomat Feb 21 '18 at 17:41
  • $\begingroup$ Yes. Please see my answer for the quaetion physics.stackexchange.com/questions/377785/… dealing with various types of symmetries $\endgroup$ – David Bar Moshe Feb 22 '18 at 9:56

Discrete gauge symmetries are always global (in the sense that the "transformation parameter" is constant).

They can arise in string theory as remnants of contnuous gauge symmetries: A ten-dimensional string model can have a large gauge symmetry (e.g. $E_8\times E_8$ and local Lorentz transformations). When that is compactified to four diemnsions, the resulting theory will have an remaining gauge symmetry that can include, besides e.g. $SU(5)$ and $SO(1,3)$ a set of $\mathbb Z_n$ factors (see e.g. https://www.sciencedirect.com/science/article/pii/055032139290195H or https://www.sciencedirect.com/science/article/pii/0550321389905890 for early papers).

As for whether they are useful: The conventional wisdom is that gravity breaks all global symmetries (related to the fact that black holes hav no hair). However, this does not apply to gauged symmetries, so a gauged discrete subgroup of $U(1)_{B-L}$ could account for baryon numer conservation. (The relation to gravity also means it's more likely to come up in string theory. There probably are pure QFT uses as well, which I don't know.)

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Well, any kind of potential (vector or scalar), like the gravitational potential in Newtonian gravity or the EM potentials, has a global symmetry that you can add any constant and the physical system is left invariant.

Also, the CPT theorem says that the S-matrix for any any Lorentz-invariant QFT must be invariant under simultaneous time, space, and charge reversal. So any physical setup will remain unchanged under CPT, and you can think of that as a "gauge symmetry". This is very practically useful, because you can immediately rule out a whole lot of potential physical processes just on the grounds that they aren't CPT invariant.

And while this way beyond my pay grade, my vague understanding is that in string theory there is a discrete global $\mathbb{Z}_2$ symmetry (related to fermion parity, although it's more complicated than that) which is declared to be an unobservable gauge symmetry, so that any states not invariant under that symmetry operation are declared "unphysical". This is called the "GSO projection".

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  • $\begingroup$ Yeah, that’s the only other example I know of though! I’m just wondering if this is ever a useful general technique. $\endgroup$ – knzhou Feb 21 '18 at 18:26
  • $\begingroup$ @knzhou I added another example $\endgroup$ – tparker Feb 21 '18 at 18:32
  • $\begingroup$ Note that the $S$ matrix knows about physical states only, so gauge symmetries are invisible in that approach. $\endgroup$ – Toffomat Feb 22 '18 at 9:42

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