# Internal (gauge) symmetries and spacetime symmetries

1. Jul 19, 2012

### TrickyDicky

Internal symmetries of the SM -U(1), SU(2), SU(3)- are usually said to belong to abstract spaces unrelated to spacetime symmetries, have there been many attempts to relate internal symmetries to spacetime symmetries, and if so how far have they gotten?

2. Jul 19, 2012

### Bill_K

Are you thinking of the Coleman-Mandula Theorem?

3. Jul 19, 2012

### TrickyDicky

Hmm, I had never heard of it, but i just looked it up on wikipedia. After reading its description it seems to have so many limitations that it is hardly a theorem that anyone could take seriously.
I was thinking about really trying to come up with ways to bring closer the SM abtract internal symmetries to the actual spacetime symmetries.
Say for instance U(1) phase invariance can be integrated within the broader spherical symmetry, or maybe SU(2) being topologically a hypersphere could somehow be related to spherical symmetry too; I'm just making up the examples so someone might get the drift of what I'm referring to. I know the particular U(1) and SU(2) of the SM are not exactly the global ones but local gauges related to weak hypercharge and weak isospin.

4. Jul 19, 2012

### tom.stoer

The Coleman-Mandula theorm says that spacetime symmetries and internal symmetries cannot be combined into one larger symmetry structure (the only possibility is the direct product which so to speak trivial). But there is one loophole, namely 'fermionic charges'. So using SUSY one can indeed combine spacetime symmetries and internal symmetries. And when gauging SUSY (i.e. when it becomes a local symmetry) you get super-gravitation SUGRA

5. Jul 20, 2012

### TrickyDicky

Yes, I was aware of SUSY, SUGRA and the Kaluza-Klein theories, and indeed all these can be classified as serious attempts to do what I asked in the OP. But I see them as theories more concentrated on unification of already known theories and interactions into a larger framework like a TOE. Besides, so far they all seem to lack any empirical support.
I was rather thinking about a more purely geometrical approach, we know by the outcome of the high energy experiments that the SM internal symmetries are there, they have to somehow be related to the known properties of our spacetime or at least be manifestations of the spacetime symmetries in the sense that there might be an underlying pattern that unifies the internal symmetries and that we are interpreting them in a fragmented way.
I'm afraid that the success of the SM may end up locking it to any further possibility to be combined with gravitation and the observed spacetime symmetries.

Last edited: Jul 20, 2012
6. Jul 20, 2012

### tom.stoer

I know about the following approaches pointing towards something like thast, namely
- string theory (which would mean unification)
- SUGRA (which would mean unification, too)
- non-commutative geometry (???)

7. Jul 20, 2012

### TrickyDicky

Ok, I'll look up the non-commutative geoemtry approach, there is something in wikipedia. If I have questions I might take them to the BSM subforum.
Thanx

8. Jul 20, 2012

### strangerep

No, no, and no. The crucial point here is that gauge transformations are unphysical. Experiments verify predictions of the SM, i.e., that the SM Lagrangian is right in a vast number of ways. But the gauge freedoms are unphysical extra degrees of freedom in the Lagrangian. All observable results are gauge-independent.

In contrast, spacetime transformations of the Poincare group are physical -- we can physically perform rotations, translations, etc.

Once this distinction is understood, the Coleman-Mandula theorem becomes more comforting, and less of a nuisance.

9. Jul 21, 2012

### tom.stoer

There are of course formulations of GR (or its extensions like Einstein-Cartan) where the Lorentz- or the Poincare group are gauged, i.e. where Lorentz symmetry becomes a local gauge symmetry (most famous: Ashtekar variables in LQG). In that sense the language in which GR and gauge theories like the SM are formulated are rather similar. GR is so to speak a gauge theory with a local Lorentz symmetry + diffeomorphism invariance and therefore with a different dynamics. But the kinematical framework is rather similar in the above mentioned formulations: local gauge symmetry, connection variables, A- and E-fields, fiber bundles, Gauss law constraint as generator of local gauge transformations, elimination of unphysical degrees of freedom (even in GR you have to project out gauge d.o.f. to reduce the 10 components of the metric to the two physical polarizations). The big difference is that in GR the gauge symmetry is intertwinded with local diffeomorphism invariance whereas in gauge theories (like the SM) formulated on flat Minkowski spacetime this aspect is trivial.

10. Jul 21, 2012

### PhilDSP

Doesn't that depend on the topology of the problem? What you say should be true if there are no boundary conditions, for instance, for a free solitary particle, but not in general don't you think?

Another approach to the original question is Penrose and Rindler's Twistor formulation. Rather than dealing with with Minkowski space directly, a mapping is made to Twistor space which works more consistently and more flexibly with SU(n). Ward seems to be an excellent guide to that methodology.

Has anyone seriously persued the possibility that space-time symmetries are emergent in U(1) (as is classical electrodynamics), but that a richer set of space-time symmetries might exist with more primitive relationships in SU(n)? I suppose Kaluza-Klein theory qualifies in that regard possibly.

Last edited: Jul 21, 2012
11. Jul 21, 2012

### TrickyDicky

That distinction is understood alright, but your reply makes clear you are conflating gauge transformations with gauge symmetries, gauge symmetries (gauge invariance) are key to understand gauge theories and therefore are physically relevant. In my previous posts I only discussed symmetries, not the transformations themselves.
On the other hand a spacetime symmetry is what makes that, for instance in the case of rotational symmetry, there is nothing that physically distinguishes one point along the spherical rotation from another.
In the case of internal symmetries the transformation is performed in an abstract space, and only in that sense is "unphysical", so I was precisey highlighting the distinction you mentioned to learn of possible ways of understanding those abstract symmetries in terms of physical symmetries.

12. Jul 21, 2012

### TrickyDicky

U(1) is the circle group, how would spacetime symmetries other than the trivial SO(2) emerge from it?

13. Jul 21, 2012

### TrickyDicky

However many people don't accept this analogy, and don't consider GR as a gauge theory. Most of them mention the distinction between passive and active coordinate transformations in dynamic theories like GR.

14. Jul 21, 2012

### TrickyDicky

15. Jul 21, 2012

### tom.stoer

That's not what we mean.

The flavor symmetry is defined in an abstract space, it's an internal symmetry, but it's not unphysical!

The color-gauge symmetry SU(3) or the U(1) symmetry in electrodynamics are internal symmetries, too, but they are indeed unphysical.

The reason is that they can be gauge-fixed which goes hand in hand with reduction of unphysical do.f. Gauge fixing in QED is well known (it does not mean to break the invariance!) and it removes the unphysical d.o.f., i.e. it eliminates 2 unphysical photons, i.e. 2 physical transversal polarizations remain.

That is the reason why gauge symmetries are unphysical; they operate on unphysical d.o.f. (of the gauge fields).

16. Jul 22, 2012

### strangerep

Afaik, one typically assumes that the gauge transformations are trivial at (eg) spatial infinity, and at the temporal endpoints of the action integral (the rationale being that the field configurations at those endpoints are physical, hence must remain unaffected by gauge transformations).

The trouble I've always had with the Twistor formulation is that you're basically dealing with the representation theory of the conformal group, or rather, it's quad-cover SU(2,2). But the conformal Casimirs are very different from Poincare, and I've yet to see an attractive way to make them "play nice" together. But this must somehow be done in order to construct suitable asymptotic states and only then confront Coleman-Mandula.

17. Jul 22, 2012

### strangerep

?????

I'd always understood the term "gauge symmetry" to be associated with a group of "gauge transformations", and "gauge invariant" to be an adjective describing quantities which remain invariant under those transformations.

Apparently, you have some other definition of "gauge symmetry" in mind, but I have no idea what that might be.

18. Jul 22, 2012

### TrickyDicky

I think we're talking at cross-purposes here. Please, read again my post, I made a clear distinction between the symmetry groups and the individual transformations. The physical versus unphysical distinction in those posts was referred to the transformations, not to the symmetries.
I do understand that when you talk about unphysicality of gauge symmetries you are merely referring to the reduction of d.o.f one can perform by fixing gauges; for instance in the analogy you make with gauge symmetry in GR, we say coordinates are unphysical in that sense because general covariance makes sure the group of isometries acts as a gauge symmetry. However you wouldn't call the active isometry unphysical since it can be performed in spacetime like strangerep commented when he mentioned translations in spacetime as something physical.
Now in the case the SM gauge groups U(1)XSU(2)XSU(3) their symmetries apparently belong to some abstract space unrelated to our spacetime. Your example of flavor symmetry is not a gauge symmetry, it is the SU(3) flavor symmetry that is related to (has as subgroups) SU(2) (isospin) and U(1) (hypercharge), these are different from the above mentioned:weak isospin SU(2), weak hypercharge U(1) (that most current references refer to just as isospin and hypercharge which could lead to confusion if the cotext is not well fixed) and color charge SU(3). My discussion was restricted as can be seen in the title of the OP to the internal gauge symmetries, not internal symmetries in general.

Last edited: Jul 22, 2012
19. Jul 22, 2012

### TrickyDicky

Don't worry we have the same concept of gauge symmetry.

My point there was that you were (probably inadvertently) not making the distiction between them to reply my post since you quoted me talking about the gauge groups, not the individual transformations that make up the group; they are certainly intimately associated, but, a gauge transformation is not a gauge symmetry group. It is part of it. This really goes nowhere since we have already cleared up in posterior posts what everyone means by gauge symmetry and by "unphysical". And I think we all agree.

20. Jul 22, 2012

### PhilDSP

Interesting, I haven't looked at it in enough detail yet to see those types of limitations. Twistor theory doesn't seem to have retained the momentum it had 10 or 20 years ago in the literature, so it may have encountered impasses too difficult to sort out at the present time.

21. Jul 22, 2012

### PhilDSP

It's not so much that U(1) would produce the symmetries but rather that U(1) would be a reduction of a richer set (from SU(n) or some other algebra). In situations where the richer symmetries are broken you get a more simple set, though not necessarily a subset. In Kaluza-Klein theory, for example, mass is an additional dimension that is factored along with space and time to get a more complex set of symmetries than space-time symmetries. If you regard c as a constant in the equation $E = mc^2$ then neither time nor space enter as dimensions on the right hand side of the equation. That should indicate the presence of other symmetry relationships somehow underlying the equation.

Last edited: Jul 22, 2012
22. Jul 22, 2012

### TrickyDicky

Ah, ok, yes that was precisely the point of the OP, to find the possible underlying symmetries. But unlike the Kaluza case within our spacetime dimensions.

23. Jul 22, 2012

### strangerep

Why do you say that? I always thought the inputs to the CM thm were reasonable.

24. Jul 22, 2012

### strangerep

Well, then I still don't understand the point of the distinction you're trying to make (even after re-reading your earlier posts). A group is simply a set of elements, with certain multiplication rules defined thereon, sure. Perhaps you're trying to distinguish between an abstract group and concrete representations thereof?

25. Jul 23, 2012

### tom.stoer

I know. I wanted to make clear that your concept "In the case of internal symmetries the transformation is performed in an abstract space, and only in that sense is "unphysical", ... " of symmetries which are unphysical b/c they act in some abstract space is wrong. Flavor symmetry acts on an abstract space and is physical; gauge symmetries act on abstract spaces as well, and they are unphysical. The fact that there is an abstract space has nothing to do with the fact that they are unphysical (b/c there are physical symmetries like flavor acting on abstract spaces, too).

I don't really understand what you intention is.