Why do gauge theories win out?

[garrett]

Thanks Marcus. I've been happy to drop in on PF and provide a different point of view on why gauge theory is ubiquitous, and a bit on what I'm working on. I don't wish to monopolize the discussion though, and should get back to the work. So I'll let you get back to regularly scheduled programming, unless folks have questions.

 

[marcus]

Thanks Marcus. I've been happy to drop in on PF and provide a different point of view on why gauge theory is ubiquitous, and a bit on what I'm working on. I don't wish to monopolize the discussion though, and should get back to the work. So I'll let you get back to regularly scheduled programming, unless folks have questions.

You haven't monopolized! On contrary I think I haven't left you enough room. Add more to this expo at any time! But in any case let's sum up, for the time being, how your proposed new world picture answers the "Why gauge?" question.

I'll make a stab at it, and urge you to say it better. Let's try to make it succinct even if there is some oversimplification. Why do gauge theories always turn up?

Because the world is a very large Lie group G which is deformed, thought of as wavy. And this G has a particular subgroup H (the master gauge group) of co-dimension 4. (Maybe that's the wrong terminology, I mean that G/H can be represented by a submanifold of G of dimension 4.) So that partly at least explains why we see space-time as 4D---because G/H is 4D. On the negative side, it leaves quantum mechanical reality unexplained. (But that's hardly a serious failing since nobody seems to have a good explanation for why QM.) And on the positive side it strongly suggests why we are always encountering gauge theories---namely we LIVE in a big one with H as the master gauge group. And the clincher is this equation:

S = V_H \int_M ( \frac{1}{4} R*R + R * EE + EE*EE + T*T )

S comes from a compellingly natural and beautifully simple action (see post#48) and this equation says that when integrated over the whole G it turns out to equal the volume of the master gauge group H multiplied by an integral over the submanifold M of some interesting curvature etc terms recalling the Einstein-Hilbert action AND the cosmological constant!
So we could say that this compellingly natural *grand* action EXPLAINS the GR action with cosmo curvature constant (as well as affirming the gauge-ness of the world.)

Yes I know, this is awfully slap-dash--almost a parody, but I need to boil things down so I can remember the important pieces and get a take-home message. So if you have time, you might say it better, more truthfully, more succinctly. I think it's a fascinating idea.

 

[marcus]

We still have Strangerep's explanation of "Why gauge?" which I quoted a couple of posts back in post#60. Basically he says, I think, that there is one big theory and we separate out parts of it which we study in isolation. And he says that "gauge" is how separate parts talk to each other.

And he says more. He says there is no one unique RIGHT way of separating the whole music up into separate instruments. there is a lot of arbitrariness about how you divide out into various components. So there HAS to be this element of physical meaninglessness we habitually see in gauge.

And he observes that his picture is similar to the one in http://arxiv.org/abs/1308.5599 just turned around, going at the same thing from the other direction.

 

[RUTA]

Rovelli sums it up nicely at the end of his paper. Why gauge? Because the world is fundamentally relational. We say the same thing in http://users.etown.edu/s/stuckeym/FOP2013.pdf. This is a paper in progress for an IOP volume on quantum gravity; I'll be adding K on the hypercube for U(1) this week, but it has the same relational structure as the Dirac K, as you might imagine (K is the difference matrix on the graph a la lattice gauge theory). We deviate from Rovelli technically in a few ways, the most important being he's still viewing the world dynamically while our fundamental axiom is adynamical. We agree gauge is key, but it's not fundamental for us -- gauge is the result of 4D relational decomposition of the worldtubes of trans-temporal objects. Specifically, we construct the worldtubes of interacting trans-temporal objects -- properties represented by sources that can be trans-temporally identified through space in Lorentz invariant fashion -- using elements of "spacetimesource," so it's background independent. Essentially, we don't think of worldtubes *in* spacetime, but rather build worldtubes *and* spacetime together via elements of spacetimesource. So, why gauge? Because the (3+1)D dynamical world is built relationally from adynamical 4D "blocks" of spacetimesource and that relationalism results necessarily in K with a non-trivial null space, i.e., gauge. Divergence-free sources then reside in the column space of this 'relations matrix' K. So, gauge is key, but not fundamental per se.

 

[garrett]

That's pretty good, Marcus. One issue with it, though, is that for larger Lie groups than Spin(1,4) we don't get a 4D G/H. But we can specify a dS submanifold as a subgroup, or by identifying Spin(1,4) as a subgroup before modding by Spin(1,3) to get dS. Here's a shot at a succinct summary:

We see gauge fields everywhere because the universe is a large Lie group that is deforming over a four dimensional submanifold. This is described by a generalized Cartan connection,
C = \frac{1}{2}\omega + E + A + \psi
deforming from the Lie group's Maurer-Cartan form, with curvature and the Yang-Mills action integrated over the larger manifold being the action for the fields, particles, and interactions in the four dimensional world we see around us.

 

[arivero]

But no: the successful theories are always gauge theories. Why is that?

In the last years I have noticed that people is not telling anymore "local gauge" and "global gauge", they say simply "gauge" to refer both to "local gauge" and to classical gauge invariance, and then all the discussion to explain the role of the covariant derivative and the "minimal coupling" is being forgotten, at least here at the amateur/divulgative level.