Why do gauge theories win out?

[marcus]

If you want to understand this derivation, and see "why gauge theories?," then the first step is to understand what the Maurer-Cartan form is for a Lie group. Have you heard of that?

Yes--heard of the MC form. Looked it up though, just now:
https://en.wikipedia.org/wiki/Maurer–Cartan_form

Differential 1-form defined on the Lie group G with values in the Lie algebra, the tangent space at the identity e.
T_g(G) -> T_e(G)

 

[garrett]

That's right. Speaking more casually, the MC form is the natural $Lie(G)$ valued 1-form over a Lie group manifold that completely describes the local geometry of the Lie group. If we are working in a coordinatized patch, the MC form is $\Theta(x) = g^- d g$, (using the group inverse and the exterior derivative). If we treat this MC form as a connection, its curvature vanishes, $F = d \Theta + \frac{1}{2} [\Theta, \Theta] = 0$, which one can check from the definition.

To construct a Cartan geometry, we first consider some subgroup, $H \subset G$, and the homogeneous space, $G/H$. We can choose a representative section, $r(x)$, corresponding to this homogeneous space, as a submanifold of $G$. The MC form pulled back to this section, $r^- d r$, is called the Maurer-Cartan connection. It has two parts, the H-connection, valued in $Lie(H)$, and the frame, valued in the complement, $Lie(G/H)$. That frame IS the frame for the homogeneous space in the usual sense. Now, to let this become a Cartan geometry, we allow the MC connection to vary over what was, but is no longer, this homogeneous space. The curvature of this Cartan connection can now be nonzero, and the frame becomes an arbitrary frame, and the H-connection becomes the arbitrary connection for an H-principal bundle, which is what $G$ has become.

Does this make sense? And can you now see a hint of how all the fundamental geometric fields of physics will come from parts of the MC form of a Lie group gone wavy? To make it more concrete, for the physical case of GR along with a scalar field, we usually choose $G = Spin(1,4)$ and $H = Spin(1,3)$. The next thing to consider is how this can generalize to larger groups, while keeping the base four dimensional.

 

[marcus]

...to let this become a Cartan geometry, we allow the MC connection to vary over what was, but is no longer, this homogeneous space. The curvature of this Cartan connection can now be nonzero, and the frame becomes an arbitrary frame, and the H-connection becomes the arbitrary connection for an H-principal bundle, which is what $G$ has become.

Does this make sense? ...

For it to make sense I want to be able to look a simple example of where G becomes an H-principal bundle over a submanifold of G, that was originally a section corresponding to the homog. space G/H. Perhaps I should look up concrete matrix forms relating to Spin(1,4) and Spin(1,3) and try to convince myself (if indeed this is true and I've understood correctly what you said) that Spin(1,4)/Spin(1,3) is 4 dimensional. IOW something that could be made to correspond to a 4d submanifold of Spin(1,4).
There may be other people here who have been thru this type of example with those Spin groups. I have not been.

I probably need to start out at a more elementary level, with more basic matrix groups G and H. Can you suggest two matrix groups G and H where I would have an easier time going thru the analogous construction?

I like the idea of turning G into a principal bundle over a submanifold M of G which initially corresponded to the cosets G/H. And that the bundle G ->M has H as the bundle group.
And that the sub manifold M of G inherits the Maurer-Cartan form which gives it the chance to have diverse curvature: more or less free-form geometry as I picture it. That's neat.

I appreciate whatever explanation you are willing to give, Garrett, but owing to gaps in my knowledge I predict that I will not be able to follow satisfactorily.

Maybe the thing for you to do is lay it out, at whatever level of sophistication is convenient for you, and let anyone here who understands any part of the exposition ask questions. IOW don't let my individual level of understanding (particularly at the level of details) limit what you have to say.

I'm already getting some intuitive idea, even though I've never worked, for example, with the group Spin(1,4).
Spin(1,3) I gather is simply SU(2) the double cover of the Lorentz group, or something like that. But there will be others better versed in that stuff than I am. So why don't you at least give an outline of the idea even if it is over my head, and see what happens. Could be fun!

 

[fzero]

I probably need to start out at a more elementary level, with more basic matrix groups G and H. Can you suggest two matrix groups G and H where I would have an easier time going thru the analogous construction?

The Hopf fibration is a really beautiful example of a fibre bundle that is rather simple to describe, yet makes contact with a wealth of geometric and topological concepts. The simplest version is the description of $S^3=SU(2)$ as the total space of an $S^1=U(1)$ bundle over $S^2$. In terms of group theory, this connects with the identification of $S^2$ with the coset space $SU(2)/U(1)$. In terms of geometry, one can write an explicit metric on the total bundle whose curvature is the Chern class of a complex line bundle over $S^2 = \mathbb{CP}^1$. Many features seen in the Hopf fibration turn up later when studying things like monopoles, gravitational instantons or local metrics on resolutions of singular manifolds (typically hyperkahler or Calabi-Yau).

 

[kof9595995]

A really simple version of the argument I've heard is the following:

Since $[ A^\mu , A^\nu]$ ~ $\eta^{\mu\nu}$, then $\langle 0 |[ A^\mu , A^\nu] | 0 \rangle$ has some negative values. But $\langle 0 |[ A^\mu , A^\nu] | 0 \rangle$ is a probability, which can't be negative. Therefore, you need some kind of symmetry to make the negative probability modes unphysical.

This argument relies on two things: The first is Quantum mechanics, which is necessary for saying there is something wrong with the negative components. The second is assuming spin one particles exist, otherwise the above would likely imply that there shouldn't be spin one particles.

I would also say that first part, $[ A^\mu , A^\nu]$ ~ $\eta^{\mu\nu}$, depends on a lot of the usual assumptions physicists make without stating. I think I read this argument in Peskin&Schroeder chapter 4.

The same argument can be without any change applied to a massive vector field theory, but we know massive vector theory is not a gauge theory.

 

[tom.stoer]

Marcus, I haven't checked all posts, but perhaps the following is interesting.

1) For a global symmetry, e.g. SU(N), you find globally conserved charges with

$[H,Q^a] = 0$
$[Q^a,Q^b] = i\,f^{abc}\,Q^c$

Of course these charges act as generators of the symmetry transformation θ = unitary operators in the canonical formalism.

$U[\theta] = e^{i\,Q^a\,\theta^a}$

2) For a local symmetry you find locally conserved "charges" = Gauß law operators

$[H,G^a(x)] = 0$
$[G^a(x),G^b(y)] = i\,f^{abc}G^c(x)\,\delta(x-y)$

You can derive a family of globally conserved charges introducing test functions f(x)

$Q^a[f] = \int_{\mathbb{R}^3}d^3x\,G^a(x)\,f(x)$

Again these charges act as generators of the symmetry transformation θ(x)

$U[\theta] = e^{i \int_{\mathbb{R}^3}d^3x\,G^a(x)\,\theta^a(x)}$

which is exactly the gauge transformation (up to topologically non-trivial transformations). In contrast to global charges the Gauß law must annihilate physical states which reflects gauge symmetry in the physical subspace.

$Q^a[f]\,|\text{phys}\rangle = 0$
$U[\theta]|_{|\text{phys}\rangle} = \text{id}$

The fact that physical states are defined as the kernel of the charges Q or as invariant subspace w.r.t. to gauge symmetry U is key when distinguishing global and local symmetries. For global symmetries the singulet states are just special states, for local symmetries only these singulet states are physical. That means that we talk about an equivalence class of states where all states differing only by unphysical vectors orthogonal to physical vectors a identified. I am not sure whether this idea fully works out simply b/c global gauge fixing, Gribov copies and all that are far from trivial.

Looking at diffeomorphism invariance the same structure applies classically (*). You have local generators as functions on phase space, they are conserved = their Poisson brackets with H vanish (modulo pure gauge), they generate a local 'algebra' - not a Lie algebra - which is not required ...

The Gribov ambiguity issue (*) is much more complicated due to the complex mapping class group (something which is widely ignored in all QG approaches). This essentially means that the above mentioned construction for Q and U may break down even classically b/c the harmless sentence "up to topologically non-trivial transformations" becomes harmful.

Quantization is not understood at all. There is no global gauge fixing procedure known for which absence of gauge anomalies can be proven.

 

[trimok]

For Marcus, $Spin(4,1)/Spin(3,1)$ should be $dS4$, the de-Sitter space of dimension $4$

The gauge ambiguity comes certainly that we do not use the most fundamental objects. For instance, the ambiguity $A_\mu \to A_\mu + \partial_\mu \phi$, can be solved, for a simple topology (See Aharonov-Bohm effect), at looking at a (Wilson ?) loop on some path C: $\int_C A = \int_C A_\mu dx^\mu$

This seems to mean, that we had better to give up space-time, and looking at a loop space, which would be more fundamental.

 

[garrett]

Marcus, fzero's suggestion is a good one -- let's work with that. But, as warm up for $Spin(1,4)$ let's use some Lie algebra isomorphisms for our group and subgroup, $Lie(G)=su(2)=spin(3)$ and $Lie(H)=u(1)=spin(2)$. And, to keep the calculations concrete but more compact, we can work at the generator level, with the algebra spanned by three rotation generators, $\gamma_{12}$, $\gamma_{13}$, and $\gamma_{23}$, which we can also treat as the equivalent Clifford bivector elements, which you're welcome to write out matrix representatives for if you like. If we choose the $u(1)$ subgroup generated by $\gamma_{12}$, then a reasonable coset representative section of $Spin(3)$ is
$r(\theta,\phi) = e^{\phi \gamma_{13}} e^{\theta \gamma_{23}} = cos(\phi) sin(\theta) + sin(\phi) cos(\theta) \, \gamma_{13} + cos(\phi) sin(\theta) \, \gamma_{23} - sin(\phi) sin(\theta) \, \gamma_{12}$
It then takes a few lines of calculations to work out the MC form,
$r^- d r =\frac{1}{2} \omega + E$
producing the spin connection and frame for this homogeneous space,
$\omega = -2 \, d \phi \, sin(2 \theta) \, \gamma_{12}$
and
$E = d \theta \, \gamma_{23} + d \phi \, cos(2 \theta) \, \gamma_{13}$
which you can happily see describe a two dimensional sphere. Yes?

 

[marcus]

Good choice of simple example. Thanks.

 

[garrett]

Great, so now you can see how this works. This MC form is allowed to vary over what was the sphere, becoming the Cartan connection, having parts corresponding to the now arbitrary $U(1)$ connection and the frame. This nicely describes how gauge theory comes from the geometry of a Lie group that goes wavy. The next steps are showing how this works for $Spin(1,4)$ and larger groups, how the action functional that's an integral over the whole group separates and reduces to an integral over spacetime, how the Cartan connection generalizes to describe fermions, etc.

Nice, eh?

 

[marcus]

...The next steps are showing how this works for $Spin(1,4)$ and larger groups, how the action functional that's an integral over the whole group separates and reduces to an integral over spacetime,...

I'm imagining that in the more advanced example where G and H are Spin(1,4) and Spin(1,3) that G/H is something like 4d deSitter space (gone "wavy").
So now (if that is roughly right) as "proof of concept" you will be wanting to show that something akin to Einstein Hilbert action can be put on this "wavy" geometry specified by the wavy connection you are talking about. I have to go to supper shortly. Looking forward to seeing more about this. It quite possibly is neat. Have to go.

 

[marcus]

BTW just as a reminder discussion of that Rovelli paper is on hold for the time being and might go to its own separate thread. The idea defended in that paper is that gauge systems have "retractible handles for coupling" sort of analogous to an airplane's retractible landing gear, or a retractible robot arm, or an amphibious vehicle's retractible propeller.

What the gauge group collapses and then people carelessly refer to as "redundant" is actually the means by which a gauge-governed system can (when the need arises) couple to another system---which itself may be governed by its own gauge theory. The collapsible coupling gear serves a purpose---and this is one of the reason for the widespread success of this type of theory.

The discussion here does not have to do with the definition of what gauge theories are (we know that already, it has to do with a group of symmetries) but instead has to do with how we think about them. Just a reminder. Back to Garrett's exposition of his nice idea.

 

[garrett]

Remember that we're starting with the Lie group, then choosing a subgroup, which determines a homogeneous space, with a corresponding representative section of the Lie group. Then we look at the Maurer-Cartan form and let the Lie group go wavy, except for the subgroup, so we get a Cartan connection. If you like, you can consider the Maurer-Cartan form to be the vacuum state of the Cartan connection. So, yes, the Maurer-Cartan form for $Spin(1,4)/Spin(1,3)$ is precisely that of de Sitter spacetime. (In the same way that the Maurer-Cartan form for $Spin(3)/Spin(2)$ is that of a sphere.) Then we let this go wavy, described by a Cartan connection, which results in de Sitter spacetime with some excitations, which is what we appear to be living in, and, I suppose, as.

For the action... given our approach above, it would be wrong-headed to just specify an action over $G/H$. Instead, we should specify an action over $G$. The best action we could hope to start with would be
$S = \int_G F * F = \int_{G/H} \int_H F * F$
generalized to our partially wavy $G$. Now, the curvature of our Cartan connection, $C = \frac{1}{2}\omega + E$, is
$F = dC + CC = (\frac{1}{2} R + EE) + T$
in which $R = d \omega + \frac{1}{2} \omega \omega$ is the Riemann curvature and $T = d E + \frac{1}{2}\omega E + \frac{1}{2} E \omega$
is the torsion. The resulting integrand, $F * F$, is $H$ invariant, so, sure enough, the action integrates to an integral over our 4D manifold,
$S = V_H \int_M ( \frac{1}{4} R*R + R * EE + EE*EE + T*T )$
which does happily include the Einstein-Hilbert action term, a cosmological constant term, and some other stuff.

 

[Paulibus]

What the gauge group collapses and then people carelessly refer to as "redundant" is actually the means by which a gauge-governed system can (when the need arises) couple to another system---which itself may be governed by its own gauge theory. The collapsible coupling gear serves a purpose---and this is one of the reason for the widespread success of this type of theory.

This reminds me of a similarity I see between gauge variables and something much more familiar to us -- money. These abstract constructs have a purely relational, non-absolute character that affects the coupling between systems they 'govern': the gauge variable 'electrical potential' of a conductor has only a relative significance to an earth. Similarly, the Dollar that governs interactions in the U.S. commercial system has only value relative to what it can buy in the U.S. (or in other countries after an exchange-rate scaling). And one's bank balance has a relational significance that let's others label you rich or poor. Even gold has no absolute value!

I suggest that the reason why 'gauge theories win out' is that just as money lubricates commerce, so gauge variables lubricate calculations. They make it easier to 'shut up and calculate'. Boundary-value problems in electrostatics are conventionally solved using Poisson's or Laplace's equation, perhaps because using a gauge variable is convenient here. I suspect that convenience is often a factor in choosing the mathematical tools for decribing a physical situation. There are many ways to skin a cat.

 

[tom.stoer]

What I wanted to indicate by my post above is that a) gauge symmetry and ordinary global symmetry are closely related, where the key difference is that gauge symmetry has to be described by infinitly many generators and that b) systems with gauge symmetry reduce to ordinary qm systems where gauge symmetry and time evolution act orthogonal, i.e. gauge symmetry acts along unphysical directions. That means that a Lagrangian with gauge symmetry acts as a generator for infinitly many different ordinary theories which are equivalent physically.

I don't know (and I assume nobody knows for sure) whether this picture survives in quantum theories with diffeomorphism invariance.

 

[MTd2]

The Gribov ambiguity issue (*) is much more complicated due to the complex mapping class group (something which is widely ignored in all QG approaches). This essentially means that the above mentioned construction for Q and U may break down even classically b/c the harmless sentence "up to topologically non-trivial transformations" becomes harmful.

I am not sure about the Gribov ambiguity issue, but Torsten's approach is all about dealing with this "up to topologically non-trivial transformations" . So, it's not all QG approaches.

 

[MTd2]

@Garrett:

So, you plan to use that "spare space" to put together 3 generations of your mode as excitations on a DS space.

 

[marcus]

... The resulting integrand, $F * F$, is $H$ invariant, so, sure enough, the action integrates to an integral over our 4D manifold,
$S = V_H \int_M ( \frac{1}{4} R*R + R * EE + EE*EE + T*T )$
which does happily include the Einstein-Hilbert action term, a cosmological constant term, and some other stuff.

This seems like a remarkably skillful and efficient way to arrive at E-H action with cosmological constant!

 

[MTd2]

Section VI, cartan geometrodynamics, of the following paper. It talks a bit of what Garrett is talking about, but with a simpler group structure. This and the other papers also they also talk about an observer space.

http://arxiv.org/abs/1111.7195v4

Spontaneously broken Lorentz symmetry for Hamiltonian gravity
Steffen Gielen, Derek K. Wise
(Submitted on 30 Nov 2011 (v1), last revised 10 May 2012 (this version, v4))
In Ashtekar's Hamiltonian formulation of general relativity, and in loop quantum gravity, Lorentz covariance is a subtle issue that has been strongly debated. Maintaining manifest Lorentz covariance seems to require introducing either complex-valued fields, presenting a significant obstacle to quantization, or additional (usually second class) constraints whose solution renders the resulting phase space variables harder to interpret in a spacetime picture. After reviewing the sources of difficulty, we present a Lorentz covariant, real formulation in which second class constraints never arise. Rather than a foliation of spacetime, we use a gauge field y, interpreted as a field of observers, to break the SO(3,1) symmetry down to a subgroup SO(3)_y. This symmetry breaking plays a role analogous to that in MacDowell-Mansouri gravity, which is based on Cartan geometry, leading us to a picture of gravity as 'Cartan geometrodynamics.' We study both Lorentz gauge transformations and transformations of the observer field to show that the apparent breaking of SO(3,1) to SO(3) is not in conflict with Lorentz covariance.

http://arxiv.org/abs/1210.0019

Lifting General Relativity to Observer Space
Steffen Gielen, Derek K. Wise
(Submitted on 28 Sep 2012 (v1), last revised 4 May 2013 (this version, v3))
The `observer space' of a Lorentzian spacetime is the space of future-timelike unit tangent vectors. Using Cartan geometry, we first study the structure a given spacetime induces on its observer space, then use this to define abstract observer space geometries for which no underlying spacetime is assumed. We propose taking observer space as fundamental in general relativity, and prove integrability conditions under which spacetime can be reconstructed as a quotient of observer space. Additional field equations on observer space then descend to Einstein's equations on the reconstructed spacetime. We also consider the case where no such reconstruction is possible, and spacetime becomes an observer-dependent, relative concept. Finally, we discuss applications of observer space, including a geometric link between covariant and canonical approaches to gravity.

http://arxiv.org/abs/1304.5430

Extensions of Lorentzian spacetime geometry: From Finsler to Cartan and vice versa
Manuel Hohmann
(Submitted on 19 Apr 2013 (v1), last revised 27 Jun 2013 (this version, v2))
We briefly review two recently developed extensions of the Lorentzian geometry of spacetime and prove that they are in fact closely related. The first is the concept of observer space, which generalizes the space of Lorentzian observers, i.e., future unit timelike vectors, using Cartan geometry. The second is the concept of Finsler spacetimes, which generalizes the Lorentzian metric of general relativity to an observer-dependent Finsler metric. We show that every Finsler spacetime possesses a well-defined observer space that can naturally be equipped with a Cartan geometry. Conversely, we derive conditions under which a Cartan geometry on observer space gives rise to a Finsler spacetime. We further show that these two constructions complement each other. We finally apply our constructions to two gravity theories, MacDowell-Mansouri gravity on observer space and Finsler gravity, and translate their actions from one geometry to the other.The fist paper of the list cites the case of SO(4,1)/(SO(3,1)) as being studied by West and Stelle and another by mansouri. Those are behind paywalls, but I checked them, and they are not quite as explicit at that.

I also found this, about discrete representations of deformations of SO(3,1):

http://www.math.utah.edu/~kapovich/EPR/def.pdf

Curiously, it uses Dehn surgery on knots to find such classifications.

That is not very different with what Torsten uses, except that he does it directly on a generic 4 manifold, through the analog of Dehn surgery in 4 dimensions, Casson Handles, and then he identifies the gauge fields with the different types of torus on tangent space:

http://arxiv.org/pdf/1006.2230.pdf

The propagating solitons are identified with propagating exotic structures here:

http://arxiv.org/pdf/1303.1632.pdf

 

[garrett]

MTd2: Stranger than that. I now see spacetime as part of a larger Lie group that deforms, with the vacuum being dS. For three generations, there will be three different dS's in the Lie group.

Marcus: Yes, I was quite happy to see that action work out so well. And, of course, the $F*F$ YM action comes out as well when working with a larger Lie group. The trickier bit will be getting the action for fermions from something that looks like $D \psi * D \psi$. This new paper is a nice step in that direction though: http://arxiv.org/abs/1308.1278

 

[MTd2]

For three generations, there will be three different dS's in the Lie group.

Do you put them by hand?

 

[marcus]

MTd2: Stranger than that. I now see spacetime as part of a larger Lie group that deforms, with the vacuum being dS. For three generations, there will be three different dS's in the Lie group.

I'm puzzled. If you want to get gravity do you not need to have just ONE 4d submanifold contained in the group, wavy, but identified with G/H

 

[garrett]

MTd2, Marcus: Recall how this works for gauge theory. Different sections of the total space correspond to different gauge choices. These are what I meant by different dS's inside the Lie group. They're related by Lie group automorphisms. At least this is how I think it will work -- we're catching up to where I am with calculations. The main new insight in this model building has been to see that dS spacetime is a submanifold (actually, a subgroup) of larger groups, rather than just a fiber bundle base. So we get to start with a large Lie group and let it go wavy, to get everything we want.

 

[marcus]

...The main new insight in this model building has been to see that dS spacetime is a submanifold (actually, a subgroup) of larger groups, rather than just a fiber bundle base...

I don't want to bug you with questions when the problem is really with my insufficient grasp (not with your ability to explain.) I look forward to seeing your article or web-posting on this. Hope it includes some introductory material and graphics as well as technical exposition.

It's not clear that my feedback, in particular, would help you progress towards this goal. But at least for now I will tell you.

I'm kind of locked in on the picture that has a "fiber bundle base". I can't visualize how you could ever get away from having a PREFERRED section which carries information about the 4d geometry and therefore about CAUSALITY.

BTW I imagine, when you say that G/H is actually a subgroup, that it must correspond somehow to "translations" in dS space. This simplest kind of subgroup, just translations. I'm near the brink of confusion and not sure this makes sense.

I fantasize that there is somehow one G/H section that is, in fact, preferred and carries causality information, but you have not mentioned this, so maybe there isn't! Maybe the causal structure somehow infuses the entire big wavy group. That then would be a radically different ONTOLOGY which would of course cause every conservative cell in my body to rise up in horror and indignation.

In any case, good luck with carrying through with this project and getting to the primary exposition stage, so you can spring it on the actual experts. It's clearly a really interesting new approach.

 

[marcus]

Just kidding in post#59 about extreme ontological conservatism--it might be really interesting to get a different grip on causality that did not depend on having one preferred 4d submanifold (or bundle base) where it emerges. Perhaps having it more diffuse. There's also the comparable idea of a Tomita-flow-induced state dependent "foliation" of the observable *algebra and the remote chance of a thematic connection.
I haven't started a separate thread for discussion of "Why Gauge?" and comment on that paper. How does it work to keep both topics in this one thread?

If anyone would prefer to have things sorted out more, please say.I'm hoping Garrett will continue "full steam ahead" expounding this G/H unification idea and we can comment both on it and on the question of why gauge-type theories are so successful and ubiquitous. If anybody (esp. Garrett) finds this awkward, inconvenient, impeding in any way please do communicate that either by post or PM.

On the "why gauge?" topic, I want to note what may have been (one of?) the most thoughtful/informative comment(s) so far, strangerep's post #26:

Just finished a first reading of Rovelli's paper...

I've always been uncomfortable with the way diffeos are described as GR gauge transformations. Certainly they constitute a symmetry of the theory, but... if a theory has a Lagrangian that's invariant under ordinary 3D rotations, we don't call that SO(3) group a gauge symmetry, but rather a geometric symmetry.

OTOH, "gauge groups" like the usual U(1) in EM arise because there's a freedom in how many ways one may decompose a composite field (electrons+photons) into an "electron field" and a "photon field" distinct from each other. Dirac gave an alternate way of decomposing the composite field such that the "new" electron field is the original electron field dressed with a coherent photon field. The new field is invariant under the U(1) transformations, unlike the original electron field. And there's definitely something important in this, since it (a) yields the correct Coulomb field for an electron, and (b) all the usual IR divergences in QED scattering are banished to all orders if we adopt the dressed field as the "physical" asymptotic states.

Come to think of it, I don't what the maximal dynamical group is for the equations of (full) classical electrodynamics, or even whether anyone has actually computed it. Dynamical groups tend to be larger than the obvious geometric symmetry groups. E.g., for the Kepler problem (and Schrodinger equation for hydrogen) we find SO(4,2).

But getting back to Rovelli's paper, the proposition (iiuc) that gauge features are associated exclusively with coupling seems reasonable -- though I'd turn it around and say those features arise because of how we attempt to decompose a nontrivial system (field) into simpler component fields.

 

[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.