Why do gauge theories win out?

[marcus]

"Gauge" is normally understood as "mathematically redundant"

Why are gauge theories so prevalent? Why do they always seem to win out in the contest to describe the world?

In a theory with some group of gauge symmetries, only the gauge-invariant quantities are considered physically meaningful.
But the mathematics that people find most convenient or true to nature allows for other, gauge-dependent quantities and these are considered physically insignificant redundancy.
So why not refine the math so as to eliminate all that meaningless "filler"? Isn't it inefficient, just "extra baggage" diluting the real physical content of the theory?

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

 

[trimok]

The problem is, considering massless gauge bosons in 3+1 dimensions, A_\mu has 4 components, and there are only 2 physical degrees of freedom. The little group ISO(2) does not transform A_\mu like a Lorentz vector (because of the "translation" part), so one has to connect A_\mu to a conserved current j^\mu, and the fermionic origin of j^\mu explains the so-called gauge-invariance, which is in fact just a mathematical redundancy.

For massless particles, helicity formalism (p_{\alpha \dot \alpha}=\lambda_\alpha \tilde \lambda_ \dot \alpha) is a first step (no need of a null constraint for p^2). Unfortunately, we have always the U(1)ambiguity : \lambda_\alpha \to e^{i\theta}\lambda_\alpha, \tilde \lambda_\dot\alpha \to e^{-i\theta}\tilde\lambda_\dot\alpha.

I think that the work made by Nima Arkani-Hamed and others on gluons amplitudes (grasmannian, twistors, etc...), is very interesting. Certainly, it means that space-time has to be redesigned.

 

[tiny-tim]

hi marcus!

isn't it emmy noether's fault?

isn't a gauge symmetry a geometrical symmetry just like any other, and so must have an associated physical "current"?

 

[marcus]

hi marcus!

isn't it emmy noether's fault?

isn't a gauge symmetry a geometrical symmetry just like any other, and so must have an associated physical "current"?

Hi tim! Noether's theorem about how conserved quantities relate to global symmetries is beautiful. At the moment I cannot relate it to the diffeomorphism group of GR, which is something that relativists consider a gauge group.

So that is an opportunity to stretch conceptually and broaden the discussion. In GR what they call a spacetime geometry is not just one metric tensor g on a given manifold. It is all the other metrics that g and accompanying matter could be mapped to by smooth maps (diffeomorphisms).

So what relativists call a geometry is an equivalence class [g] under diffeos, of all the metrics that are equivalent to g by smooth mappings.

The distinction between two metrics that only diffeo-differ (say merely by a smooth CHANGE OF COORDINATES) is not considered to be physically significant. It is just "gauge". It doesn't count.
That is the essence of Einstein's principle of General Covariance and it kind of means that points of space and time do not have absolute meaning or objective reality, all that matters is the relationships among things.

You know all this and I think it's fairly common knowledge. The reason I repeat it is to make the point that GENERAL RELATIVITY IS ALSO A GAUGE THEORY.
It is just that the gauge group isn't linear transformations like in flat Minkowski space, the gauge group is mooshy morphy nonlinear transformations.

So adding GR to the list, along with Maxwell Electrodynamics equation, and YangMill, or what Trimok was discussing, seems to suggest that ALL successful contemporary theories are gauge theories. Which if that's right does ask for an explanation.

 

[DimReg]

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.

 

[marcus]

Hi DimReg, the question I'm asking is why does it seem that all successful theories--the ones that have won out as reasonably fundamental/accurate descriptions of the world--seem to be gauge theories?

In the sense that General Relativity is a gauge theory, and Electrodynamics (say classical version with U(1) gauge symmetry), and other examples which you can give from particle theory. Is that a fair impression? If so, can you suggest some general reason for the overall prevalence?

 

[julian]

"Gauge" is normally understood as "mathematically redundant"

Why are gauge theories so prevalent? Why do they always seem to win out in the contest to describe the world?
?

When was the last time a new gauge principle was experimentally verified? What happened to GUTS? Supersymmetry?? Higgs boson may have been discovered recently but that goes back to theory how many decades ago??

 

[atyy]

In gauge theories, the "true" objects are loopy, but we usually like the non-loopy description, which is redundant.

So why are the true objects loopy? Because of string theory:) Alternatively, the true objects are local, but they interact in such a way that the low energy degrees of freedom are loopy, like spin ice which has an emergent gauge field. http://arxiv.org/abs/1112.3793

 

[Pythagorean]

"Gauge" is normally understood as "mathematically redundant"

Why are gauge theories so prevalent? Why do they always seem to win out in the contest to describe the world?

In a theory with some group of gauge symmetries, only the gauge-invariant quantities are considered physically meaningful.
But the mathematics that people find most convenient or true to nature allows for other, gauge-dependent quantities and these are considered physically insignificant redundancy.
So why not refine the math so as to eliminate all that meaningless "filler"? Isn't it inefficient, just "extra baggage" diluting the real physical content of the theory?

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

Is the other stuff actually filler though? My limited understanding is that only the Lagrangian is invariant. The specific geometry will be frame-dependent (where a particular particle is at a particular time).

 

[julian]

What emmy noether's found was that if you do the same experiment at the same time but at a different position but the dynamics was the same this implies via the Lagrangian that momentum is conserved, did the same experiment at the same place but at a different time and the dynamics was the same that the energy is conserved.

How does this compare to gauge symmetries which are when the `state' can be changed without changing the `physical state' (mathematical redundancy)? You can move your apparatus from one place to another but is there an actual physical process by which you can make an U(1) transformation? Well maybe the Aharonov–Bohm effect, but aside from that.

 

[S.Daedalus]

This is inspired by Rovelli's recent paper, I presume? I haven't yet had time to digest it, but I usually find these more conceptual papers of his very enjoyable (was a big fan of his discussion of entropy in his recent FQXi contest entry).

 

[julian]

In gauge theories, the "true" objects are loopy, but we usually like the non-loopy description, which is redundant.

So why are the true objects loopy? Because of string theory:) Alternatively, the true objects are local, but they interact in such a way that the low energy degrees of freedom are loopy, like spin ice which has an emergent gauge field. http://arxiv.org/abs/1112.3793

In non-general covariant theories the loopy description is a discrete approximation, in GR the loopy description is actually exact and well-defined, maybe describing truncated degrees of freedom though, precisely because of the non-local nature of diffeomorphisms - making it mathematically rigorous/finite. I think the non-loopy description is an approximation in the low energy limit - and relates to the r-Fock description type issues.

 

[DimReg]

Hi DimReg, the question I'm asking is why does it seem that all successful theories--the ones that have won out as reasonably fundamental/accurate descriptions of the world--seem to be gauge theories?

In the sense that General Relativity is a gauge theory, and Electrodynamics (say classical version with U(1) gauge symmetry), and other examples which you can give from particle theory. Is that a fair impression? If so, can you suggest some general reason for the overall prevalence?

I do think I answered your question to some extent though. Once you have lorentz invariance, quantum mechanics, and spin one particles, you're forced to have a gauge symmetry. Given that these are ingredients that we have a high degree of faith must be fundamental, it leads us to believe that the only promising fundamental theories also have these features, hence gauge theories are also the most successful theories we know of.

If you agree with the argument I gave in my previous post, then the answer to your question would just be that spin one particles exist therefore gauge theories are necessary. (Off the top of my head, theories that have lorentz symmetry breaking could get around this, but those don't tend to do very well experimentally)

General Relativity is a different story, since it doesn't require spin one particles. However it's symmetries are more inevitable: the gauge group of the standard model seems much more arbitrary than the "gauge group" of GR. For example, local lorentz invariance is a result of formulating GR as the theory of the metric on a semi-riemannian manifold -> gauging the lorentz group is something you would do even if you had never heard of gauge theory. Gauging SU(3) X SU(2) X U(1) on the other hand... For this reason, I'm hesitant to think that the gauge elements of GR and the gauge elements of the SM are of the same origin.

My overall point is that gauge redundancies fit pretty well into things we know to be true. In my opinion, understanding "why gauge theories" would likely also include understanding "why quantum mechanics", which hopefully you agree would be biting off more than one can chew. I think there are some opinions of experts along the lines of "gauge symmetries show that we don't use the right degrees of freedom", but I don't know who those people might be.

 

[marcus]

This is inspired by Rovelli's recent paper, I presume? I haven't yet had time to digest it, but I usually find these more conceptual papers of his very enjoyable (was a big fan of his discussion of entropy in his recent FQXi contest entry).

Exactly! I've been trying to understand his reasoning. It seems to be not about theories of separate phenomena in a stand-alone mode, but about how theories of separate phenomena couple together.

When we couple a particle to geometry we don't couple it to the geometry itself, the equivalence class. We pick a representative of that class, a particular metric g on a particular manifold with particular coordinates. And we couple the particle to THAT with all its arbitrary and physically meaningless detail.

 

[marcus]

Hi julian, DimReg, you have some good points. There's more than I can respond to right now. I'm also trying to read the august paper .5599 which is about this question of why gauge theories should be so ubiquitous. http://arxiv.org/abs/1308.5599.

Note that when gr is considered as a gauge theory the gauge group is the diffeomorphisms which can really stir things around a lot on the manifold and even map from one manifold to another. they are not merely "nonlinear changes of coordinates" and not just local Lorentz symmetries.

If you have two manifolds and a diffeo mapping between them and on one you have a solution of GR equation consisting of a metric g and a certain distribution of matter flowing around, then if you map all that over to the other manifold you get another metric and matter flow that is ALSO A SOLUTION of GR equation. Even if you imagine them looking quite different the two solutions represent the same reality. So that's general covariance---reality doesn't have a preferred structured layout---only relationships of stuff&geometry relative to stuff&geometry, and you can moosh it around as long as you moosh both the distance function and the matter consistently in the same manner. this may be a bad verbal translation, it's hard to express verbally (for me at least.)

 

[atyy]

I do think I answered your question to some extent though. Once you have lorentz invariance, quantum mechanics, and spin one particles, you're forced to have a gauge symmetry. Given that these are ingredients that we have a high degree of faith must be fundamental, it leads us to believe that the only promising fundamental theories also have these features, hence gauge theories are also the most successful theories we know of.

If you agree with the argument I gave in my previous post, then the answer to your question would just be that spin one particles exist therefore gauge theories are necessary. (Off the top of my head, theories that have lorentz symmetry breaking could get around this, but those don't tend to do very well experimentally)

General Relativity is a different story, since it doesn't require spin one particles. However it's symmetries are more inevitable: the gauge group of the standard model seems much more arbitrary than the "gauge group" of GR. For example, local lorentz invariance is a result of formulating GR as the theory of the metric on a semi-riemannian manifold -> gauging the lorentz group is something you would do even if you had never heard of gauge theory. Gauging SU(3) X SU(2) X U(1) on the other hand... For this reason, I'm hesitant to think that the gauge elements of GR and the gauge elements of the SM are of the same origin.

My overall point is that gauge redundancies fit pretty well into things we know to be true. In my opinion, understanding "why gauge theories" would likely also include understanding "why quantum mechanics", which hopefully you agree would be biting off more than one can chew. I think there are some opinions of experts along the lines of "gauge symmetries show that we don't use the right degrees of freedom", but I don't know who those people might be.

I think you and trimok gave the textbook answers. Does it not extend to spin 2? The usual thing I've heard is that the special relativistic quantum theory of massless spin 2 is uniquely GR, automatically including the EP and general covariance, whereas the EP plus general covariance does not imply GR. Something like in Hinterbichler's review http://arxiv.org/abs/1105.3735.

In non-general covariant theories the loopy description is a discrete approximation, in GR the loopy description is actually exact and well-defined, maybe describing truncated degrees of freedom though, precisely because of the non-local nature of diffeomorphisms - making it mathematically rigorous/finite. I think the non-loopy description is an approximation in the low energy limit - and relates to the r-Fock description type issues.

Maybe we can consider it inexact in GR also, since GR seems non-renormalizable - unless LQG or asymptotic safety work?

 

[marcus]

This is inspired by Rovelli's recent paper, I presume? ...

Exactly! I've been trying to understand his reasoning. It seems to be not about theories of separate phenomena in a stand-alone mode, but about how theories of separate phenomena couple together...

Here's a key passage, from page 5. He distinguishes between gauge invariants on the one hand and gauge variables on the other. The former are observables in Dirac's sense of being predictable within the theory, and having a definite numerical value regardless of how we act on the system with the gauge symmetries. The latter, gauge variables are exactly that, their values depend, so to speak, on the circumstances (perhaps the presence of some other system brought into contact with the one under study.) There is an element of arbitrariness and variability not governed by the system itself--so the gauge variable is not a Dirac observable.
==quote http://arxiv.org/abs/1308.5599 ==
That is, we can “measure” a gauge variable (below are several examples). But this number cannot be predicted by the dynamical equations of the system S. Quantities that can be measured but not predicted were denoted “partial observables” in [13].
The conclusion that gauge-variables can be measured does not contradict Dirac’s original argument according to which considering a gauge variable as “observable” leads to indeterminism. In fact, it confirms it. Dirac used “observable” in the sense of predictable quantity. We can turn Dirac’s argument around: every time that the physics is such that a certain measurable quantity entering the description of a system is unpredictable in terms of the system’s dynamics alone (like the time, or the position of a single starship), it must be described in the formalism by a gauge variable. This is what gauge variables are.
==endquote==

So if this is correct there must be gauge variables in physics (at least as long as physics must deal with two or more separate systems which, in some instances, need to be hooked together).
A very simple example, or "parable", of this is given in the paper. It is of a line of spaceships which as far as we know is alone in the universe. Location-wise, all that matters are the, say, 9 numbers x_i between the 10 spaceships. These suffice to tell the ships' locations (anything more would be redundant or meaningless). A ship's location is only known relative to the other ships because AFAWK there is nothing else in the universe. So that's like one selfcontained physical theory. Then another fleet appears with its separation distances y_j. In order to relate the two we need one addtional datum, which might the distance between x₁₀ and y₁. Measurable, but not predictable within either of the two separate theories.

 

[DimReg]

I think you and trimok gave the textbook answers. Does it not extend to spin 2? The usual thing I've heard is that the special relativistic quantum theory of massless spin 2 is uniquely GR, automatically including the EP and general covariance, whereas the EP plus general covariance does not imply GR. Something like in Hinterbichler's review http://arxiv.org/abs/1105.3735.
Maybe we can consider it inexact in GR also, since GR seems non-renormalizable - unless LQG or asymptotic safety work?

I definitely gave the textbook answer, as I pointed out I read it in a textbook! I can't claim to have done too much original thinking...

I don't think I agree with that review's characterization of the way Einstein derived GR, but I think his statements about spin 2 systems are correct.

Edit: Rovelli's paper is more interesting than my regurgitation of textbook explanations of gauge theory, I now wish I had read that paper before posting...

 

[atyy]

But is Rovelli right? His discussion is entirely classical. Usually we say that classical electromagnetism can be completely described by using the E and B fields which are "real", and we don't need the vector potential which is "gauge". Do classical E, B fields and charge count as "gauge" in Rovelli's language?

How about a Turing machine? What is "gauge" in a Turing machine?

 

[marcus]

I definitely gave the textbook answer, as I pointed out I read it in a textbook! I can't claim to have done too much original thinking...

I don't think I agree with that review's characterization of the way Einstein derived GR, but I think his statements about spin 2 systems are correct.

Edit: Rovelli's paper is more interesting than my regurgitation of textbook explanations of gauge theory, I now wish I had read that paper before posting...

No harm done! Your post contributed a lot to our understanding of one sector of the problem. I hope you comment on the recent paper as well!

But is Rovelli right? ...

Well, he's an astute dude. Also these are fairly elementary matters. The electrodynamics vector potential is indeed "gauge", but look at equation (1) where interaction with a fermion is written using A. It seems to follow the pattern he's indicating.

He's starting off by giving two classic examples: classical GR and classical ED. (Don't see the point of asking if E, B, and charge are "gauge". Classically I see no indication that anybody thinks they are gauge, E and B are what he calls the field F.)

EDIT: I see I referred to the Maxwell potential as "vector potential"(which could be misunderstood). It's a four-vector and also sometimes called a "four-potential". To reduce chance of confusion I'll here quote from the paper and try to use its terminology consistently.
==quote 1308.5599 page 1==
Electromagnetism can be expressed as a U(1) gauge theory for the Maxwell potential A. The gauge-invariant content of the theory, whose evolution is predicted by the Maxwell equations, is captured by the electromagnetic field F = dA. This leads us to say that what exists in Nature is only the gauge invariant content of A. However, we describe the coupling of the electromagnetic field with a fermion field ψ in terms of the interaction lagrangian density...
==endquote==

EDIT FOR ATYY: Rovelli does not claim that gauge theory is necessary in any universe with at least two interacting objects. You must have misread. He gives an example of a universe with many objects described by a theory with no gauge component. Don't understand your saying "My skepticism is about the claim that gauge is necessary in all universes with at least two interacting objects"

 

[atyy]

Well, he's an astute dude. Also these are fairly elementary matters. The electrodynamics vector potential is indeed "gauge", but look at equation (1) where interaction with a fermion is written using A. It seems to follow the pattern he's indicating.

He's starting off by giving two classic examples: classical GR and classical ED. (Don't see the point of asking if E, B, and charge are "gauge". Classically I see no indication that anybody thinks they are gauge, E and B are what he calls the field F.)

If gauge is by definition a redundancy of our description, then his argument is wrong by definition. So he must mean something other than gauge. My skepticism is about the claim that gauge is necessary in all universes with at least two interacting objects. I do agree with the claim that all observables are relational.

 

[garrett]

Maybe gauge theory has been so successful because our universe is a deforming Lie group.

 

[Paulibus]

Gauge interactions describe the world because Nature
is described by relative quantities that refer to more than
one object. In a sense, this is a step along the direction
devised by Galileo, when he stressed that velocity is a
quantity that does not refer to a single object, but to
two objects: the velocity of an object is only defined in
relation to another object.
Gauge is ubiquitous. It is not unphysical redundancy
of our mathematics. It reveals the relational structure of
our world.

Nice to have this question resolved. I've always been puzzled by the significance of electrical potential - it's not there just to make life easier for physicists like Maxwell! Of course the smart birds that perch on power lines with impunity have known about this all along.

 

[S.Daedalus]

How about a Turing machine? What is "gauge" in a Turing machine?

The Turing machine! (If it's computationally universal.) For any (universal) Turing machine T executing a program p, there exists for every Turing machine T' a program sp such that T' executing sp is equivalent to T executing p (if all else fails, by sp being a program that directly simulates T executing p on T'). So any given computation is really an equivalence class of programs on different Turing machines giving rise to the same 'phenomenology'.

(I'll try and read the rest of the thread later, got to get to work now...)

 

[marcus]

Paulibus, good to hear from you! Yes, about the birds on the power line Maybe you can get some other pertinent quotes.
Yes it is a nice paper! I want to make the fairly obvious point that it uses the standard definition of what a gauge theory IS and focuses on how we *understand* it. So the aim is to improve our understanding of what such theories do and why we have them.

Correct me if I'm mistaken but the standard definition involves specifying a group--transformations of some sort--that leave the dynamics unchanged. Say the Lagrangian is unchanged, or the main equation of the theory.

So the DEFINITION depends on specifying a gauge group. "Gauge" is not "mathematical redundancy by definition". We are talking about a specific kind of redundancy described by a group of symmetries.

Now once the concept is DEFINED people can propose ways to understand it.
according to one opinion it merely represents redundancy, applying a gauge transformation (an element of the group) makes no physical difference. For those people "gauge" represents nothing about the world, it represents physical meaninglessness. That's not the DEFINITION that's the way some people LOOK at it.

And the paper is suggesting to us to look at gauge symmetry is a somewhat different way---suggesting something about the world. Not mere meaninglessness. Examples are given where what is redundant in a closed isolated system serves as a HANDLE which enables coupling two systems together, when the need arises.

The metaphor of the handle comes up several times, and several examples of this are given.
It's bedtime here so I'll sign off. I'll find some quotes to illustrate this tomorrow.

 

[strangerep]

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.

 

[Paulibus]

The perspective from which Rovelli's paper is written is manifestly relativistic, and he ends with the conclusion that the structure of our world is relational. Should we therefore perceive, describe and understood our world only from this relativistic perspective? I feel uneasy here, especially as there is no discussion of the other side of this coin; the absolute. This word is used by Rovelli only once (l.7,p.3).

Yet the world contains both relational and absolute stuff, stuff that just is (whatever that means; for those who prefer it, given). For example in physics c and h seem to me absolute and without any relational flavour, as do prime numbers in mathematics. And Nature's fundamental entities, like electrons, qua electrons, seem so far better to fit in the category of absolute stuff. Is Rovelli discussing in a deep and interesting but tautological way only 'interactions', that are perforce relational?

 

[Physics Monkey]

It may be worth observing that every theory can be written in "gauge" variables, i.e. redundant variables. At the same time, every gauge theory has a gauge invariant description. I have in mind, let's say, lattice models with a local hilbert space.

So it seems to me that the question shouldn't "why gauge?" but instead "why deconfined gauge?". That is, why are gauge fields often found in a relatively weekly fluctuating semiclassical ground state?

And in fact, in condensed matter context, deconfined gauge fields seem relatively rare so far, although this may be an accident of history or our limited understanding of how to probe such systems.

But if deconfined gauge dynamics is rare, perhaps the answer in the context of the standard model is anthropic?

 

[atyy]

The Turing machine! (If it's computationally universal.) For any (universal) Turing machine T executing a program p, there exists for every Turing machine T' a program sp such that T' executing sp is equivalent to T executing p (if all else fails, by sp being a program that directly simulates T executing p on T'). So any given computation is really an equivalence class of programs on different Turing machines giving rise to the same 'phenomenology'.

Is that like saying there are many ways to compute 2X2, ie. the Turing machine is specified by its computational capability, and so does not identify a specific physical object?

Presumably, a specific Turing machine in the class is a particular physical object? Is there any necessary gauge then?

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.

Is there a difference between gauge and duality? For duality, I'm thinking of something like AdS/CFT (it's probably not be strictly a duality, since the bulk theory may not be defined, but just to give an idea of what I mean). Or can we say that duality is a highly non-trivial form of gauge?

EDIT FOR ATYY: Rovelli does not claim that gauge theory is necessary in any universe with at least two interacting objects. You must have misread. He gives an example of a universe with many objects described by a theory with no gauge component. Don't understand your saying "My skepticism is about the claim that gauge is necessary in all universes with at least two interacting objects"

Yes, I must be misreading. Here's an attempt to interpret what he's saying as sensible.

What can be "observed" or is "real" is always relational. Therefore a term that couples two gauge variant quantities can be gauge invariant, since it is relational.

Example 1. Is velocity absolute or relative in Newtonian physics? One answer is that it is relative to the choice of inertial frame. However, if we are able to specify a particular inertial frame, then velocity is absolute, eg. the reading of the speedometer in the car is an observable, and in turn it gives the speed of the car relative to the inertial frame in which the car was initially designated to be at rest.

Example 2. Is the universe really exapnding? One answer is that it is not, since that is not the only possible description of the FLRW solution. However, if we specify the class of observers as those at rest in the Hubble flow and having the global symmetry of the solution, then with respect to that class of observers the expansion of the universe is absolute.

Assuming that gauge is by definition a redundancy of the description, then the question is why use those variables. As Physics Monkey says, a redundant description is always possible. So gauge by itself is trivial. The question is what properties make particular descriptions convenient? In the case of gauge in relativistic QFT (where the notional invariant objects are nonlocal loops), the answer that trimok and DimReg gave is that it respects locality (and unitarity). So Rovelli's first two examples with the vector potential and GR adds detail to their answer in saying that even when we go to the interacting case and describe relations between objects, we like to preserve locality (of some form).

 

[S.Daedalus]

Is that's like saying there are many ways to compute 2X2, ie. the Turing machine is specified by its computational capability, and so does not identify a specific physical object?

Presumably, a specific Turing machine in the class is a particular physical object? Is there any necessary gauge then?

Well, I'm not sure how exactly to draw the analogy, to be honest. (May I ask, what prompted you to ask the question in the first place?)

But presumably, if one takes seriously the idea that our universe either 'is' a computation, or can at least be modeled as one, then one runs into a bit of a problem if one wants to find out its 'true' fundamental description. For, take the idea that we're really living in a simulation: how do you find out if, say, that simulation is run on a PC, a Mac, or some completely different architecture? And the point is, you can't: the same program can be run on completely different architectures and works equally (in theory, if not always in practice); that's how things like emulation and so on work. But then, this must mean that the phenomenology within the program must in some sense be independent of the underlying architecture. So let's for concreteness take the class of Turing machines, which we know is denumerable, so that we can parametrize it by a single natural number; then, phenomenology should be invariant under change of Turing machine, which at least morally looks like a kind of gauge transformation: you 'port' to a different Turing machine, which necessitates the modification of the underlying program, just as you make a (local) U(1) transformation, which necessitates a change in the potential A_\mu.

(That's not the only way of looking at this situation, mind: David Deutsch has claimed that it entails 'giving up on science', precisely because we would be unable to get at the fundamental details.)

Regardless, I'm not really sure this is more than a cute analogy (though I do sometimes think that maybe all those dualities and similar things are nothing but mappings between different Turing machines; but then again, don't we all have dark thoughts in the night sometimes?).

In any case, having now read Rovelli's article, I must confess that I'm unsure what to think. In a way, I emphasize with his relational viewpoint; but I'm not sure I haven't just missed what's especially deep about it... But first, regarding the following:

Yet the world contains both relational and absolute stuff, stuff that just is (whatever that means; for those who prefer it, given). For example in physics c and h seem to me absolute and without any relational flavour, as do prime numbers in mathematics. And Nature's fundamental entities, like electrons, qua electrons, seem so far better to fit in the category of absolute stuff.

I don't think that a relational view necessarily entails abandoning definite, or absolute, facts---if x and y stand in the relation R, then that x and y stand in relation R is an absolute fact (which you, of course, can relativize again, or else would have to regress ad infinitum---in which case, the infinite regress would be a definite fact*).

But relational views, of course, have some great problems, most of all that a set of relations simply doesn't suffice to pick out a set of things---any two sets of the same cardinality can be endowed with the same structure (i.e. set of relations), since after all, relations are just sets of entities, and thus, a structure is just a subset of some set's powerset. That means that the same structure can be realized using entirely different objects---the relations between differently-colored marbles can be equivalently modeled by differently-sized people, or sportscars of different power, etc. In philosophy, this is known as Newman's problem, and as far as I know, nobody has yet found a good counterargument. So in the case of such a relational world, really the only thing open to empirical investigation would be its cardinality, which seems too poor to lead to the variety we are actually met with in life (but which is surprisingly parallel to the simulation-world I've outlined above). So the merits of such a complete relativization seem to be questionable at least.

As for gauge variables, usually what's used to argue for their reality is the Aharonov-Bohm effect and its variants. I've read part of Richard Healey's Gauging what's Real, which I think is probably as close to a canonical reference on this question as you'll get (but I don't seem to remember much of it). However, there's a cute argument by Sunny Auyang (probably in How is Quantum Field Theory possible?), where she uses the AB effect in order to motivate gauge symmetries: the introduction of a vector potential will shift interference fringes, as will local phase transformations; if we want our theory to be invariant under such transformations, we must thus necessarily include an electromagnetic potential. This is of course just a heuristic way of looking at the usual derivation of the covariant derivative, but it provides a nice illustration of the relation between local symmetries and gauge quantities.

Rovelli essentially argues that gauge quantities pertain to our relative localization, either quite directly, or in a more abstract sense as in the case of a fiber bundle. Perhaps, then, calling the gauge variable itself a 'mathematical redundancy' doesn't do justice to the situation, as we can't make do without at least the relative specification of its value. To loop back to the state of this post, we can describe physics using an arbitrary Turing machine, but we do need some specific program (that obviously we haven't found yet); the description of the world is then relative to the underlying computing machinery.

But this is all getting a little postmodern, perhaps...




*I probably shouldn't use the term 'definite fact', after all, there aren't really any 'indefinite facts', but to me, it just seems to be the natural counterpoint to relational facts for some reason.

 

[friend]

"Gauge" is normally understood as "mathematically redundant"

Why are gauge theories so prevalent? Why do they always seem to win out in the contest to describe the world?

I wonder if this is the same as asking why is reality part of the SU(3)SU(2)U(1) gauge theory symmetry group? Is any and every gauge theory applicable to physics? Or are you asking what selection principle would pick out the gauge theories that do find application?

 

[marcus]

Maybe gauge theory has been so successful because our universe is a deforming Lie group.

This is a good chance for me (and maybe some others of us) to get clear about what exactly a gauge theory is. Garrett may not be dropping back by here soon enough to explain what he said, but there are others here who know the rigorous side of gauge theory and can correct me if I am wrong.

Probably almost everyone reading the thread here has at one time pictured the tangent bundle on a manifold. A vector field is where at each point you pick a tangent vector, in a smooth way. Picking a tangent vector at each point can be described as taking a section of the tangent bundle--a slice through it. In case anyone has not encountered the concept, a FIBER bundle is just a generalization of the tangent bundle idea, where the fiber over a point doesn't have to be the tangent vector space. A "section" of a fiber bundle is just the analogous idea, a choice at each point. If you have a GROUP then that group (acting on itself as a homogeneous space) can serve as the fiber and that defines the PRINCIPAL bundle of that group on the underlying base manifold.

I wanted to mention the fiber bundle language in part because S.D. just brought it up in his last post #30, in connection with Rovelli's paper. Is the language of fiber bundles and groups necessary to saying clearly what is meant by a gauge theory? Maybe not, maybe yes in some, if not all, cases. It would be great if someone wishes to provide more clarification.

The one thing that does seem obvious is that (at least in Rovelli's paper) whenever a gauge theory is mentioned a group is always specified. In electrodynamics the group is U(1), the unit circle of rotations. In GR the group is diffeomorphisms (smooth change of coordinates, and the like), and so on. There is some formal idea of what a gauge theory consists of that involves a group (the "gauge symmetries")

I think we can distinguish between what a gauge theory actually is and the loose SLANG use of the word "gauge" to mean "redundancy". A gauge theory is not merely a theory with mathematical redundancy. To say something is a gauge theory implies more: it has a certain structure involving some specific group of symmetries.

The slang usage of "gauge" either as adjective or noun synonym for "redundant" or "redundancy" is I think just symptomatic of the way a lot of people think about gauge theories. It does not define/delimit the theories themselves---rather it expresses a particular view/attitude about the theories. There could be other ways to understand their significance and why we have them.

Anyway that's what I'm hoping is the case.

 

[garrett]

Hi Marcus, thanks for the followup. I certainly agree that the mathematical language of fiber bundles is essential to gauge theory. I'd like to clarify my previous oblique comment about deforming a Lie group. There is a nice way of deriving the geometric pieces of GR (Spin(1,3) spin connection and frame over a 4D base) by allowing the Spin(1,4) Lie group to deform, as described by Cartan geometry. Here, spacetime may be imagined as part of Spin(1,4) that has become arbitrarily wavy while a Spin(1,3) subgroup maintains its structure. This reproduces the fiber bundle picture in which Spin(1,3) is the typical fiber over a 4D base, along with a spin connection and frame that evolves from Spin(1,4)'s Maurer-Cartan form. I'm currently working out a generalization of Cartan geometry to larger groups, including a description of fermions, which is going well. Since I've just finished up construction of phase one of the Pacific Science Institute here in Maui, I have some more time to work and talk about this stuff. If Cartan geometry and its generalization is relevant to this thread, then great, I'm happy to contribute; otherwise, I'll get back to working on it as a paper.

 

[marcus]

Hi Marcus, thanks for the followup. I certainly agree that the mathematical language of fiber bundles is essential to gauge theory. I'd like to clarify my previous oblique comment about deforming a Lie group. There is a nice way of deriving the geometric pieces of GR (Spin(1,3) spin connection and frame over a 4D base) by allowing the Spin(1,4) Lie group to deform, as described by Cartan geometry. Here, spacetime may be imagined as part of Spin(1,4) that has become arbitrarily wavy while a Spin(1,3) subgroup maintains its structure. This reproduces the fiber bundle picture in which Spin(1,3) is the typical fiber over a 4D base, along with a spin connection and frame that evolves from Spin(1,4)'s Maurer-Cartan form. I'm currently working out a generalization of Cartan geometry to larger groups, including a description of fermions, which is going well. Since I've just finished up construction of phase one of the Pacific Science Institute here in Maui, I have some more time to work and talk about this stuff. If Cartan geometry and its generalization is relevant to this thread, then great, I'm happy to contribute; otherwise, I'll get back to working on it as a paper.

Go for it, Garrett. You are most welcome to expound Cartan geometry ideas and generalizations thereof.
If I feel the need to discuss Rovelli's paper specifically, at greater length, I don't need this thread to do it in. I can start a thread on that "Why gauge?" paper in particular.
Your time is valuable if you are working on a paper, so please give us only what you can readily spare.

It has been bothering me that I don't see how the diffeomorphism group can fit into the more conventional gauge theory format (where you usually have a Lie group). The diffeo group is too big and doesn't seem to contain a replica of the manifold. I haven't thought much, just experienced generic bafflement. And yet GR is clearly a prime example of a gauge theory! The dynamics (equation, action etc.) has the diffeo symmetry! So perhaps you are talking about something (the spin connection + tetrad formalism?) which can help us see the familiar gauge theory format in Gr more clearly. In any case, go for it. You are cordially welcome. Naturally

I just got in from walk up the hill behind the Greek Theatre, to east of the UC Berkeley campus, and saw your post. Wife playing Mozart. Nice afternoon.

 

[garrett]

Well, thanks Marcus. And yes, the Berkeley hills are spectacular, especially when there's a clear view out over the Bay. There are strongly different points of view on diffeomorphism invariance, so I don't want to be pushy about mine, but I can share it. I don't consider diffeomorphisms to be a symmetry group of nature in the same way Spin(1,3) or SU(3) is. Rather, diffeomorphism invariance is a consequence of general covariance -- i.e. of coordinate independence. And, as such, you shouldn't "gauge" diffeomorphisms, but merely make sure your theory is coordinate independent. And the modern mathematical description of fiber bundles with connections is already set up to be coordinate independent, so we're good as long as we use modern tools.

If you hold this view, then all the geometric pieces we've seen so far can be described by a fiber bundle including the spin connection, gravitational frame, Standard Model gauge fields, Higgs, and fermions. And the dynamics comes from an action built from contractions of their connections, curvatures, and covariant derivatives, all Hodged together using the frame. It is kind of a mess... A good theoretical model starts with something simpler and derives these. The nicest theoretical model I've been working on is to start with a large Lie group and its Maurer-Cartan form, then let it go wavy in a way similar to how that is described using Cartan geometry, and get ALL of those geometric pieces as parts of a generalized Cartan connection, and use an action that's the Yang-Mills action over the Lie group. 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?

 

[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}}<br /> = 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.