# Gauge theory of Gravitation

Hi,
while I'm going deeper in my SR/GR knowledge, having LQG unrderstanding as main goal ( my QM background and maths is a bit stronger than GR's one, til now ) I came across some interesting youtube lectures about Gauge theory of Gravitation:

I've roughly touched gouge theories in my QFT incursions, and I found a possible link between gouge theories and GR in the fantastic Baez book "Gauge Fields, Knots and Gravity". So, gauge theories has always fascinated me ( even if I'm still trying to strengthen by knowledge on Lie groups/algebra and bundles ).

This post is about my main doubt about how a gauge could be chosen to encompass GR between gauge theories. Wikipedia ( which is not always a good place to understanding such deep things ) is mentioning mainly 2 theories about it, i.e.:

1) http://en.wikipedia.org/wiki/Gauge_gravitation_theory, which should be the one analysed in the above lectures ( which are following mainly this book http://www.worldscientific.com/worldscibooks/10.1142/p781 )

2) http://en.wikipedia.org/wiki/Gauge_theory_gravity, which seems a bit different from the above one.

From what I have understood, both theories are currently keeping the spacetime as flat introducing a suitable gauge in that canvas. Is this correct?

Also, I was guessing, the graviton can be actually seen as a gauge particle in a GR gauge theory context? I've read somewhere the gauge choosing is somehow a version of the equivalence principle for GR, but I'm not fully sure can be interpreted like that, even if cool as it would be.

A minor doubt I have is how much this version of theory could actually really help me in landing to LQG understanding.

May you help me a bit, as always, in removing a bit of fog from this particualr topic, please?

thanks, regards

EDIT: probably this is the link to my doubt with LQG: https://www.physicsforums.com/showthread.php?t=179731

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member 11137
Well I am absolutely not a specialist but what my eyes have seen here and on the web until now say: first learn GR intensively, then have a look at the ADM procedure, make a few meters byside to learn about the new Ashtekar Barbero variables and perhaps, at this stage, you shall stand up one morning and start in wondering LQG' beach.

Hi Ricardo, you have started from a long way back with Baez' s wonderful book on 'Gauge Fields, Knots and Gravity.

May I suggest that a a route to LQG would be:
Gambini and Pullin: A First Course in Loop Quantum Gravity
Rovelli : Quantum Gravity
Thiemann: Modern Canonical Quantum General Relativity
Rovelli - Introduction to LQG (draft): http://www.cpt.univ-mrs.fr/~rovelli/IntroductionLQG.pdf
The International Loop Quantum Gravity Seminar series site has useful links and materials:
http://www.loopquantumgravity.org/

Marcus on this forum also maintains a number of fantastic threads for LQG. Keep up-to-date with the quarterly best picks.

atyy
There is some discussion of the similarities between current LQG proposals and SU(2) Yang-Mills in http://arxiv.org/abs/1205.2019 (p22, paragraph after Eq 49).

thanks for all answers you gave me. Nevetheless, this post have been slightly diverging from my original intention. LQG is just one of my goals and I know starting from Baez's book is a bit far from LQG. The real question was about the one depicted in the topic: what are the difference between the 2 graviation gauge theories I've quoted from wikipedia.
As Blackforest suggested to me, I'm strengthening my GR foundations, before going on ( and I'm studying on Hurtle's, on Gravitation and on main GR classics ). So a long path has to be done before landing on LQG.
My real question was about that online course about gauge theory of gravitation and about the book which inspired it. I'd like to understand whether gauge theory of gravitation could make my understanding about GR deeper, or it's a just a digression which would take no actual application to current GR evolutions ( LQG included ).

Thanks

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member 11137
thanks for all answers you gave me. Nevetheless, this post have been slightly diverging from my original intention. LQG is just one of my goals and I know starting from Baez's book is a bit far from LQG. The real question was about the one depicted in the topic: what are the difference between the 2 graviation gauge theories I've quoted from wikipedia.
As Blackforest suggested to me, I'm strengthening my GR foundations, before going on ( and I'm studying on Hurtle's, on Gravitation and on main GR classics ). So a long path has to be done before landing on LQG.
My real question was about that online course about gauge theory of gravitation and about the book which inspired it. I'd like to understand whether gauge theory of gravitation could make my understanding about GR deeper, or it's a just a digression which would take no actual application to current GR evolutions ( LQG included ).

Thanks
I am certain that some people on these forums can give you a quite better answer than I can but, as you evocated it in your opening thread, I can at least try to remove the fog.

Bibliography
First suggestion concerning a book: “Gravitation” (Misner, Thorne and Wheeler, alias MTW) is certainly a reference, a must; even if it weighs more than a kilo.

Why a gauge theory
The reason why many scientific brains are working on the topic “a formulation of the theory of relativity as a gauge theory” is (so far my understanding) rooted in the search for unity in physics.

More precisely: the Einstein’s theory of relativity (ETR) is until now the unique theoretical model which cannot be mathematically formulated under the formalism of a gauge theory. The standard model is a successful gauge theory; consequently it has been believed that if the ETR could be written under that formalism, then we would have reached the final unification between the diverse descriptions of all interactions (gravitation for the ETR and the others for the standard model).

The graviton
One possible mathematical demonstration introducing the graviton is based on a formal comparison between EM and gravitational waves and is developed in considering plane waves (it’s easier), that is in fact the linearized version of the Einstein’s field equations (EFE). These plane waves are in some way assimilated to perturbations of the metric (hλμ). These perturbations are symmetric and respect the condition (Dalambertian hλμ = 0). All this is sufficient to find the formalism of the gravitational waves and to state that they do have 6 degrees of freedom. In scrutinizing the behavior of the obtained relations in a gauge transformation, it becomes clear that only 2 degrees of freedom survive.

Can we say, because of that demonstration, that the graviton is a gauge particle? I would not answer with yes. We just have discovered the real number of degrees of freedom for the wave under study when we were transforming the equations in a gauge. Don’t put the car before the cow.

The two approaches
Gauge theory gravity, as said inside the text of reference that you have proposed is an attempt to cast in the mathematical language of “Geometric algebra”. The other approach prefers “differential geometry”. So far my actual understanding the two branches are perhaps already connected via recent progresses in mathematical physics.

My proposition
But I leave more sophisticated explanations to specialists!

haushofer

http://arxiv.org/abs/1011.1145

It also treats the rel. case. The book on Sugra by Van Proeyen could also be useful.

I am certain that some people on these forums can give you a quite better answer than I can but, as you evocated it in your opening thread, I can at least try to remove the fog
Thanks Blackforest, this sounds good to me.

Bibliography
First suggestion concerning a book: “Gravitation” (Misner, Thorne and Wheeler, alias MTW) is certainly a reference, a must; even if it weighs more than a kilo.
Yes, I own already that bible and I'm studying it altogether with the Hurtle, someother texts from Wheeler, Zee's Gravity in a nutshell, the above mentioned Baez's jem, and others ( I'd like to collect physics and math books ).
Incidentally, since you mentioned, Gravitation, does it contain any reference to gauge thoeries of gravitation? ( If yes, I missed them, probably ).

Why a gauge theory

More precisely: the Einstein’s theory of relativity (ETR) is until now the unique theoretical model which cannot be mathematically formulated under the formalism of a gauge theory. The standard model is a successful gauge theory; consequently it has been believed that if the ETR could be written under that formalism, then we would have reached the final unification between the diverse descriptions of all interactions (gravitation for the ETR and the others for the standard model).
OK, so basically this is "just" a different roadmap towards the unification. Roughly, I think this is related to different choices of gauges, more or less exactly like LQG does. So, does the roadmap to LQG could traverse gauge theories as well, I gues...

The graviton
One possible mathematical demonstration introducing the graviton is based on a formal comparison between EM and gravitational waves and is developed in considering plane waves (it’s easier), that is in fact the linearized version of the Einstein’s field equations (EFE). These plane waves are in some way assimilated to perturbations of the metric (hλμ). These perturbations are symmetric and respect the condition (Dalambertian hλμ = 0). All this is sufficient to find the formalism of the gravitational waves and to state that they do have 6 degrees of freedom. In scrutinizing the behavior of the obtained relations in a gauge transformation, it becomes clear that only 2 degrees of freedom survive.

Can we say, because of that demonstration, that the graviton is a gauge particle? I would not answer with yes. We just have discovered the real number of degrees of freedom for the wave under study when we were transforming the equations in a gauge. Don’t put the car before the cow.
Yes, I got the point.

The two approaches
Gauge theory gravity, as said inside the text of reference that you have proposed is an attempt to cast in the mathematical language of “Geometric algebra”. The other approach prefers “differential geometry”. So far my actual understanding the two branches are perhaps already connected via recent progresses in mathematical physics.
OK, probably I would prefere differential geometry since my knowledge is a bit deeper compared to geometry algebra.
Incidentally, may you suggest some good textbook about gauge theories which doesn't go deep too much with particles? I mean, I know it could sound strange and I know the standard model is actually a mix of gauge theories with different groups ( U(1), SU(2) etc ) but I'd like to have a more general point of view without messing around the particle zoo born from it.

http://arxiv.org/abs/1011.1145

It also treats the rel. case.
Thanks haushofer, it sounds interesting even if a but too advance to me, at least for now

The book on Sugra by Van Proeyen could also be useful.
thanks, that Sugra book seems really interesting and not too advanced to me. I'm going to give it a chance for sure.

member 11137
OK, probably I would prefere differential geometry since my knowledge is a bit deeper compared to geometry algebra. Incidentally, may you suggest some good textbook about gauge theories which doesn't go deep too much with particles? I mean, I know it could sound strange and I know the standard model is actually a mix of gauge theories with different groups ( U(1), SU(2) etc ) but I'd like to have a more general point of view without messing around the particle zoo born from it.
After your last post I have made a small inspection of my modest collection and looked in the tables of content where the word “gauge” appeared. What a surprise.

There is effectively nothing directly related to the topic in “gravitation”.

I tried with “Advanced general relativity” (J. Steward – Cambridge monographs on mathematical physics; first published in 1991 (I am not so young you know)) but just discovered a very short text (pp. 23-24) explaining what is a “choice of gauge” and, by side, mentioning that that locution is an old-fashioned one (in extenso: that intuitive concept can be replaced by a mathematical procedure). The interesting point was the positioning of the topic (the gauge) inside the chapter “differential geometry > integrals and Lie derivatives”.

This is confirming another approach made in the “Weber and Arfken: Essential mathematical methods for physicists – published in 2004”, positioning the “gauge invariance” problematic directly in the introduction of chapter 4, p. 229, titled “Group theory”.

So, I think that you get the arena where all this is playing… without being obliged to walk with the entire zoo (particles).

You also get a collection of propositions (pure mathematics) in googling with the words "books on gauge theory"; e.g. on the amazon website (free publicity!).

first published in 1991 (I am not so young you know)
ahahah. I'd really bet I'm currently older than you, since I'm 43.

You also get a collection of propositions (pure mathematics) in googling with the words "books on gauge theory"; e.g. on the amazon website (free publicity!).
thanks to your googling suggestion above, I was finally able to find this really cool book:

https://www.amazon.com/dp/0521378214/?tag=pfamazon01-20&tag=pfamazon01-20

It really covers gauge theories, differential geom and GR. I really think I'm going to buy it.

EDIT: If you look at chapter 4 yuo can see it states some hints on how choosing gauges in GR

Another one, found in my these days quest, is:

https://www.amazon.com/dp/0750306068/?tag=pfamazon01-20&tag=pfamazon01-20

I already own Frankel one but I agree with one of the reviewers telling Frankel's seems a bit more demanding than this one ( where for demanding I mean it's not modular with selfcontained chapters ).

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1 person
atyy
In the most general sense, a gauge theory is simply one in which the variables used to describe the system are redundant - different values of the variables correspond to the same physical state. Both general relativity and Yang Mills are gauge theories in this sense.

In the more restricted sense, a gauge theory means specifically a gauge theory of Yang Mills form. General relativity is not a gauge theory in this restricted sense, though the there are formalisms like the Ashtekar variables and the Holst action that make GR resemble Yang Mills.

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In the most general sense, a gauge theory is simply one in which the variables used to describe the system are redundant - different values of the variables correspond to the same physical state. Both general relativity and Yang Mills are gauge theories in this sense.

In the more restricted sense, a gauge theory means specifically a gauge theory of Yang Mills form. General relativity is not a gauge theory in this restricted sense, though the there are formalisms like the Ashtekar variables and the Holst action that make GR resemble Yang Mills.
Hi atyy,
thanks for you effective explanation. The gauge choosing for U(1) is actually clear to me, even without disturbing fibre bundles. this is explained superbly here ( thanks to WBN for having shared that doc ):

http://scistud.umkc.edu/psa98/papers/weinstein.pdf

Here you can see he's raising a lot of problems telling the invariance under diffeomorphism ( typical in GR ) cannot be put and interpreted like other gauges used within Yang-Mills theories. I found that article really enligthing.

I really think I have to refresh my knowledge about fibre bundles to understand deeper how gauge are working, and, as far as I remember, the best and coincise approach to it
was on Baez's book.
Anyway, again, the above mentioned doc contains some pearls like this one at pag.6:

"The vectors in the U(1) case are just the local phases. If the connection is such that the phase undergoes a change when moving around a closed loop, this means the curvature [...] is non-zero in the vicinity of the loop, i.e., there's an electromagnetic field there."

I've one main doubt, however. If you read at the very beginning about U(1) local Vs global gauge invariance, mainly at pag.3 of that pdf. It's clear to me using just e^iθ is a constant ( and hence global ) phase transformation, which doesn't change the physics at all. It's also clear introducing a space dependent phase variation, like e^iθ(x) we face a local phase transformation, which changes the physics, since it's connected to momentum changing, I mean quoting:

"Then the transformed states will NOT be equivalent, for although they willl have the same probability distribution for position, they will have a different probability distribution for momentum ( because the phase of a wave function in configuration space effectively encodes the momentum of the particle )"

My doubt is more trivial: why do we choose that kind of local transformation? I mean: which is the sense to LOCALLY impose the phase to be such this at point x1, while such that in another point? which is the physical meaning of changing the phase in that way? is for letting different momenta at different points?
Hope my question was clear.

atyy
My doubt is more trivial: why do we choose that kind of local transformation? I mean: which is the sense to LOCALLY impose the phase to be such this at point x1, while such that in another point? which is the physical meaning of changing the phase in that way? is for letting different momenta at different points?
Hope my question was clear.
Why do we choose to use redundant, "unphysical" variables, in which more than one value of the variable corresponds to the same physical situation? This is because in the case of gauge theories, the physical variables are holonomies or Wilson loops which are "nonlocal", and it is easier to use the redundant, nonphysical variables to make the physics look "local". The book by Baez discusses quite a lot how to do things using the physical, gauge invariant variables, rather than choosing gauge variant variables and fixing a gauge. There is also a detailed explanation in Arkani-Hamed's lecture https://video.ias.edu/pitp-2011-arkani-hamed1 (around 20 minutes).

Why do we choose to use redundant, "unphysical" variables, in which more than one value of the variable corresponds to the same physical situation? This is because in the case of gauge theories, the physical variables are holonomies or Wilson loops which are "nonlocal", and it is easier to use the redundant, nonphysical variables to make the physics look "local". The book by Baez discusses quite a lot how to do things using the physical, gauge invariant variables, rather than choosing gauge variant variables and fixing a gauge. There is also a detailed explanation in Arkani-Hamed's lecture https://video.ias.edu/pitp-2011-arkani-hamed1 (around 20 minutes).
Thanks for your answer and for the video. Anyway my question was more oriented to the example I quoted from the doc, i.e. U(1) global gauge invariance Vs local one ( i.e. e^iθ Vs e^iθ(x) ).
Anyhow I will look carefully at Baez's and to the video.

atyy
Thanks for your answer and for the video. Anyway my question was more oriented to the example I quoted from the doc, i.e. U(1) global gauge invariance Vs local one ( i.e. e^iθ Vs e^iθ(x) ).
Anyhow I will look carefully at Baez's and to the video.
Global U(1) invariance is a true symmetry, and leads to charge conservation.

Local U(1) invariance is a gauge symmetry, and not a true symmetry. It is the use of redundant, unphysical variables in which more than one value of the variable describes the same physical situation. We use these unphysical variables in order to make locality manifest.

Global U(1) invariance is a true symmetry, and leads to charge conservation.

Local U(1) invariance is a gauge symmetry, and not a true symmetry. It is the use of redundant, unphysical variables in which more than one value of the variable describes the same physical situation. We use these unphysical variables in order to make locality manifest.
OK atyy, I got the point, it's almost clear, even if I have to dig deeper through examples, after having refreshed fiber bundles section as well, of course.
I've seen the video yuo sent me, it's really really interesting. Incidentally I followed some other video lectures by Arkani-Hamed on other topics and he's really passionate in his presentations.
Unfortunately the level at which that video is developing is a bit too advanced to me, at least following the details, but I was able to follow the whole picture.
I was wondering if he wrote some books with the same "new" approach is following ( or at least different from any other approach I've seen ). I will look forward to him about it.

haushofer
To understand this redundancy, you could look at the Stueckelberg mechanism, e.g. in Hinterbilcher (great!) notes on massive gravity on the arxiv. :)

Btw, Nakahara is also a nice book on geometry, treating also gauge theories.

Haelfix
There are several not necessarily equivalent notions of what a 'gauge' theory of gravity is. Beware this loaded language, it is very easy to slip up and learn two different methods and to then get really confused. Each approach is filled with subleties and the analogy with more traditional gauge theories is always somewhat stretched.

Make sure to always identify exactly what is being gauged in the first place. In the literature there are formalisms that gauge the connection variables, there are those that gauge the Poincare group, there are those that gauge the translation group, there are those that gauge the asymptotic symmetry algebras and there are those that linearize gravity and draw analogies with Yang Mills theory.

The two most successful attempts were written down by Utiyama-Kibble and Feynman-Deser respectively. It is these two that most professionals work with.

To understand this redundancy, you could look at the Stueckelberg mechanism, e.g. in Hinterbilcher (great!) notes on massive gravity on the arxiv. :)

Btw, Nakahara is also a nice book on geometry, treating also gauge theories.
Hi hausofer,
I've given the Hinterbilcher doc a rough first look, and it seems a bit too advanced to me ( the main problem I'm facing there are no good (at least to me, of course) sources on gauge theories at a undergrad level, I mean all discussion run quickly in sophisticated evolution of it, or implementation in parallel use ). Nevertheless I will try to go deeper in it, thanks.
Hope Nakahara will give a hand in that as well.

haushofer
My understanding of GR as a gauge theory is as follows:

* Take the global symmetries of spacetime (the symm. of a point particle, say), i.e. the Poincare group {P,M}, consisting of translations P and LTs M
* Gauging this algebra (see e.g. Van Proeyen notes) gives you two gauge fields, e(P) and w(M), along with their transformations and curvatures F(P) and F(M)
* Now you put F(P)=0. With this the field w(M) becomes dependent on e, and with some identities you can show that the translations P become linear combinations of gct's and M, with field dependent parameters (you obtain a so called soft algebra)
* The Einstein Hilbert action can be written down in terms of R(M). If you like, you can postulate the Vielbein postulate to make contact to the usual formulation of GR. The field e can be identified as Vielbein, the field w as spin connection. F(P)=0 then becomes the torsionless condition of GR.

haushofer
BTW, What you should know about gauge theories, I think, is all in the Van Proeyen sugra notes (online) or in his book :)

BTW, What you should know about gauge theories, I think, is all in the Van Proeyen sugra notes (online) or in his book :)
This will be for sure the very next book I'm going to buy this month. Incidentally, I've start to read this one I was mentioning in one of post above:

I strongly suggest it if you are intersted in gauge theories ported to GR as well. The mood is a graduate one, but even if formal the approach it's not too math addict ( like some Springer book, but this is because I'm not too math addict, always in between Physics and Maths ).
The book is really readable and start with a good chapter about exterior derivative. It then goes through Einstein/Cartan theories going deep into some gauges for GR.
Really cool ( at least to me, since I'm facing this part ).

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There are several not necessarily equivalent notions of what a 'gauge' theory of gravity is. Beware this loaded language, it is very easy to slip up and learn two different methods and to then get really confused. Each approach is filled with subleties and the analogy with more traditional gauge theories is always somewhat stretched.

Make sure to always identify exactly what is being gauged in the first place. In the literature there are formalisms that gauge the connection variables, there are those that gauge the Poincare group, there are those that gauge the translation group, there are those that gauge the asymptotic symmetry algebras and there are those that linearize gravity and draw analogies with Yang Mills theory.

The two most successful attempts were written down by Utiyama-Kibble and Feynman-Deser respectively. It is these two that most professionals work with.
sorry haelfix,
I've just noticed today your post ( don't know why the system has not notified it to me ).
I've seen you are mentioning Feynman-Deser theory...does it appear within the "Feynman's lectures on gravitation?"