Development of Loop Quantum Gravity

[marcus]

This is a thread for collective learning of the classical (i.e. non-quantum) theory of Gravity-----which means General Relativity.

Reference material is Chapter 2 (pages 21-71) of Rovelli's book
"Quantum Gravity" and John Baez GR Tutorial

The title of Rovelli's book notwithstanding, the first half of the book develops classical relativity and a portion of ordinary quantum mechanics. The development of Loop Quantum Gravity starts in the second half, around page 160. So the book's first half provides a convenient modern synopsis including an interesting philosophical take on General Relativity. Rovelli's August 2003 draft is online:

http://www.cpt.univ-mrs.fr/~rovelli/book.pdf

I'm going to try to paraphrase some stuff out of Rovelli Chapter 2, the modern treatment of GR, and use Baez, which is the old-fashioned exposition in a light style, for backup references if and where needed. If you are learning GR and using these materials, ask questions. Baez GR tutorial links are:

General Relativity Tutorial: Short Course Outline
http://math.ucr.edu/home/baez/gr/outline1.html

Long Course Outline
http://math.ucr.edu/home/baez/gr/outline2.html

Oz stories of General Relativity
http://math.ucr.edu/home/baez/gr/oz1.html

The Meaning of Einstein's Equation
http://www.math.ucr.edu/home/baez/einstein.html

 

[marcus]

Some quotes to start with

Rovelli page 19:

"In classical general relativity, a classical trajectory of the gravitational field has the structure of a (pseudo) Riemannian manifold. Therefore even if the dynamics of the theory has no preferred time variable, we nevertheless have a notion of spacetime for each given solution.

In quantum theory there are no trajectories. The theory deals only with transition amplitudes. Formally, these can be computed summing over all trajectories..."

 

[marcus]

In treating classical (non quantum) gen. relativity, essentially the first half of the book, Rovelli uses a 4D manifold M and a different font M for Minkowski space.

[edit: something I'm unclear about is the identification of the tangent space T_m at a point of the manifold with Minkowski space. this may be part of the language of fiber bundles that I am not used to]



For rovelli, the gravitational field is just an M - valued 1-form on M

at every point m in M it tells you a linear map T_m --> M whose matrix if you express it in coordinates is the jacobean of a change of coordinates into a locally inertial (or "free-fall") frame

I believe that, technically, so as to describe the smoothness or differentiability of the map, one would need to introduce a Lorentz-group bundle P with fiber M But Rovelli's style is initially rather intuitive and informal so he does not
describe the field technically in bundle terms until page 44---after he has given a lot of examples like buckets of water and rotating stars and clouds of galaxies.

[edit: I have gone back to page 44 and don't understand rovelli's language at the top of the page. he is describing what the field is in terms of differential forms and in terms of a "principal G-bundle" where G is the Lorentz group. At the moment I cannot give an adequate paraphrase.]

I hope Lethe goes ahead with his differential forms tutorial! The gravitational field e describes geometry and is what things freefall and rotatate and accelerate with respect to. It is the "distant stars" with respect to which the water in Newton's bucket turns. And this field is a differential form---one of the simplest type: a mere 1-form.

The variables of the (classical, non-quantum) theory are (e,ω) the gravitational field itself and the associated connection. The Einstein equation---the dynamical model of how the field evolves---can be written in terms of these two variables together with the matter stuff.

I should get some more quotes from rovelli.

 

[Lonewolf]

Does Minkowski space necessarily have the metric

/-1 0 0 0 \
| 0 1 0 0 |
| 0 0 1 0 |
\ 0 0 0 1 /

or are there other Minkowski spaces? Just want to be sure that this is the only one.

 

[marcus]

Originally posted by Lonewolf
Does Minkowski space necessarily have the metric

/-1 0 0 0 \
| 0 1 0 0 |
| 0 0 1 0 |
\ 0 0 0 1 /

or are there other Minkowski spaces? Just want to be sure that this is the only one.

yes the metric is diag(-1,1,1,1)
with the convention that the timecoordinate is x0
and x1,x2,x3 are spatial

those are the conventions he is using.
In other books you may see it set up with the
first three coordinates spatial and the last one time
and the metric being diag(1,1,1,-1), or some other way

but relativists like Rovelli is very often use the convention
he uses---it is maybe standard, anyway for this book at least, yes

 

[selfAdjoint]

Originally posted by Lonewolf
Does Minkowski space necessarily have the metric

/-1 0 0 0 \
| 0 1 0 0 |
| 0 0 1 0 |
\ 0 0 0 1 /

or are there other Minkowski spaces? Just want to be sure that this is the only one.

They will all be diagonal, and the only difference is in the signs. This is called choosing a signature, and the one you have shown is (-,+,+,+), in this signature, as Rovelli says, the "time" vector will have negative norm.

The alternative signature is to give the time the positive norm and the negative norms to the spacelike vectore: (+,-,-,-). There is absolutely no difference in the underlying physics, but the formulas have different signs in them. In this sense Minkowski spacetime is unique.

 

[marcus]

Originally posted by Lonewolf
Does Minkowski space necessarily have the metric

/-1 0 0 0 \
| 0 1 0 0 |
| 0 0 1 0 |
\ 0 0 0 1 /

or are there other Minkowski spaces? Just want to be sure that this is the only one.

maybe if Lonewolf asks more questions and selfAdjoint takes over some explaining, as in this case just now, this could be quite an enjoyable thread. Since we are on the subject of 4x4 matrices and I have this copyable format I will show you a horrible 4x4 monstrosity

we know that SO(3,1) is the proper lorentz group of 4x4 real matrices that have det =1 and preserve this bilinear form that LW just wrote

we also know that there is a group homomorphism--the "double cover"
φ : SL(2, C) --> SO(3,1)

so that we very often perhaps more times than not USE the 2x2 complex matrices! But WHEN WAS THE LAST TIME YOU ACTUALLY SAW THAT MORPHISM? Can you off-hand write down how 4 complex numbers transform a vector of 4 real numbers?

Maybe someone has a prettier formula for the double cover homomorphism and would like to write it down, but here is mine and it is ugly as hell.

a matrix in SL(2, C) is 4 complex numbers a,b,c,d, ad-bc = 1

/ a  b \
\ c  d /

and it gets mapped to

/(aa*+bb*+cc*+dd*)/2   (aa*-bb*+cc*-dd*)/2  -Im(ab*+cd*) Re(ab*+cd*)\
|(aa*+bb*-cc*-dd*)/2   (aa*-bb*-cc*+dd*)/2  -Im(ab*-cd*) Re(ab*-cd*)|
|Im(ac*+bd*)           Im(ac*-bd*)           Re(ad*-bc*)  Im(ad*+bc*)|
\Re(ac*+bd*)          Re(ac*-bd*)         -Im(ad*-bc*)   Re(ad*+bc*)/

and this is in SO(3,1) and preserves that diag(-1,1,1,1) bilinear form.

 

[selfAdjoint]

Cool! I've never seen it before. And no wonder!

 

[marcus]

Originally posted by selfAdjoint
Cool! I've never seen it before. And no wonder!

selfAdjoint, I must confess that I copied out of Mark Naimark's "Normed Rings" he being the Naimark of Gelfand-Neumark (various spellings). It might be presented more elegantly by a modern author, perhaps you have such a description in one of your books. There is a blood-curdling directness about this Russian-circa 1960 way of brute-force writing out the matrix.
I concur----no wonder we rarely see it presented so!

Note the "double cover" feature that both

/ a  b \
\ c  d /       and its negative

/ - a  - b \
\ - c  - d /

get mapped to the same 4x4 real matrix

 

[marcus]

the basic GR formalism---rovelli page 21

[edit: Lethe has pointed out some lack of clarity so I went back to my 9 August (8AM) post and edited to reduce the chance that my paraphrase of rovelli is the cause. I will edit here too if I can see how to.]

M is a 4D manifold, points of M are labeled m.
systems of coordinates labeled x = (x0, x1, x2, x3)
the tangent space T_m at each point is identified with Minkowski space M

There are two important one-forms-----e and ω
e is the gravitational field and ω is the connection.

At each point e:T_m --> T_m
is a vector-valued one-form.

[but somehow T_m is identifed with Minkowski space M, so that pointwise e is also a map T_m -->M. This is currently giving me problems understanding]

If you pick coordinates x = {x^μ} you can look
at e expressed as a bunch of numbers
and think of it as a jacobian matrix of a change of coordinates to free-fall coordinates as he describes page 42
or you can look at the same thing with the indices showing:
e^α(x) = e^α_μ dx^μ


Another oneform, ω, has values in the Lorentz algebra so(3,1). The so(3,1) matrices are skewsymmetric. At any point in any coordinates ω^{αβ} = - ω^{βα}

Following rovelli nearly verbatim, the connection omega defines a gauge covariant exterior derivative D on forms. For example, given a vectorvalued one-form u^α

D u^α = du^α + ω^α_β /\ u^β

A torsion two-form T^α =
D e^α = de^α + ω^α_β /\ e^β

Given e, we can set T = 0 and solve for ω. that is, the gravitational field e determines a unique torsion free connection ω

The pair (e, ω) are the new variables or Ashtekar variables. Restricted to a 3D manifold the "triad" and the connection are written (E, A), familiar in LQG, but these are the
4D versions. The 4D classical version of the gravitational field should probably be thought of like the trajectory of a particle. Something that quantum mechanics teaches us does not exist.
Instead we have transition probabilities---presumably calculable as a sum-over-histories as in a Feynman path-integral---between 3D states. But in the classical situation a whole 4D trajectory is calculated when the field equation is solved over the 4D manifold M.

[edit: in part I am paraphrasing raw material from rovelli here which I have not assimilated. Some of it is giving me problems.
Will keep trying.]

What impresses me about this right now is that all these equations are differential form equations----oneforms and twoforms. And since they are vector and matrix valued forms, I have to figure out what the wedge of two of them means.
I will post this and see how it looks, maybe edit later.

 

[lethe]

marcus:

i didn t know what rovelli was talking about with minkowski vectors versus lorentz vectors, indices of which he labels with greek letters at the beginning of the alphabet and the middle of the alphabet, respectively (unless i ve got it backwards).

can you explain that to me?

 

[marcus]

Originally posted by lethe
marcus:

i didn t know what rovelli was talking about with minkowski vectors versus lorentz vectors, indices of which he labels with greek letters at the beginning of the alphabet and the middle of the alphabet, respectively (unless i ve got it backwards).

can you explain that to me?

Lethe,
first of all I must make sure my paraphrase is not contributing to the confusion. I will go back and clean up the sloppy paraphrasing. Maybe selfAdjoint will help.

secondly, I hope you were able to print off the relevant pages of rovelli's draft book. it's better to focus on what he says than my (possibly mistaken) version of it.

things will hopefully get clearer with time. I just recently encountered the book---for all I know it wasnt even on the web prior to 1 August. and it is a draft: rovelli himself may eventually have to clear up some ambiguities of wording if there are some on his part

what impresses me is that he introduces GR (not via the metric but) using the ashtekar "new variables" right from the start. and he makes the frame field intuitive
as a change of coordinates to coordinates which are locally inertial or free-fall
I found the heuristic presentation useful on pages 41, 42 (large cloud of galaxies, each one has a locally inertial frame---formulas all very basic and Newtonian---only the galaxy in the center of the cloud has coordinates that are not in fact accelerating but you can't tell if you are at the center or off to one side)

I will try to re-edit my earlier paraphrase to reduce the confusion. Please tell me if you are looking at rovelli's actual draft and if you can point to particular things he says.

 

[lethe]

Originally posted by marcus

I will try to re-edit my earlier paraphrase to reduce the confusion. Please tell me if you are looking at rovelli's actual draft and if you can point to particular things he says.

i am.

 

[marcus]

Originally posted by lethe
i am.

I'm glad of that, for several reasons.
I can't understand paragraph at top of page 44.
There is the idea of a "principal G-bundle" and G here
would be the Lorentz group
so one would say "principal Lorentz-bundle"
indicating what group is acting on the fibers

and the fiber seems to be identified with minkowsi space

I feel sure that with some patience we can get clear
about it but for now I regret to say I cannot give
a good paraphrase

 

[Lonewolf]

Given e, we can set T = 0 and solve for ω.

Sorry, how can we set T = 0? Can the torsion take an arbitary value? Or is this just so we can get a torsion free connection, and if so, what are the advantages in that? Won't it be affected if torsion is introduced?

 

[marcus]

Originally posted by Lonewolf
Sorry, how can we set T = 0? Can the torsion take an arbitary value? Or is this just so we can get a torsion free connection, and if so, what are the advantages in that? Won't it be affected if torsion is introduced?

I think you are right
0. with an arbitrary connection the torsion can take
arbitrary values.
1. we set T=0 to solve for a particular connection which is THE
connection compatible with e in the sense that it is torsion free
2. the pair (e,ω) is a solution of the field equation for all spacetime given certain conditions (say involving matter) and changing the conditions will affect the solution

To quote from page 22 just before equation 2.6,
"A tetrad field e determines uniquely a torsion free spin connection ω[e], called compatible with e;
that is ... determined by..."

It looks as if ω is just a convenient auxilliary with all the essential information already contained in the gravitational field e

Notice on page 25, equation 2.35, he gives the
"Lagrangian of the world" which includes a gravity contribution
S_GR[e, ω]
he obviously likes having ω around, it must make writing equations easier, even though unless I'm mistaken it is redundant since you can derive it from the field.

BTW there is a typo--the letters GR reversed

 

[marcus]

Originally posted by lethe
marcus:

i didn t know what rovelli was talking about with minkowski vectors versus lorentz vectors, indices of which he labels with greek letters at the beginning of the alphabet and the middle of the alphabet, respectively (unless i ve got it backwards).

can you explain that to me?

It's a shock encountering all this notation.
The importance of keeping track of the indices is impressing
itself on me. I am becoming keenly aware that
α and β are "internal Lorentz" indices
and μ (presumably with other mid-alphabet ones) is
a tangentspace and manifold coordinates index.

The internal Lorentz indices are very interesting and I will
discuss them in a moment, but first let's dispose of the mundane μ! You get this index whenever you choose a coordinate patch on the manifold---so partials, and differential dx's as well, are indexed with that

∂_μ , a basis for the conventional tangent space

dx^μ

With only this much machinery we could have the usual sort of fun with tangent and cotangent bundles---and could do many nice things

but there would be no Lorentz group action engaged with it.

Somehow, we have to tie in the group. Permanently enmesh it in our affairs. It seems that to do this we must permanently and more or less arbitrarily choose an identification of the tangent space with Minkowski space.

I am unsure about the ramifications of this, but it seems as if we must once and for all choose a basis for the tangent space. Then we can identify it with M
This gives some machinery to use in calculation. This is what "internal indices" are about.
There is a diag(-,+,+,+) metric on M that can be used for raising and lowering indices
and there is a pre-decided group action setup. I think of it as a utility more than anything else. Like getting your DSL line hooked up.

I believe that selfAdjoint is thoroughly at home with these "internal indices" and could say a few reassuring words that would make the situation more agreeable.

But I still feel that it is awkward to have both alpha and mu---distinct sets of indices---and I would like to understand it in terms of a G-bundle, or if necessary in terms of two bundles in association. My suspicion is that bundle language offers a possible way to say these things nicely.

But meanwhile, as we wait for some of the fog to clear, isn't it great that there are all these one-forms and two-forms? the universe seems to be made out of them!

 

[jeff]

Originally posted by lethe
marcus:

i didn t know what rovelli was talking about with minkowski vectors versus lorentz vectors...

can you explain that to me?

Originally posted by marcus
... ω is just a convenient auxilliary with all the essential information already contained in the gravitational field e...it must make writing equations easier, even though unless I'm mistaken it is redundant since you can derive it from the field.

The point of these extra indices is to allow coupling of spinors to gravity:

The standard formulae of GR - i.e. those using g rather than e - require that matter fields form representations of GL(n,R) under coordinate transformations. For example, V′^μ → V′^μ = Z^μ_νV^μ in which the matrix Z^μ_ν = ∂x′^μ/∂x^ν is an element of GL(n,R). Now, GL(n,R) always gives an SO(n) rep by restriction since SO(n) is a subgroup of GL(n,R). But spinors form reps of SO(n) which do not arise from reps of GL(n,R). Therefore we need a modified framework in which Z^μ_ν is replaced by an SO(n) matrix (or SO(n-1,1) matrix, depending on the signature).

This is achieved by using the principle of equivalence to introduce a local lorentz symmetry on the spacetime manifold, that is, a lorentz symmetry in the tangent space at each of it's points. Explicitly, we replace the metric g_μν(x) by the vierbein e^a_μ(x) which satisfies η_abe^a_μ(x)e^b_ν(x) = g_μν(x) where η_ab is the minkowski metric. The vierbein transforms under local lorentz transformations as e′^a_μ(x) = Λ^a_b(x)e^a_μ(x) in which the matrix Λ^a_b(x) is an element of the homogeneous lorentz group SO(n-1,1) and replaces Z^μ_ν. Thus, since spinors do form reps of the lorentz group, we simply couple them to gravity by tying them to the local lorentz indices.

In complete analogy to yang-mills, for this lorentz symmetry to be a true local symmetry requires the introduction of an associated connection ω called the spin connection. In order that the content of GR not be modified, we simply demand that the vierbein satisfy a compatibility condition analogous to that satisfied by the metric: D_μe^a_ν = ∂_μe^a_ν-Γ^λ_μνe^a_λ+ω_μ^a_be^b_ν.

With the aid of the spin connection it is simple to couple spinors to gravity. For example, the covariant derivative of a spinor ψ is just D_μψ = ∂_μψ+(1/2)ω_μ^abΣ_abψ with Σ_ab being the generators of the lorentz group in the spinor representation. This is completely analogous to yang-mills.

A few comments. The vierbein may be used to couple tensorial indices to either spacetime or local lorentz indices, it's your choice. But coupling to the latter type allows us to take better advantage of the machinery developed for use with yang-mills theories, and in fact this is the origin of the power of the ashtekar reformulation of GR (recall that the canonical pair (Aⁱ_a(x),E_i^a(x)) in LQG are viewed as a yang-mills connection and it's conjugate electric field). In particular, the critical SU(2) symmetry of LQG just denotes the freedom we have in choosing a vierbein satisfying e^a_μe_aν = g_μν. Finally, some people argue that the gauge group of GR is really local lorentz symmetry and not diffeomorphism invariance which isn't a gauge symmetry in the conventional sense.

 

[lethe]

Originally posted by jeff

The vierbein transforms under local lorentz transformations as e′^a_μ(x) = Λ^a_b(x)e^a_μ(x) in which the matrix Λ^a_b(x) is an element of the homogeneous lorentz group SO(n-1,1) and replaces Z^μ_ν. Thus, since spinors do form reps of the lorentz group, we simply couple them to gravity by tying them to the local lorentz indices.

oops, a little typo there, you need a b on the RHS. no problem though.

ok, so i should be thinking of a and b as analogous to the indices counting the components of the connection of the gauge group in yang-mills theory...? the internal gauge degrees of freedom.

ok, but umm... i don t see how we get spinors out of this. wouldn t a and b have to be some kind of spinorial indices? i mean, Λ is not a member of the lorentz group, it is a member of an irrep of the lorentz group right?

wait... i think i get it. we need the gravitational field to transform like SO(n) (or the lorentz group), not GL(n), because we can only get spinorial (is that the correct word? spinorial? it looks funny...) representations from SO(n), not GL(n).

umm.. so why can t i get a spin representation from GL(n)? i mean, it contains SO(n) as a subgroup, so can t i have a represent of GL(n) for each rep of SO(n)?

no, wait. each representation of the quotient group G/H induces a representation on G. i guess it s not true that reps of H induce reps of G. nevermind. (plus, i don t know if SO(n) is a normal subgroup)

In complete analogy to yang-mills, for this lorentz symmetry to be a true local symmetry requires the introduction of an associated connection ω called the spin connection. In order that the content of GR not be modified, we simply demand that the vierbein satisfy a compatibility condition analogous to that satisfied by the metric: D_μe^a_ν = ∂_μe^a_ν-Γ^λ_μνe^a_λ+ω_μ^a_be^b_ν.

and the familiar formula for Γ could be derived from this, and the relation between e and g, i assume.

what is a good text to read more about this notion of e as the field instead of g? i found rovelli too brief.

With the aid of the spin connection it is simple to couple spinors to gravity. For example, the covariant derivative of a spinor ψ is just D_μψ = ∂_μψ+(1/2)ω_μ^abΣ_abψ with Σ_ab being the generators of the lorentz group in the spinor representation. This is completely analogous to yang-mills.

A few comments. The vierbein may be used to couple tensorial indices to either spacetime or local lorentz indices, it's your choice. But coupling to the latter type allows us to take better advantage of the machinery developed for use with yang-mills theories, and in fact this is the origin of the power of the ashtekar reformulation of GR (recall that the canonical pair (Aⁱ_a(x),E_i^a(x)) in LQG are viewed as a yang-mills connection and it's conjugate electric field). In particular, the critical SU(2) symmetry of LQG just denotes the freedom we have in choosing a vierbein satisfying e^a_μe_aν = g_μν. Finally, some people argue that the gauge group of GR is really local lorentz symmetry and not diffeomorphism invariance which isn't a gauge symmetry in the conventional sense.

SU(2)? i thought we decided that the symmetry group was SO(n-1,1)?

anyway, thanks for that explanation. it helped a lot.

 

[marcus]

Just a brief item, not to break the flow of conversation.

continuing to try to understand Rovelli's notation
I found it refers to some standard constructions in differential
geometry----a "principal bundle" and "associated bundle"

the fiber of the principal bundle looks like the Group (e.g. SO(1,3) or GL(4)) and specially useful examples of principal bundle are the "frames bundle" and the "bundle of bases" (of the tangent space at each point).

The "associated bundle" to a principal is particularly interesting in our case because if one is given a group action on some space (e.g. SO(1,3) acting on Minkowski space) one gets an associated bundle whose fiber is the space of the action.

So there is a seemingly rather natural construction that gives two closely related bundles----one where the fiber looks like the Lorentz group and the other where the fiber looks like Minkowski space.

Rovelli has this basic setup and so he has a 1-form (defined on ordinary tangent space as usual but) with values in Mink. and he also has a 1-form with values in G. And he can "multiply" the two forms together by having the G-value act on the Mink. value.

Some educational material is on EricWeisstein MathWorld: articles on "principal bundle" "associated bundle" "frame bundle", by Tod Rowland.

Also google search with keywords such as "associated fiber bundle" gives SPR articles by Baez and others.

I will try to keep this post to a minimum and put the definitions
and links in a "principal bundle/bundle of bases" footnote.

 

[lethe]

Originally posted by marcus
----one where the fiber looks like the Lorentz group and the other where the fiber looks like Minkowski space.

ahh... i see. GR as i know it is formulated as geometry on a tangent vector bundle of some metric space.

with rovelli, we are instead working with a G-bundle instead?

i m not exactly sure what a "principle" bundle is, though i know what a bundle is. i guess i ll take a look at your other thread, and any questions can go there.

 

[jeff]

Originally posted by lethe
ok, so i should be thinking of a and b as analogous to the indices counting the components of the connection of the gauge group in yang-mills theory...? the internal gauge degrees of freedom.

Yes, local lorentz indices should be viewed as internal.

Originally posted by lethe
Λ is not a member of the lorentz group, it is a member of an irrep of the lorentz group right?

Λ(x) is a local lorentz transformation, i.e. an element of SO(3,1) on the tangent space at spacetime point x.

Originally posted by lethe
i don t see how we get spinors out of this. wouldn t a and b have to be some kind of spinorial indices?

No. I should've shown the spinorial indices in the example above as follows: D_μψ_α = ∂_μψ_α+(1/2)ω_μ^ab[Σ_ab]_{αβ}ψ^β. The main point is that if we list all the reps of GL(n), among them we'll find reps of, say, SO(n-1,1), but none of these are spinorial. So we "do SR in the tangent spaces" at each point where we can couple spinors as reps of the lorentz group induced by the principle of equivalence as a local gauge group and not merely as a restriction of GL(n) on the spacetime manifold.

Originally posted by lethe
SU(2)? i thought we decided that the symmetry group was SO(n-1,1)?

This point was specific to LQG, but I'd been discussing covariant GR in the main body of the post. Sorry about the confusion, that was my fault.

Originally posted by lethe
and the familiar formula for Γ could be derived from this, and the relation between e and g, i assume.

Yes.

Originally posted by lethe
what is a good text to read more about this notion of e as the field instead of g? i found rovelli too brief.

The main points are laid out in Weinberg's or Wald's texts on GR, as well as the text's on string theory by green schwartz and witten, polchinski, or johnson, and also in some text's on supersymmetry.

 

[jeff]

Originally posted by lethe
i m not exactly sure what a "principle" bundle is...

It's a bundle whose fibre is it's structure group.