How area in LQG transforms under Lorentz boosts

[marcus]

It is an interesting issue.
Right at the end of the seminal paper of Livine (Projected Spin Networks for Lorentz connection: Linking Spin Foams and Loop Gravity)
that selfAdjoint recently referred to as "Projected..."
at the end of the Conclusions and Outlooks section, he says:

"Finally, using an explicitly covariant formalism opens the door to the systematic study of space-time related issues such as transformations of areas under Lorentz boosts, as studied in LQG[23]."

this shows that Livine thinks the issues raised by Rovelli and Speziale in reference [23] are important ones, so I looked at their paper, which is 12 pages and dated May 2002: "Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction."

So in this thread I'm talking about a 12-page paper ("Reconcile...") and a 15-page paper ("Projected...")

http://arxiv.org/gr-qc/0205108
http://arxiv.org/gr-qc/0207084

Naively stated the concern is that, since quantum gravity has a minimal length (or area), it must be incompatible with local coordinate change to a boosted observer who would see lengths contracted.
This may have encouraged the illusion that loop gravity is doomed to fail. It may seem self-evident to some people that any theory with a minimum length can't be right because it can't survive boosts. So this connection to controversy makes the issue extra piquant.

But it is kind of interesting anyway, apart from that. How is a surface (whose area is to be observed) determined in a diffeo-invariant theory? You have to have some matter---the area is a physical desktop, say. And then how about the spectrum of the operator which corresponds to observing that desktop's area? How does the spectrum change, if at all, under boosts? Notice that the area can change without the spectrum of the operator changing---at least no a priori reason it couldn't AFAIK. Anyway it's interesting.

So I want to look at these two papers, one of which is by way of being a reply to the other: Livine "Projected..." responds in a certain sense to Rovelli/Speziale "Reconcile..."

 

[marcus]

From Rovelli/Speziale's introduction:

"Here, we show how the apparent conflict between Lorentz contraction and Planck scale discreteness is resolved in loop quantum gravity [3] (for a review and extended references, see [4,5].)

Within loop quantum gravity, a minimal length appears characteristically in the form of a minimal (nonzero) value A₀ of the area of a surface [6,7]. Here we show that in loop quantum gravity a boosted observer O' does not observe a Lorentz contracted A₀. The minimal (nonzero) area that the boosted observer O' can observe is still A₀. We show that Planck scale discreteness is compatible with a certain implementation of local Lorentz invariance, and we study the transformation properties of the area operator under an infinitesimal local boost."

I think what matters most here is the intuitive grasp gained by considering this. It is not just a technical issue. Here's what Rovelli/Speziale say immediately after that:


"A. The basic idea

The key to understand how this may happen is the fact that in loop quantum gravity a minimal length does not appear as a fixed structural property of space geometry. Space geometry, indeed, has no fixed structural property at all in this approach. The geometry of space comes from a quantum field, the quantum gravitational field. Therefore the observable properties of the geometry, such as, in particular, a length, or an area, are observable properties of a quantum physical system. A measurement of a length is therefore a measurement in the quantum mechanical sense. Generically, quantum theory does not predict an observable value: it predicts a probability distribution of possible values..."

That's one of the things I like about Rovelli---that he would begin a paper with a heading like "The basic idea".

Given two observers O and O', and a surface like that of a physical desktop, some real material thing moving in spacetime, we have two different surface area operators A amd A' corresponding to the two different observers measuring the area of the desktop. The technical point, says the authors, is that these two operators do not commute.

 

[marcus]

That's the technical point but there's an intuitive one that goes like this:

People naively imagine that LQG has a picture of space as woven of little loops, like medieval chain mail, or as a gigantic irregular latticework or spiderweb.

Ashtekar likes the word "polymer" but he doesn't say space is a polymer, he says the EXCITATIONS of geometry, the quantum states of the gravitational field, the wavefunctions of geometry, are polymeric.
There is a subtle difference

A drumhead can be imagined as smooth, but the different ways the drumhead can be excited and made to vibrate can be depicted and listed as rather latticey or polymer-looking diagrams. In a rough analogy, the mode of excitation of space is not the same as space--one may be a stick-figure and the other not.

Just to be sure, I will quote how Rovelli/Speziale say it:

...The key to understand how this may happen is the fact that in loop quantum gravity a minimal length does not appear as a fixed structural property of space geometry. Space geometry, indeed, has no fixed structural property at all in this approach. The geometry of space comes from a quantum field, the quantum gravitational field. Therefore the observable properties of the geometry, such as, in particular, a length, or an area, are observable properties of a quantum physical system...

If people mistakenly believe that loop gravity says space IS a spin network or that spacetime IS a spin foam (the 4D version of a spin network), they may well enter into a critique on the basis of that misconception.

Loop gravity can be formulated with the gauge group SL(2,C) or with the more restricted gauge group SU(2). Spin foams are formulated in various styles including the Lorentzian Barrett-Crane model. Loop gravity tends to have the same kind of symmetry as GR---the use of networks to define an orthonormal basis of quantum states need not interfere with that, or the use of spin foams to describe quantum histories of one state evolving into another.

Livine's thesis is interesting because it constructs COVARIANT loop gravity using the non-compact gauge group SL(2,C), and then uses the covariant theory as a bridge between ordinary loop gravity and spin foam models.

So I want to look at Livine's paper "Projected Spin Networks for Lorentz connection: Linking Spin Foams and Loop Gravity" and see what he thinks about this and how he continues the train of thought started by Rovelli/Speziale

 

[selfAdjoint]

Excellent discussion and important insights, for us groundlings. Please carry on!

 

[marcus]

Originally posted by selfAdjoint
Excellent discussion and important insights, for us groundlings. Please carry on!

us groundlings what has our feet on the ground have to do our own explication these days---the elite are too busy to stop and explain

hope you had a good trip and that offspring was enjoyable road company

BTW I know you will step in and add clarifications whenever, and they are greatly welcomed

 

[marcus]

We have a straight shot from Livines
July 2002 "Projected..." to Alexandrov/Livine "SU(2) LQG seen from Covariant Theory" (which the earlier paper refers to as "in progress") and from there to Livine's thesis Chapter 8.

July 2002 "Projected..."
September 2002 "SU(2) LQG from Covariant"
September 2003 Thesis

I am told Livine is 23 (b.1980). He built up to his PhD thesis by doing papers with some pretty good people: Freidel, Oriti, I think Rovelli, Alexandrov, can't think who else---and he seems to have planned it well so all those papers provided groundwork for the thesis.

I'll tell you what is slowing me down here. I wonder how important it is to have the theory covariant (full Lorentz group) when it seems to work OK with SU(2). There seems to be an overarching covariant theory from which the SU(2) version can be derived by giving up some symmetry.

But at present the covariant theory looks to be more complicated. The spin networks are more heavily labeled, the measure is more trouble to define.

There is the thought that, yes it often pays to do things right from the start, but maybe this covariant loop gravity is still too new.
It may get streamlined and be more approachable a year or so from now.
And then on the other hand the very fact that it is new is a chance to get in on it and see the process of streamlining take place

The covariant theory (Livine, Freidel, Alexandrov and whoever else) is overarching and includes the spin foam model as well as customary Loop Gravity. That in itself is interesting. It's a big tent--it specializes down to two different lines of theoretical development.
I like that.
But I am slowed down by the additional complication. To take a simple example, the REPRESENTATIONS of SL(2,C) are labeled by (n,rho) where rho is a real number. This immediately makes me nostalgic for SU(2) where they were all finite dimensional, one rep for each integer or half-integer. The labels on the spin networks were just half-integers.

I guess what I would like is an as yet unstated "Livine's Theorem" which says that it is OK to use the SU(2) theory that I understand better. One of the big theorems nobody remembers how to prove but is very useful so one takes a deep breath and invokes it and says seven Hail Marys and all is well. Such is mathematics.

And another very strange thing is that in the covariant theory the Immirzi parameter seems to go away, it drops out of the equations and no longer seems physically meaningful. How can this be?

Anyway, here we are on the verge of becoming covariant--the theory is about to go into warp. Douglas Adams probably has some suggestions about how to prepare this.

 

[ranyart]

Originally posted by marcus


And another very strange thing is that in the covariant theory the Immirzi parameter seems to go away, it drops out of the equations and no longer seems physically meaningful. How can this be?

Anyway, here we are on the verge of becoming covariant--the theory is about to go into warp. Douglas Adams probably has some suggestions about how to prepare this.

Marcus, I think not?..the immirzi-parameter does not go away?

 

[marcus]

Originally posted by ranyart
Marcus, I think not?..the immirzi-parameter does not go away?

Right. There are some signs that it might. But in these new papers it plays a reduced role and doesn't disappear entirely.

I will try to snag a few page references about this.

 

[marcus]

Livine's thesis page 109

talking about covariant loop gravity: "Finally one obtains
a quantum theory and an area spectrum independent of the Immirzi parameter..."

(Livine's italics here, not mine")

Also in the september 2002 paper by Alexandrov and Livine
"SU(2) Loop Quantum Gravity seen from Covariant Theory"
on page 9:

"This means that there is a unique loop quantization preserving all classical symmetries of general relativity. Moreover for this choice of connection the area spectrum does not depend on the Immirzi parameter..."

the links are:
http://arxiv.org/gr-qc/0209105
http://arxiv.org/gr-qc/0309028

 

[marcus]

the crux of it

here's one way to look at it
in any theory you have a choice of what the variables are going to be
(changing the variables is familiar from calculus)
and so far most Loop Gravity research has been done using
the "Ashtekar-Barbero" connection as the main variable that describes the geometry of the spatial manifold being studied. The A-B connection is constructed using the Immi-Pa.

The A-B connection replaced the metric as the main way the geometry was represented and as the main thing one tried to quantize.
One wants to quantize "geometry" so for a while people tried to quantize the metric (which would have been great if it had worked) and then they shifted attention to quantizing the A-B connection.

But it was always recognized that there were other possible connections and in fact Ashtekar himself had proposed at least one alternative and other people occasionally tried out others.

In the end one wants to find a representation of the geometry---a form of the main variable of the theory---a connection, perhaps---that "Nature likes". She will show us that she likes it, when we finally arrive at it, by its simplicity and its success at resolving problems and getting results (like the spectra of area and volume operators, or like a bridge to other theories, like correct dynamics...) and when I say "we" I mean "they"

the Ashtekar-Barbero connection is a SU(2)-valued connection

Some people (dangerous radicals, revolutionaries?) have been engaged in constructing a SL(2,C)-valued connection
and they have succeeded in constructing such a connection which is really two (like twins or doppelgaengers) and one of the two does
not involve or anyway SEEMS not to involve the Immirzi
this is not necessarily bad thing, but definitely unsettling

 

[marcus]

As I get deeper into it I become more convinced that Livine's new brand of Loop Gravity is worth pursuing---might lead to a more intuitive approach

You fix attention on a directed graph Γ with E edges and V vertices. What could be simpler? Two numbers E and V---a bunch of points (V of them) connected by some lines (E of those). The graph is abstract, it does not have to be embedded in any continuum, the edges do not have any pre-ordained lengths. The graph is just a list of E relationships among V abstract entities, and it has no particular geometry.

The space of discrete G-valued connections on Γ is simply G^E/G^V
a space of E-tuples of group elements (g1, g2, ...gE) modulo
gauge equivalence. A gauge action amounts simply to the Adjoint
action of the gauge group G that you get by assigning a group element to each vertex.
Sometimes Livine and friends write G_Γ for the space of discrete G-valued connections on Γ, and sometimes alternatively A_Γ

The relaxing part of this approach, or one of its relaxing aspects, is that you don't have to LABEL the graph with irreducible representations and intertwiners. that part is PUT OFF until it is time to choose an orthonormal basis of a hilbertspace. Which for gods sake is already so technical that who is going to notice a little more jargon? And it is morally the right place because of Plancherel's theorem. You WANT to introduce the reps of G at that point because you want to write cylinder functions in some systematic basis. OK, but until it is actually time to construct a special efficient basis to catalog a special hilbertspace you do not have to worry about labeling edges with G-reps.

Another relaxing thing is you don't have to go thru differential geometry and manifolds to define a general purpose infinitesimal smooth "connection". NO thank heaven a discrete connection is a very elementary thing--merely some finite stick-figure graph with group elements on each stick telling how things twist and turn as you go along that segment.

Ultimately we are going to take an infinite number of these graphs Γ and say "Fockspace" and the infinitude of finite graphs in all the postures that they can diffeomorph into will give us a handle on the geometry of space---but for now all is peaceful and quiet

And Livine appears and with a polite smile he finesses into existence a MEASURE on the space of connections living on the graph.
Before you know it, a measure, to allow integrating gauge-invariant functionals, is established on

G_Γ= G^E/G^V

 

[marcus]

Originally posted by ranyart
Marcus, I think not?..the immirzi-parameter does not go away?

Ranyart, it is worth keeping an eye on the Immirzi parameter. A new thesis appeared today that has a section on "The role of the Immirzi parameter in spin foam models" [Oriti section 4.3.5 pages 167-169]

It is an unresolved question. Maybe the parameter is physically meaningful, or maybe it will go away with an improvement in how GR is quantized. Oriti is one of a younger generation of quantum gravity researchers (under 30s, like Livine, Louapre, Girelli, Bojowald, Sahlmann, Oeckl, Perez, Corichi,I can't remember all their names maybe Freidel belongs to that wave). I want to listen to what they are saying as possibly different from what more established but still comparatively young people like Rovelli Smolin Ashtekar.

Oriti's thesis "Spin foam models of quantum spacetime" is 340 pages so I will just quote what he says, instead of giving the link.

If the Immirzi does ultimately go away in some newer model of LQG this is both good and bad, or maybe neither good nor bad: just how it turns out. It is hard even to guess at this point. I will go find that quote.

 

[marcus]

Daniele Oriti "Spin Foam Models of Quantum Spacetime" (exerpt)

Here is an exerpt from pages 167-170 of Oriti's thesis:
-------------

4.3.5 The role of the Immirzi parameter in spin foam models

We would like to discuss briefly what our results suggest regarding the role of the Immirzi parameter in the spin foam models, stressing that this suggestion can at present neither be well supported nor disproven by precise calculations. ...

...

... Nevertheless, it suggests that a “canonical” interpretation of a spin foam might not be as straightforward as it is believed. Instead of taking as spatial slice an SO(3, 1) spin networks by cutting
a spin foam, we might have to also project the SO(3, 1) structure onto an SU(2) one; the resulting SU(2) spin network being our space and the background SO(3, 1) structure describing its space-time embedding.

...

...what seems to happen is that the symmetry is broken by a particular choice of quantization procedure, and that a fully covariant quantization, like the spin foam quantization or the manifestly Lorentzian canonical one, does not give rise to any one-parameter ambiguity in the physical quantities to be measured, i.e. no Immirzi parameter.

...

A crucial step is the appearance of the same kind of simple
spin networks that appear in the Barrett-Crane model in the context of covariant canonical loop quantum gravity, based on the full Lorentz group, obtained in [185], about which we will say more later on.
-----------------------------------------------

His reference [185] is probably to an ER Livine paper. I will check that and get back.
Yeah, reference [185] is to an article published in Phys. Rev. Series D this year by Alexandrov and Livine. The preprint link is
http://arxiv.org/gr-qc/0209105
"Loop Quantum Gravity as seen from the Covariant Theory"
we've talked about that article here at PF some