Is the SO(2) Connection in Ashtekar's General Relativity Real or Complex?

[marcus]

Loop gravity---two questions

A poster in another thread writes:

A question: Then the SO(2) connection used like a variable in Ashtekar's general relativity is a real connection or a complex connection? There are papers that say that is real and others that is complex...

I've just read that loop quantum gravity violates the "weak energy condition" at short distances, when the granularity of spacetime becomes significant. I've don't have the foggiest idea of what is the weak energy condition...
--------------------------

Comments invited.
My understanding is that Ashtekar's new variables for GR initially involved a complex connection and that physical solutions required imposing some additional conditions called "reality conditions".

The original suggestion for making the connection real from the start came, i believe, from Barbero and involved an undetermined (real number) parameter called the Barbero-Immirzi parameter.

I don't recall hearing about a "weak energy condition" that loop gravity fails to satisfy.

Anyone who can step into shed addtional light on either of these two questions (from Meteor) is most cordially welcome.

Meteor, I'm trying to use that other thread as a sticky in which to collect useful links and general information, conferences etc., source material. Let's start a thread here with your questions in case they lead to discussion.

 

[selfAdjoint]

The weak energy condition (WEC) is
$$u^\mu u^\nu T_{\mu\nu} \geq 0$$
At every point, where u is an arbitrary timelike vector and T is the momentum energy tensor.

Quoting from Matt Visser's wonderful book Lorentzian Wormholes:
"The physical significance of this condition is that it forces the local energy density as measured by any timelike observer to be positive."

 

[nonunitary]

Marcus,

Your understanding about the original Ashtekar connection being complex is correct. I think that Ashtekar himself considered the real connections from the beginning but with a B-I parameter equal to 1, since that is the self-dual connection for Euclidean gravity.

Barbero was the first to consider the free real parameter giving rise to the one parameter family of Ashtekar-Barbero, real, SU(2) connections.

Immirzi was the one to realize the ambiguity in the spectrum of the geometric operators. Thus the name of Barbero-Immirzi parameter.

The issue of the weak energy condition does not really make sense.
General relativity in its metric fromulation admits solutions that violate the WEC, and that does not make the theory patological or anything like that. The role of the WEC is a physical input that one requieres for a solution to be admissible and with an interpretation.

As far as I kinow there is not a precise implementation of WEC (or any other) in LQG (so far, which does NOT mean that it is not possible).

 

[marcus]

selfAdjoint and nonunitary,
Thanks to you both!

there may be more questions
I hope from Meteor or others
(including myself, I too have questions
but I cannot think of one just now )

it continues to bother me that there
does not seem to be a natural choice of
a beginning textbook in loop gravity.
I think Meteor alluded indirectly to this
lack when he mentioned getting in at
the Schaum Outline Diff. Geometry level.
the beginnings are a big hurdle and
anything that helps is worth sharing

 

[meteor]

I've read that LQG violates the weak energy condition in this paper:
"Contrasting quantum cosmologies"
http://arxiv.org/abs/gr-qc/0312045
Roughly, the Weak energy condition is a constraint that don't permit you travel faster than light. If it is violated, there's a causality problem
It seems that form part of a series of constraints that appear in general relativity, the other constraints take names like strong energy condition, null energy condition, and others

 

[nonunitary]

Originally posted by marcus
selfAdjoint and nonunitary,

it continues to bother me that there
does not seem to be a natural choice of
a beginning textbook in loop gravity.
I think Meteor alluded indirectly to this
lack when he mentioned getting in at
the Schaum Outline Diff. Geometry level.
the beginnings are a big hurdle and
anything that helps is worth sharing

Marcus, I agree with you that there is no beginning textbook
about LQG that I particularly like.
The new book by Rovelli is a very nice read but you have to
read many things before getting to LQG and even then, this simple things as real vs. complex, immirzi parameter and the like is not extremely clear.
Someone should write a concise introduction!

 

[selfAdjoint]

There are two levels that such a book could fill. One is the "Elegant Universe" level which apeals to the numerate but untrained public, and the other is a level that really doesn't exist in today's market that is aimed at the semi-trained group. This is a large public, if you can reach it. It's ten years since it was published that over a million Americans had taken calculus. If you could aim at people who have at least a dim memory of trig functions and the chain rule, you could maybe do something new and worthwhile.

What would you have to convey?

Networks are not the spacetime, their amplitudes are (string theorists have a similar misconception to overcome).

Basic function theory (Marcus, your neat explanation of the spectrum)

Basic Hamiltonian/Lagrangian theory.

Quik intro to complex analysis, up to $$e^{iz}$$ Cauchy thm? Continuation? (?)

Quik into to Lie groups and algebras (?) If we try to do representations we'll lose every body, but if you don't can you honestly present the subject? And if you can't motivate "Lie Algebra valued connection" how can you ever motivate the Ashtekar varaiables? And how can you do other than a schlock version without them?

Basic principle bundles and connections like the thread I tried to do but (natch) better.

Basic diff eq. Duals without Hodge - means only impressions.

The Ashtekar variables.

Sigh, Hilbert spaces. What a dense set is. And all that.

Course you could always settle for a schlock quicky book for next Christmas, lots of pictures and colored diagrams to give the buyers the illusion of understanding.

Then there's the undergraduate textbook. But my understanding was that was what Rovelli's book is.

 

[marcus]

Originally posted by selfAdjoint
Then there's the undergraduate textbook. But my understanding was that was what Rovelli's book is.

In his preface Rovelli puts a short paragraph (right at the end) saying who the book is for.

First of all it is for researchers interested in working in quantum gravity.

But also for "a good PhD student".

Thirdly, for scholars who are curious to learn about the field---but not necessarily to begin research in it.

He doesn't say anything about undergrads.

The way I read the relevant sentence or two in the preface he is targeting

1. postdocs and already established researchers who are interested in moving into the field

2. grad students looking for PhD thesis topics in loop gravity

3. people on the sidelines with curiosity and sufficient general knowledge, but no ambition to do original research
----------------------

Perhaps the next thing that could happen is someone in effect adds a few quantum chapters on at the end of an undergraduate textbook in General Relativity.

Take your favorite GR textbook (at the undergrad level) and imagine it re-issued in a new edition with a "part II, quantum GR" section at the end.

...I know... you have already indicated the difficult obstacles facing any treatment of loop gravity at the undergrad level. But that's one way to visualize how the problem could be addressed.

 

[marcus]

Oh yes, the million people who've taken calculus!
That is a real interesting group of readers.
Looks to me like there is room for soft-core
treatments of quantum gravity at several wide-audience levels.

Azimov would have already done one. Non-academic treatments
can be really smart books. I like your outline
for a non-textbook smart loop book.

 

[stephenlenane]

Unified Theory of Proportionality

I read the article. And God bless Scientific American for publishing those semi-annual 'special editions' which are always irresistibly amusing and annoying.

Loop Gravity reminded me of a mid-eighties theory of mine. In it, I proposed the boundary conditions of the universe were pure energy, in which matter simply condensed, not exploded into existence from some infinitely dense singularity... Simple and self-contained.

If you can imagine, a single moment; A line, a radius to a circle, and a universe created in an instant constructed of ubiquitous positively curved matter and the remaining negatively curved energy. Energy which does not orbit, but flows endlessly across this finite but unbounded terrain. Directed, but never fully contained.

An idea which required no messenger particles or multiple dimensions. A simply geometry which could be explored, and begin to address what no theory has ever attempted; Predicting, justifying the absolutes and relationships of matter and energy, and explaining why things are the way they are, (Why does E=MC2. Why is there gravity and electromagnetism? Why does metal conducts where wood does not...)

I'm not sure whether the full text is still available from the review archives, but if I've piqued anyone's curiosity, I'd be glad to discuss this further.

Thank you for your time,

Steve Lenane

 

[clothoid]

An intro "book" on spin foams

Daniele Oriti's PhD thesis is pretty good in my opinion...Check it out:

http://xxx.lanl.gov/abs/gr-qc/0311066

 

[MathematicalPhysicist]

meteor isn't the only one who asked about the issue of lqg vilation of GR postulate.

i too have asked about it, here's a link to thread i have opened:
https://www.physicsforums.com/showthread.php?s=&threadid=4193