The Mathematics of LQG Theory: Topology, Manifolds, and More

[MathematicalPhysicist]

i think i asked it before already, but if i recall correctly no one has given me a specific detailed answer.
what i want to know is what maths does LQG theory use?
i mean do you need to know topology,manifolds, what are the main subjects that needed to be learned?

btw, as far as i know (im not really up to date here) the theory of LQG as of now isn't the same as say before 5 years ago, right? what changes had occurred during the past 5 years?

p.s
i know the irony, that my username is lqg, but quite frankly i don't know what currently is going on with the theory.

 

[marcus]

i think i asked it before already, but if i recall correctly no one has given me a specific detailed answer.
what i want to know is what maths does LQG theory use?
i mean do you need to know topology,manifolds, what are the main subjects that needed to be learned?

as far as I know (others may correct me or add to this) the simplest way to state the main prerequisites is to say you need whatever you need to understand GR and QM

that is, if you do a good junior/senior undergrad course in GR and same in QM you will be exposed to many, though not all, of the math tools that you will need.

to go one step better, in preparation, acquire some of the math toolkit that you'd meet in a QFT course or textbook----other terms that come to mind are "fiber bundles", "Lie algebras", "group representation". Sounds awfully abstract doesn't it?

My take on it is that the math tools you need are only TRIVIALLY DIFFERENT FROM THOSE YOU NEED TO LEARN ordinary GEN REL AND QFT TO THE SAME LEVEL. There are some novel technical devices---you won't see a SPIN NETWORK or a spinfoam in conventional QFT courses, but those things are normally explained by the Loop/Spinfoam authors.

I can think of one exception, which I will explain in a minute.

btw, as far as i know (im not really up to date here) the theory of LQG as of now isn't the same as say before 5 years ago, right? what changes had occurred during the past 5 years?

I DONT THINK THE MATH PRE-REQUISITES HAVE CHANGED IN THE PAST 5 YEARS. With the exception of this one thing I will tell you in a minute, whatever math tools and sophistication it took to read Loop/Spinfoam 5 years ago, it takes about the same bunch of stuff today. Like I say advanced undergrad GR and QM plus whatever you can manage from QFT.
Loop/Spinfoam QG is not the SAME as conventional physics, but it uses a similar bag of math tools.
And, as you correctly say, Loop/Spinfoam QG have indeed changed and progressed a lot in the past 5 years, but AFAIK you still need about the same bag of math tools. The field can progress without the math pre-requisites changing much.

The one exception I can think of is the DOUBLY SPECIAL RELATIVITY stuff.
"DSR" also called "deformed special relativity". This involves quantum groups, Hopf algebras, tools overlapping with what is encountered in Non-commutative Geometry.

this represents a problem. I don't think you would meet the math tools you need to understand DSR just by doing a physics major or some beginning grad student courses. Pedagogically, it is an unsatisfactory situation. Our elders and betters have not prepared the ground for us yet. It is fairly new. I don't know what would be an efficient way to approach the subject.

but DSR is not BASIC to non-string QG or to Loop/Spinfoam QG in particular. It is currently more of a "side-show"----an auxiliiary to the main circus.

that reminds me, even tho DSR is not an essential PREREQUISITE for post-string QG, we should try to have a tutorial introduction to it, or pester John Baez to point us to the right tutorial introduction.

the reason is that probably the most exciting new results of Loop/Spinfoam are of the form "Loop/Spinfoam IMPLIES DSR"
and "DSR IMPLIES energy-dependent speed of light"
and "energy-dependent speed of light can be TESTED in 2007".
and this is CONTROVERSIAL---there are people who argue against the alleged implications. It is a hot topic. I will restate these controversial implications:

Loop/Spinfoam => DSR

DSR => energy-dependent speed of light

energy-dependent speed of light TESTABLE in 2007

this appears to mean that a considerable chunk of QG models run the risk of falsification near-term
=====================
for a non-technical discussion of DSR and much else, there is Smolin's new book.
it is hugely successful in explaining a whole lot of stuff about both string and non-string QG without using equations, it is amazing how clear it comes across in non-technical language.

for more technical treatment there is Smolin's paper "generic predictions" which you can find on arxiv. and the references to other papers that he gives.

however DSR is still, in my estimation, very rough and hairy. we badly need a "nice" introduction to it that could go into some kind of tutorial.

 

[kakarukeys]

i think i asked it before already, but if i recall correctly no one has given me a specific detailed answer.
what i want to know is what maths does LQG theory use?
i mean do you need to know topology,manifolds, what are the main subjects that needed to be learned?

btw, as far as i know (im not really up to date here) the theory of LQG as of now isn't the same as say before 5 years ago, right? what changes had occurred during the past 5 years?

p.s
i know the irony, that my username is lqg, but quite frankly i don't know what currently is going on with the theory.

You are glad you meet me.
1. Differential Geometry (A must) the more proficient the better, this includes manifold, differentiable forms, connection, curvature, cartan's theory (tetrad formalism), holonomy, a bit of topology, a bit of fiber bundle theory.
2. Functional Analysis (A must) without this you couldn't understand how gravity is quantized to obtain LQG, this includes topological vector space, hilbert space, distribution theory, rigged Hilbert space, C*-algebra, abit of measure theory.
3. Group theory, represention theory, without this you couldn't understand self-dual connection, spin network, this includes Lie group, Lie Algebra, Schurs Lemma, CG-decomposition, Peter-Wyl Theorem.

 

[kakarukeys]

Oh one more thing:
http://sps.nus.edu.sg/~wongjian/lqg.html
this webpage gives you the quickest path to learn LQG.

 

[selfAdjoint]

The quantization of LQG, in the Ashtekar formalism, requires a lot of beyond-undergraduate work with Hilbert spaces and operator algebras. I don't know if there are specific courses on this material, as versus seminars, or textbooks, other than, e.g., Thiemann's text on LQG which is online, and I believe linked in one of Marcus's bibliographic threads. Ashtekar-variables LQG theorists also tend to review the material they are going to use in their longer papers. But I have found that you need a deeper understanding of the material than you get from this kind of casual exposure, in order to make sense of the critiques of LQG offered by leading string theorists.

 

[neutrino]

Here's something pictorial...

http://math.ucr.edu/home/baez/PUB/toe.gif

...and this is just to learn the "basics."

 

[MathematicalPhysicist]

well, thanks to all.

 

[f-h]

That's a great list kakarukeys, if I come across any of the missing things or nice additions I'll keep it in mind!

 

[ccdantas]

I hope http://christinedantas.blogspot.com/2006/02/basic-curriculum-for-quantum-gravity.html" is of some help...

Christine