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Sep 6, 2003, 7:27:53â€¯PM9/6/03

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Also available at http://math.ucr.edu/home/baez/week198.html

September 6, 2003

This Week's Finds in Mathematical Physics - Week 198

John Baez

I recently got back from a summer spent mostly in Hong Kong.

It was interesting being there. Since I wasn't there long,

most of my observations are pretty superficial. For example,

they have a real commitment to public transportation. Not

only is there a wonderful system of subways, ferries, buses

and green minibuses where you can pay for your ride using a

cool high-tech "octopus card", the local gangs run their own

system of *red* minibuses. These don't run on fixed schedules,

and they don't take the octopus card, but they seem perfectly

safe, and they go places the others don't.

Another obvious feature is the casual attitude towards English,

which is still widely used, but plays second fiddle to Cantonese

now that the Brits have been kicked out. Menus feature strange

items such as "mocked eel" and "mocked shark fin soup", which

bring to mind the unsettling image of a cook ridiculing hapless

sea creatures before cooking them. Also, perfectly nice people

wear T-shirts saying things that wouldn't be wise where I come

from, like

Lost Pig

or

I SEE

WHY

YOU SUCK

On a more serious note, it was interesting to see the effects

of the July 1st protest against Article 23 - an obnoxious piece of

security legislation that Tung Chee-Hwa was trying to push through.

About 8% of the entire population went to this demonstration. It

stopped or at least delayed passage of the current version of this

bill, and seems to have invigorated the democracy movement. Time

will tell if it leads to good effects or just a crackdown of some

sort. The police have placed a large order for tear gas.

While in Hong Kong, I received a copy of a very interesting book:

1) David Corfield, Towards a Philosophy of Real Mathematics,

Cambridge U. Press, Cambridge, 2003.

I should admit from the start that I'm completely biased

in favor of this book, because it has a whole chapter

on one of my favorite subjects: higher-dimensional algebra.

Furthermore, Corfield cites me a lot and says I deserve

"lavish praise for the breadth and quality of my exposition".

How could I fail to recommend a book by so wise an author?

That said, what's really special about this book is that it

shows a philosopher struggling to grapple with modern mathematics

as it's actually carried out by its practitioners. This is what

Corfield means by "real" mathematics. Too many philosophers of

mathematics seem stuck in the early 20th century, when explicitly

"foundational" questions - questions of how we can be certain of

mathematical truths, or what mathematical objects "really are" -

occupied some the best mathematicians. These questions are fine

and dandy, but by now we've all heard plenty about them and not

enough about other *equally* interesting things. Alas, too many

philosophers seem to regard everything since Goedel's theorem as

a kind of footnote to mathematics, irrelevant to their loftier

concerns (read: too difficult to learn).

Corfield neatly punctures this attitude. He calls for philosophers

of mathematics to follow modern philosophers of the natural sciences

and focus more on what practitioners actually do:

[...] to the extent that we wish to emulate Lakatos and

represent the discipline of mathematics as the growth of

a form of knowledge, we are duty bound to study the means

of production throughout its history. There is sufficient

variation in these means to warrant the study of contemporary

forms. The quaint hand-crafted tools used to probe the

Euler conjecture in the early part of the nineteenth century

studied by Lakatos in "Proofs and Refutations" have been

supplanted by the industrial-scale machinery of algebraic

topology developed since the 1930s.

He also tries to strip away the "foundationalist filter" that

blinds people into seeing philosophically interesting mathematics

only in the realms of logic and set theory:

[...] Straight away, from simple inductive considerations,

it should strike us as implausible that mathematicians

dealing with number, function and space have produced

nothing of philosophical significance in the past seventy

years in view of their record over the previous three centuries.

Implausible, that is, unless by some extraordinary event

in the history of philosophy a way had been found to *filter*,

so to speak, the findings of mathematicians working in core

areas, so that even the transformations brought about by

the development of category theory, which surfaced explicitly

in 1940s algebraic topology, or the rise of non-commutative

geometry over the past seventy years, are not deemed to merit

philosophical attention.

To me, it's a breath of fresh air just to see a philosopher

of mathematics *mention* non-commutative geometry. So often

they seem to occupy an alternate universe in which mathematics

stopped about a hundred years ago! Elsewhere in the book

we find interesting discussions of Eilenberg-MacLane spaces,

groupoids, the Ising model, and Monstrous Moonshine. One

gets the feeling that the author is someone one might meet

on the internet instead of the coffeehouses of fin-de-siecle

Vienna, and that he writes using a word processor instead of

a fountain pen.

The book consists of chapters on loosely linked subjects,

some of which seem closer to "real mathematics" than others.

The chapters on "Communicating with automated theorem provers"

and "Automated conjecture formation" are mildly depressing,

given how poor computers are at spotting or proving truly

interesting conjectures without lots of help from humans -

at least so far. True, Corfield describes how in 1996 the

automated theorem prover EQP was the first to crack the

Robbins conjecture. This states that a Boolean algebra is

the same as a set equipped with an commutative associative

binary operation "or" together with a unary operation "not"

for which one mind-numbing axiom holds, namely:

not(not(p or q) or not(p or not(q)) = p

All the rest of Boolean logic is a consequence! But proving

this seems more like a virtuoso stunt than the sort of thing

we working mathematicians do for a living. This is actually

part of Corfield's point, but I find it a somewhat odd choice

of topic, unless perhaps philosophers need to be convinced

that the business of mathematics is still a mysterious process,

not yet easily automated.

Apart from the one on higher-dimensional algebra, the chapters

that make me happiest are the ones on "The importance of

mathematical conceptualisation" and "The role of analogy in

mathematics".

The first is a marvelous study of the so-called "conceptual

approach" in mathematics, which emphasizes verbal reasoning using

broad principles over calculations using symbol manipulation.

Some people are fond of the conceptual approach, while others

regard it as "too abstract". Corfield illustrates this split

with debate over "groupoids versus groups", with the supporters

of groupoids (including Grothendieck, Brown and Connes) taking

the conceptual high road, but others preferring to stick with

groups whenever possible. As a philosopher, Corfield naturally

leans towards the conceptual approach.

The second is all about analogies. Analogies are incredibly

important in mathematics. Some can be made completely precise

and their content fully captured by a theorem, but the "deep"

ones, the truly fruitful ones, are precisely those that resist

complete encapsulation and only yield their secrets a bit at a

time. Corfield quotes Andre Weil, who describes the phenomenon

as only a Frenchman could - even in translation, this sounds like

something straight out of Proust:

As every mathematician knows, nothing is more fruitful

than these obscure analogies, these indistinct reflections

of one theory into another, these furtive caresses, these

inexplicable disagreements; also nothing gives the researcher

greater pleasure.

I actually doubt that *every* mathematician gets so turned on by

analogies, but many of the "architects" of mathematics do, and

Weil was one. Corfield examines various cases of analogy and

studies how they work: they serve not only to discover and

prove results but also to *justify* them - that is, explain why

they are interesting. He also examines the amount of freedom

one has in pushing forwards an analogy. This is a nice concrete

way to ponder the old question of how much of math is a free

human creation and how much is a matter of "cutting along the

grain" imposed by the subject matter.

The analogy he considers in most detail is a famous one between

number fields and function fields, going back at least to Dedekind

and Kummer. By a "number field", we mean something like the set

of all numbers

a + b sqrt(-5)

with a,b rational. This is closed under addition, subtraction,

multiplication, and division by anything nonzero, and the usual

laws hold for these operations, so it forms a "field". By a

"function field", we mean something like the set of all rational

functions in one complex variable:

P(z)/Q(z)

with P,Q polynomials. This is again a field under the usual

operations of addition, subtraction, multiplication and division.

Sitting inside a number field we always have something called the

"algebraic integers", which in the above example are the numbers

a + b sqrt(-5)

with a,b integers. These are closed under addition, subtraction,

multiplication but not division so they form a "commutative ring".

Similarly, sitting inside our function field we have the "algebraic

functions", which in the above example are the polynomials

P(z)

This is again a commutative ring.

So, an analogy exists. But the cool part is that there's a

good generalization of "prime numbers" in the algebraic integers

of any number field, invented by Kummer and called "prime ideals"...

and prime ideals in the algebraic functions of a function field

have a nice *geometrical* interpretation! In the example given

above, they correspond to points in the complex plane!

The analogy between number fields and function fields has been

pushed to yield all sorts of important results in number theory

and algebraic geometry. In Weil's hands it led to the theory of

adeles and the Weil conjectures. These in turn led to etale

cohomology, Grothendieck's work on topoi, and much more. And

the underlying analogy is still far from exhausted! But if we

ever get it completely nailed down, then (in the words of Weil):

The day dawns when the illusion vanishes; intuition

turns to certitude; the twin theories reveal their

common source before disappearing; as the Gita teaches

us, knowledge and indifference are attained at the same

moment. Metaphysics has become mathematics, ready to

form the material for a treatise whose icy beauty no

longer has the power to move us.

Or something like that.

Anyway, I hope this book shows philosophers that modern mathematics

poses many interesting questions apart from the old "foundational"

ones. These questions can only be tackled after taking time to

learn the relevant math... but what could be more fun than that?!

I also hope this book shows mathematicians that having a well-

informed and clever philosopher around makes math into a more

lively and self-aware discipline.

(The same is true of physics, of course. I listed a few good

philosophers of physics in "week190".)

Someday I'd like to say more about the analogy between number

fields and function fields, because I'm starting to study

this stuff with James Dolan... but it will take a while

before I know enough to say anything interesting. So instead,

let me say what's going on with spin foam models of quantum gravity.

I've already talked about these in "week113", "week120", "week128"

and "week168". The idea is to calculate the amplitude for spacetime

to have any particular geometry. An amplitude is just a complex

number, sort of the quantum version of a probability. If you know

how to calculate an amplitude for each spacetime, you can try to

compute the expectation value of any observable by averaging its

value over all possible geometries of spacetime, weighted by their

amplitudes. When you do this to answer questions about physics at

large distances scales, the amplitudes should almost cancel except

for spacetimes that come close to satisfying the equations of general

relativity. This is how quantum gravity should reduce to classical

gravity at distance scales much larger than the Planck length.

But in a spin foam model, a spacetime geometry is not described

by putting a metric on a manifold, as in general relativity.

Instead, it's described in a somewhat more "discrete" manner.

Only at distances substantially larger than the Planck length

should it resemble a metric on a manifold.

How do you describe a spacetime geometry in a spin foam model?

Well, first you take some 4-dimensional manifold representing

spacetime and chop it into "4-simplices". A "4-simplex" is

just the 4-dimensional analogue of a tetrahedron: it has 5

tetrahedral faces, 10 triangles, 10 edges and 5 vertices.

Then, you label all the triangles in these 4-simplices by numbers.

These describe the *areas* of the triangles. Here the details

depend on which spin foam model you're using. In the Riemannian

Barrett-Crane model, you label the triangles by spins j = 0, 1/2,

1, 3/2.... But in the Lorentzian Barrett-Crane model, which

should be closer to the real world, you label them by arbitrary

positive real numbers. Either way, a spacetime chopped up into

4-simplices labelled with numbers is called a "spin foam".

To compute an amplitude for one of these spin foams, you first use

the labellings on the triangles and follow certain specific formulas

to calculate a complex number for each 4-simplex, each tetrahedron,

and each triangle. Then you multiply all these numbers together

to get the amplitude!

In "week170", I mentioned some mysterious news about the Barrett-Crane

model. At the time - this was back in August of 2001 - my collaborators

Dan Christensen and Greg Egan were using a supercomputer to calculate

the amplitudes for lots of spin foams. The hard part was calculating

the numbers for 4-simplices, which are called the "10j symbols" since

they depend on the labels of the 10 triangles. They had come up with

an efficient algorithm to compute these 10j symbols, at least in the

Riemannian case. And using this, they found that the 10j symbols were

*not* coming out as an approximate calculation by Barrett and Williams

had predicted!

Barrett and Williams had done a "stationary phase approximation" to

argue that in the limit of a very large 4-simplex, the 10j symbols

were asymptotically equal to something you'd predict from general

relativity. This seemed like a hint that the Barrett-Crane model

really did reduce to general relativity at large distance scales,

as desired.

However, things actually work out quite differently! By now

the asymptotics of the 10j symbols are well understood, and they're

*not* given by the stationary phase approximation. If you want to

see the details, read these papers:

2) John C. Baez, J. Daniel Christensen and Greg Egan,

Asymptotics of 10j symbols, Class. Quant. Grav. 19 (2002) 6489-6513.

Also available as gr-qc/0208010.

3) John W. Barrett and Christopher M. Steele, Asymptotics of

relativistic spin networks, Class. Quant. Grav. 20 (2003) 1341-1362.

Also available as gr-qc/0209023.

4) Laurent Freidel and David Louapre, Asymptotics of 6j and 10j

symbols, Class. Quant. Grav. 20 (2003) 1267-1294. Also available as

hep-th/0209134.

The physical meaning of this fact is still quite mysterious. I could

tell you everyone's guesses, but I'm not sure it's worthwhile. Next

spring, Carlo Rovelli, Laurent Freidel and David Louapre are having

a conference on loop quantum gravity and spin foams in Marseille.

Maybe after that people will understand what's going on well enough

for me to try to explain it!

I'd like to wrap up with a few small comments about last Week.

There I said a bit about a 24-element group called the "binary

tetrahedral group", a 24-element group called SL(2,Z/3), and

the vertices of a regular polytope in 4 dimensions called the

"24-cell". The most important fact is that these are all the

same thing! And I've learned a bit more about this thing from here:

5) Robert Coquereaux, On the finite dimensional quantum group

H = M_3 + M_{2|1}(Lambda^2)_0, available as hep-th/9610114 and at

http://www.cpt.univ-mrs.fr/~coque/articles_html/SU2qba/SU2qba.html

Just to review: let's start with the group consisting of all the

ways you can rotate a regular tetrahedron and get it looking the

same again. You can achieve any even permutation of the 4 vertices

using such a rotation, so this group is the 12-element group A_4

consisting of all even permutations of 4 things - see "week155".

But it's also a subgroup of the rotation group SO(3). So,

its inverse image under the double cover

SU(2) -> SO(3)

has 24 elements. This is called the "binary tetrahedral group".

As usual, the algebra of complex functions on this finite group

is a Hopf algebra. But the cool thing is, this Hopf algebra is

closely related to the quantum group U_q(sl(2)) when q is a third

root of unity - a quantum group used in Connes' work on particle

physics because of its relation to the Standard Model gauge group!

In short: the plot thickens.

I'm not really ready to describe this web of ideas in detail,

so I'll just paraphrase the abstract of Coquereaux's paper and

urge you to either read this paper or look at his website:

We describe a few properties of the non-semisimple associative

algebra H = M_3 + M_{2|1}(Lambda^2)_0, where Lambda^2 is the

the Grassmann algebra with two generators. We show that H

is not only a finite dimensional algebra but also a

(non-cocommutative) Hopf algebra, hence a "finite quantum

group". By selecting a system of explicit generators, we

show how it is related with the quantum enveloping algebra

of U_q(sl(2)) when the parameter q is a cubic root of unity.

We describe its indecomposable projective representations as

well as the irreducible ones. We also comment about the relation

between this object and the theory of modular representations

of the group SL(2,Z/3), i.e. the binary tetrahedral group.

Finally, we briefly discuss its relation with the Lorentz group

and, as already suggested by A. Connes, make a few comments

about the possible use of this algebra in a modification of

the Standard Model of particle physics (the unitary group of

the semi-simple algebra associated with H is U(3) x U(2) x U(1)).

Quote of the week:

"The enrapturing discoveries of our field systematically conceal,

like footprints erased in the sand, the analogical train of thought

that is the authentic life of mathematics" - Gian-Carlo Rota

-----------------------------------------------------------------------

Previous issues of "This Week's Finds" and other expository articles on

mathematics and physics, as well as some of my research papers, can be

obtained at

http://math.ucr.edu/home/baez/

For a table of contents of all the issues of This Week's Finds, try

http://math.ucr.edu/home/baez/twf.html

A simple jumping-off point to the old issues is available at

http://math.ucr.edu/home/baez/twfshort.html

If you just want the latest issue, go to

Sep 7, 2003, 6:04:44â€¯PM9/7/03

to

John Baez wrote:

> True, Corfield describes how in 1996 the

> automated theorem prover EQP was the first to crack the

> Robbins conjecture. This states that a Boolean algebra is

> the same as a set equipped with an commutative associative

> binary operation "or" together with a unary operation "not"

> for which one mind-numbing axiom holds, namely:

>

> not(not(p or q) or not(p or not(q)) = p

>

> All the rest of Boolean logic is a consequence!

As a side note, a few years ago Stephen Wolfram conducted a computer

search for the smallest possible axiom systems for Boolean algebra

expressed in terms of "nand". The candidates

1. (ab)(a(bc))=a, ab=ba

2. ((bc)a)(b((ba)b))=a

3. (b((ab)b))(a(cb))=a

(if I have made no typographical error) turned out to be correct; you

can easily find proofs yourself by plugging them into Waldmeister (an

automated theorem prover). See pages 808-9 of "A New Kind of Science".

Sep 7, 2003, 6:46:10â€¯PM9/7/03

to

John Baez wrote:

>

> ...

> now that the Brits have been kicked out.

A cavil: the British rented Hong Kong for a fixed period of time and

left when that period of time expired. There was no, and no need for

any, "kicking out".

--

G.C.

Note ANTI, SPAM and invalid to be removed if you're e-mailing me.

[Moderator's note: I'll remove this jocular comment on the website

version. Followups set to the more freewheeling newsgroups. - jb]

Sep 7, 2003, 7:28:05â€¯PM9/7/03

to

David Hillman wrote:

> John Baez wrote:

You were one of the instigators!

http://www.cs.unm.edu/~veroff/BA/

http://www.cs.unm.edu/~veroff/BA/candidates.html

http://mathworld.wolfram.com/WolframAxiom.html

--

Uncle Al

http://www.mazepath.com/uncleal/eotvos.htm

(Do something naughty to physics)

[Moderator's note: followups directed to the most relevant

newsgroup, sci.math. - jb]

Sep 9, 2003, 5:43:31â€¯PM9/9/03

to sci-physic...@ucsd.edu

On Sat, 6 Sep 2003, John Baez wrote:

> However, things actually work out quite differently! By now

> the asymptotics of the 10j symbols are well understood, and they're

> *not* given by the stationary phase approximation. If you want to

> see the details, read these papers: ...

>

> The physical meaning of this fact is still quite mysterious.

John has done the mathematical part, so let me explain the physical

meaning now because it is not so hard to understand, it is quite

illuminating and the message is important. The physical meaning of these

results is, roughly speaking, that these mathematicians and physicists

have found approximately the 39th example of a general fact that if one

defines a random and generic discrete quantum mechanical model and studies

its low-energy limit, it can never reduce to general relativity or its

extensions unless the model is equivalent to a subsector of

string/M-theory. The fact that one needs a major modification of physics

at the ultra-short distances (that is generally defined to be

string/M-theory - which has unfortunately recently become a very large

"landscape" of possibilities) can be seen from the divergent structure of

the canonically quantized General Relativity.

In other words, the scientists have showed that loop quantum gravity

should not be called "gravity" despite the fact that the equations of

General Relativity were used to derive the discrete model itself.

Consequently, loop quantum gravity should probably be renamed. It is

however a clear progress that the researchers in this field started to

reveal, using a language that they can understand, another inconsistency

of loop quantum gravity as a theory of quantum gravity - something that

has been known to most of their colleagues who study quantum gravity for

nearly two decades. This observation shows that the conjecture of loop

quantum gravity was essentially a _scientific_ conjecture because it could

have been ruled out.

During the last year, another piece of evidence that loop quantum gravity

cannot be a theory of quantum gravity has been found. Loop quantum gravity

could be consistent with physics of gravity at long distances only if it

predicted the correct Bekenstein-Hawking black hole entropy, first

calculated by Hawking using semiclassical physics and thermodynamic

considerations. In loop quantum gravity, the first known semi-heuristic

calculations gave a wrong result - by a multiplicative factor of

ln(2)/pi.sqrt(3). In order to save loop quantum gravity, it has been

therefore argued that Newton's constant can be, in fact, multiplied or

renormalized by an arbitrary constant which is called Barbero-Immirzi

parameter gamma today.

Such an ad hoc modification of the theory could only be acceptable if one

could calculate this number from some independent considerations; these

considerations can only be independent if they relate physics at long and

short distances. Olaf Dreyer from the Perimeter Institute was brave

enough to do it, and he proposed a relation - a relation as justified as

the rest of loop quantum gravity - based on a numerical coincidence

between the required parameter gamma=ln(3)/2.pi.sqrt(2) - well, he had to

switch the numbers 2,3 a bit, but it could have been justified by

replacing the group SU(2) by SO(3) - and the asymptotic frequency of the

so-called quasinormal modes.

Quasinormal modes are damped (and oscillating) solutions of the linearized

equations around the black hole background such that they are purely

outgoing (and exponentially growing) both at infinity as well as at the

horizon. Only some discrete choices of the complex frequencies are

possible for these modes. The imaginary part of these allowed frequencies

can be arbitrarily large, but the real part turns out to converge to a

value that is proportional to the desired value of the Barbero-Immirzi

parameter.

This has led Dreyer to conjecture that the black hole entropy can be

calculated from the asymptotic real part of the quasinormal frequencies.

In the case of the Schwarzschild black holes, such a statement reduces to

a numerical conjecture by Hod from 1998 that was proved by me in December

2002. A conjecture that the limiting quasinormal frequencies equal a

specific constant proportional to ln(3). At that moment, loop quantum

gravity practitioners were very happy. Edward Witten, a leading

theoretical physicist nowadays, explained that he understood why they were

excited, and John Baez wrote an article to Nature about it.

However, further investigation of other black holes - the

Reissner-Nordstrom (charged) black holes that we studied with Andrew

Neitzke (see also his own newer paper); higher-dimensional Schwarzschild

black holes (the same reference); and recently also the Kerr (rotating)

black holes - see a recent paper by Cardoso, Onozawa, Berti, and Kokkotas

- showed that the conjecture relating the quasinormal modes and the black

hole entropy predicted by loop quantum gravity is incorrect for all black

holes except for the four-dimensional Schwarzschild ones. Equivalently,

loop quantum gravity predicts an incorrect black hole entropy for most

black holes if the relation to the quasinormal modes is taken seriously.

This shows that the agreement about the constant ln(3) in this special

case was a single numerical coincidence which is unlikely to have some

deep consequences. In fact, it is very easy to see why the constant must

be a logarithm of some familiar number (such as 3).

Loop quantum gravity therefore can't really predict the black hole

entropy, and - if a development that was believed by most of the

practitioners in that field is taken seriously - it has even predicted an

incorrect value, much like the incorrect asymptotic behavior of the 10-j

symbols that John has described previously. Well, these things should not

happen in a consistent theory. It is still possible that some

people will try to revive loop quantum gravity, but let me not speculate

because I have no idea what could they use as an argument. Too many things

don't work.

Even though these developments might be disappointing for many of us who

have worked on loop quantum gravity, they are also a piece of good news

because they show, from a very external perspective, the uniqueness of the

quantum theory of gravity. Although string/M-theorists have found a lot of

uniqueness and/or interconnectedness within "their" theory, they could

never prove rigorously that there is no completely different quantum

theory of gravity. The failure of loop quantum gravity shows, at least,

one more example of the general wisdom that all good ideas about the

spacetime and its physics are contained in string theory. Well, it turns

out that this theory contains too many things, but let us postpone this

topic into another thread.

You can find the relevant articles at

http://www.slac.stanford.edu/spires/hep/

For example, you can enter the query

find author motl and author neitzke

Best wishes

Lubos

______________________________________________________________________________

E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/

phone: work: +1-617/496-8199 home: +1-617/868-4487

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Superstring/M-theory is the language in which God wrote the world.

Sep 26, 2003, 6:59:14â€¯PM9/26/03

to

In article <Pine.LNX.4.31.03090...@feynman.harvard.edu>,

Lubos Motl <mo...@feynman.harvard.edu> wrote:

Lubos Motl <mo...@feynman.harvard.edu> wrote:

>On Sat, 6 Sep 2003, John Baez wrote:

>> However, things actually work out quite differently! By now

>> the asymptotics of the 10j symbols are well understood, and they're

>> *not* given by the stationary phase approximation. If you want to

>> see the details, read these papers: ...

>>

>> The physical meaning of this fact is still quite mysterious.

>John has done the mathematical part, so let me explain the physical

>meaning now because it is not so hard to understand, it is quite

>illuminating and the message is important. The physical meaning of these

>results is, roughly speaking, that these mathematicians and physicists

>have found approximately the 39th example of a general fact that if one

>defines a random and generic discrete quantum mechanical model and studies

>its low-energy limit, it can never reduce to general relativity or its

>extensions unless the model is equivalent to a subsector of

>string/M-theory.

I don't have time for a detailed rebuttal of this post by Motl, but

I will say one thing. Here we are talking about a model in which

spacetime is assumed to be made of "4-simplices", where a 4-simplex

is the 4-dimensional analogue of a tetrahedron. The hope is that

these 4-simplices will on average be very small - roughly the Planck

length in size - and that the usual continuum model of a spacetime

satisfying Einstein's equations, perhaps with some matter included,

will emerge only as an approximation that's good on much larger distance

scales.

On the other hand, the asymptotics of the 10j symbols describe

what happens in the limit where a 4-simplex becomes very BIG.

Thus, it's far from obvious that the asymptotics of the 10j symbols

are relevant to the physics of the model. What matters most for

physics is not really the behavior of a single large 4-simplex,

but a lot of little 4-simplices!

So, the fact that the 10j symbols behave in an unexpected way for

a large 4-simplex is not the death knell for this model, much as

Motl would enjoy that.

Indeed, if getting the 10j asymptotics to work out in some particular

way were the crucial ingredient of a successful model, we could

simply *define* them so that they had these asymptotics. That would

be very easy! In reality, we need to create something like a

renormalization group theory for these models, to see what effects

the fundamental Planck-scale physics has at larger distance scales.

People have known this for a long time, but it'll take a lot of work.

The main reason we studied the 10j asymptotics is that this is something

much easier to do.

Sep 29, 2003, 7:12:06â€¯PM9/29/03

to

On Fri, 26 Sep 2003, John Baez wrote:

> ...physics is not really the behavior of a single large 4-simplex,

> but a lot of little 4-simplices!

I am pretty curious about your "other" amplitude for the large 4-simplex

in spacetime created from the small simplices, especially because they

have mathematical reasons to be equal.

Of course, it is possible for loop quantum gravity researchers to say that

the true amplitude should be calculated differently after finding a new

type of discrepancy, and one can simply keep on "improving" the formalism

continuously. Once an infinite number of incorrect terms is removed and

the theory is made equivalent to string theory, then - of course - one

will obtain a consistent theory of gravity.

> So, the fact that the 10j symbols behave in an unexpected way for

> a large 4-simplex is not the death knell for this model, much as

> Motl would enjoy that.

Of course that it is. The question whether the model provides us with a

finite definition of gravity can only be decided at long distances, and

your paper shows that it is not finite. Instead, the spacetime in this

model is equally crumpled at long distance as it is in hundreds of other

attempts to discretize gravity.

> In reality, we need to create something like a renormalization group

> theory for these models, to see what effects the fundamental

> Planck-scale physics has at larger distance scales.

That's very correct. Because you derived your model to be formally equal

to Einstein's equations at the Planck scale, and because all the terms in

the action run with the scale, you are essentially guaranteed not to get

Einstein's equations at low energies. Instead, the spacetime in this model

is going to crumple like in these hundreds of other approaches to

discretization of gravity, and there is no flat space limit at all. The

crumpled spacetime can easily be seen by the dominating contribution of

singular simplices in your paper.

A finite theory could never lead to dominant singular contributions for

any physical amplitude.

If you had a finite discrete model, it would have to be valid at all

scales. It is only the effective description that can run - and the

running always requires (logarithmic) ultraviolet divergences. Because

your model is constructed in such a way that it can't have these

divergences, it can't run. Any discrepancy is therefore serious.

Consider another (more precisely "the") example - string theory. It is

also claimed to be the full description at all scales. However, the low

energy limit can be checked to coincide with (super)Einstein's equations.

Any discrepancy here would be a serious problem for string theory, and the

cosmological constant, for example, is a relatively serious one. But it is

still solvable in principle because the observed Lambda is not zero, and

one might get the right value in some model. The inability to obtain any

smooth spacetime at long distances would be a much more severe problem.

Oct 28, 2003, 3:03:09â€¯AM10/28/03

to

In article <Pine.LNX.4.31.030927...@feynman.harvard.edu>,

Lubos Motl <mo...@feynman.harvard.edu> wrote:

Lubos Motl <mo...@feynman.harvard.edu> wrote:

>On Fri, 26 Sep 2003, John Baez wrote:

>> ...physics is not really the behavior of a single large 4-simplex,

>> but a lot of little 4-simplices!

>I am pretty curious about your "other" amplitude for the large 4-simplex

>in spacetime created from the small simplices, especially because they

>have mathematical reasons to be equal.

No, they shouldn't be equal, any more than the amplitude for

a large complicated Feynman diagram with 3 external legs should

equal the amplitude for this diagram:

/

/

----

\

\

in phi^3 theory.

The only situation where you can chop a simplex into lots of bits

and have its amplitude *equal* the product of the amplitudes of

the little bits - summed over labellings of internal faces - is

when you have a TOPOLOGICAL quantum field theory. In a topological

field theory there are no local degrees of freedom, so we can chop

a big simplex into lots of bits without changing anything. There are

lots of examples of these, like 3d Riemannian quantum gravity with

an appropriate sign of the cosmological constant - but we don't expect

4d gravity to be a topological quantum field theory.

>Of course, it is possible for loop quantum gravity researchers to say that

>the true amplitude should be calculated differently after finding a new

>type of discrepancy, and one can simply keep on "improving" the formalism

>continuously.

It's possible we need to change the Barrett-Crane model because its

amplitude for large 4-simplices doesn't match the semiclassical

approximation... but it's also quite possible that we don't, since

the world is not actually made of large 4-simplices.

>Once an infinite number of incorrect terms is removed and

>the theory is made equivalent to string theory, then - of course - one

>will obtain a consistent theory of gravity.

Yeah, yeah. Even if you're right and we're all mixed up

and need to change the model drastically and when we do it

turns into something like string theory, we'll still have

something interesting, namely a manifestly background-free

formulation of string theory as a state sum model. I would

not mind this at all.

>> So, the fact that the 10j symbols behave in an unexpected way for

>> a large 4-simplex is not the death knell for this model, much as

>> Motl would enjoy that.

>Of course that it is. The question whether the model provides us with a

>finite definition of gravity can only be decided at long distances, and

>your paper shows that it is not finite.

No it doesn't. I wish it did - that would be a definitive death blow

to the theory, and then I could forget it and think about something else!

But we only consider the behavior of a single huge 4-simplex, which is

completely different from a spacetime made of zillions of tiny Planck-sized

4-simplices. It's much less relevant to the actual physics of the model,

but much easier to study.

>Instead, the spacetime in this model is equally crumpled at long

>distance as it is in hundreds of other attempts to discretize gravity.

That could be, but nothing we did showed that.

>> In reality, we need to create something like a renormalization group

>> theory for these models, to see what effects the fundamental

>> Planck-scale physics has at larger distance scales.

>That's very correct.

I'm going to print this out and frame it: Motl said something

I said is correct!

>Because you derived your model to be formally equal

>to Einstein's equations at the Planck scale and because all the terms in

>the action run with the scale, you are essentially guaranteed not to get

>Einstein's equations at low energies.

If you start with a discretized theory with the Einstein-Hilbert

action at some length scale L, what sort of effective action do you

get at some length scale AL, and how do the terms run as A -> +infinity?

>If you had a finite discrete model, it would have to be valid at all

>scales. It is only the effective description that can run - and the

>running always requires (logarithmic) ultraviolet divergences. Because

>your model is constructed in such a way that it can't have these

>divergences, it can't run. Any discrepancy is therefore serious.

If I could understand all this and - here's the catch - also believe it,

I would be a happy man.

Oct 31, 2003, 6:30:59â€¯PM10/31/03

to

"John Baez" <ba...@galaxy.ucr.edu> schrieb im Newsbeitrag

news:bnl7rt$d81$1...@glue.ucr.edu...

> In article <Pine.LNX.4.31.030927...@feynman.harvard.edu>,

> Lubos Motl <mo...@feynman.harvard.edu> wrote:

news:bnl7rt$d81$1...@glue.ucr.edu...

> In article <Pine.LNX.4.31.030927...@feynman.harvard.edu>,

> Lubos Motl <mo...@feynman.harvard.edu> wrote:

> >Once an infinite number of incorrect terms is removed and

> >the theory [LQG] is made equivalent to string theory, then - of course -

one

> >will obtain a consistent theory of gravity.

>

> Yeah, yeah. Even if you're right and we're all mixed up

> and need to change the model drastically and when we do it

> turns into something like string theory, we'll still have

> something interesting, namely a manifestly background-free

> formulation of string theory as a state sum model. I would

> not mind this at all.

I find it really interesting that the following looks like a way to "change

the model" and indeed turn it into string theory:

So let's suppose we want to formulate our fundamental theory in terms of

functional states on a space of gauge connections A_mu taking values in some

Lie algebra. We want to be really background free. In LQG one does away with

background _fields_ on spacetime, but one still does need a manifold to set

up the theory. Let's do away with the assumption of a

(topological/differentiable) manifold, too.

Without a manifold it makes no longer sense to have the A_mu(x) be functions

of coordinates. Therefore let's assume, being very naive, that the

connection is _independent_ of any coordinates.

As we learn from LQG, functions (observables) on the space of connections

are spanned by generalized Wilson lines ("networks", graphs - I'll avoid the

word "spin" for the moment). In ordinary LQG these are embedded into a

manifold. But since we have just done away with this manifold we now have to

evaluate our connections on abstract networks that are not embedded into any

a priori structure. The natural way to do that is to equip the network that

comes with a given state (function on the space of connections) with a

D-tuple valued (piecewise defined) 1-form k and compute the holonomy of

Sum_mu k^mu A_mu

along the edges of the network, intertwining at the vertices as desired and

finally tracing over the result.

In other words, in this manifold-independent formulation of "Ashtekar

geometry" a network state is given not just by a coloring of edges {e} by

representations {r} (and coloring of vertices by intertwiners {i}) but

instead by a coloring by representations _and_ D-tuple valued 1-forms k.

(The information that was previously contained in the coordinates of a given

edge has now moved into the extra piece of data k.) So a basis of states

should now be of the form

{ psi_{e,r,i,k} }

where each element psi_{e,r,i,k} is associated with an abstract

combinatorial graph e colored by r,i, and k.

This basis spans the kinematical Hilbert space. Now we need dynamics.

Personally I feel that in LQG kinematics is very beautiful but that as soon

as the ordinary dynamics enters the game things become rather awkward. A

fundamental theory is not supposed to look awkward, so let's slighly modify

the ordinary LQG dynamics. Instead of using the action of B^F theory on the

space of connections A we'd rather use the simplest action quadratic in the

curvature:

S = Tr F^2 = Tr [A,A]^2 .

In other words, the slight modification of LQG that I am proposing here is a

theory whose configurations are given by _constant_ gauge connections A and

whose observables (correlation functions) are

<psi_{e1,r1,i1,k1} psi_{e2,r2,i2,k2}...psi_{en,rn,in,kn}>

=

int DA psi_{e1,r1,i1,k1}(A) ...psi_{en,rn,in,kn}(A) exp(-S(A))

with S and psi_{...} as defined above. The two major modifacations as

compared to ordinary LQG are the absence of the manifold background and the

switching from an action linear in the curvature to the simplest one

quadratic in the curvature.

The point of all this is the following: In 1996 the authors N. Ishibashi, H.

Kawai, Y. Kitazawa and A. Tsuchiya have proved for us (see hep-th/9908038

and references given there) that

if we identify Wilson lines (network edges) in the above theory with

fundamental strings

and if we use U(N>>>1) as the gauge group then the above action for the

connection induces on these Wilson line, which are now also regarded as

functionals (states) on the configuration space of the fundamental string,

the equations of motion of string field theory!

Voila.

I spent some time at the "Strings meet loops" symposium trying to find LQG

people who would find this as interesting as I do. That's because my

impression is that maybe the true value of this IKKT model is not properly

appreciated in the string community, which might have to do with its radical

background independence. Lubos mentioned that timelike T-duality which does

away with time (!) looks suspicious. But maybe it is just what we need, I

wonder. In any case, when looked at it from the proper perspective (as I

have tried to demonstrate above) the IKKT model looks much more like LQG

than like string theory. Of course, looked at it from another perspective it

completely looks like string theory. Great.

Luckily, I found several very friendly and open minded LQGists who did find

this interesting. I am looking forward to hearing what they come up with

when studying this in detail.

Nov 4, 2003, 2:45:18â€¯AM11/4/03

to

"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote in message news:<3fa2f0b3$1...@news.sentex.net>...

> Without a manifold it makes no longer sense to have the A_mu(x) be functions

> of coordinates. Therefore let's assume, being very naive, that the

> connection is _independent_ of any coordinates.

>

> As we learn from LQG, functions (observables) on the space of connections

> are spanned by generalized Wilson lines ("networks", graphs - I'll avoid the

> word "spin" for the moment).

The spin networks are wavefunctionals (states),

not observables. In the observable algebra there

are things corresponding to "multiplying by a

spin network" but there are also surface integrals

(fluxes) of the cotetrad (electric) field, which

is canonically conjugate to the connection, in a

sense (one must not forget the weird

noncommutativity of these fluxes).

> In ordinary LQG these are embedded into a

> manifold. But since we have just done away with this manifold we now have to

> evaluate our connections on abstract networks that are not embedded into any

> a priori structure. The natural way to do that is to equip the network that

> comes with a given state (function on the space of connections) with a

> D-tuple valued (piecewise defined) 1-form k and compute the holonomy of

>

> Sum_mu k^mu A_mu

>

> along the edges of the network, intertwining at the vertices as desired and

> finally tracing over the result.

This is somewhat ad hoc, as one needs

far less states to span the function

space of a single triple of Lie algebra

elements.

> Instead of using the action of B^F theory on the

> space of connections A we'd rather use the simplest action quadratic in the

> curvature:

>

> S = Tr F^2 = Tr [A,A]^2 .

Is this supposed to be the action or

the Hamiltonian? Because if you modify

the action you modify the kinematics.

If it is the Hamiltonian, how do you

solve the time problem? Do you expect

it be a constraint?

> The two major modifacations as

> compared to ordinary LQG are the absence of the manifold background and the

> switching from an action linear in the curvature to the simplest one

> quadratic in the curvature.

In fact, absense of manifold background

is quite easy to do in LQG on the

kinematics level: you just don't equip

your networks with an embedding modulo

diffeomorphism into any space. It's also

easy in dynamics, if you understand spin

foams by "dynamics".

> That's because my

> impression is that maybe the true value of this IKKT model is not properly

> appreciated in the string community, which might have to do with its radical

> background independence.

How background independant is it, though? The "fields" still

carry a flat space index. It is true that them being

noncommutative matrices the issue is made more subtle. Moreover,

I had a thought as a quantum gravity theory in asymptotically flat

spacetime should carry a Poincare group action, it might also

include some kind of position observables, as these are present in

all reps of the Poincare group - which are essentially Minkwoski

space particle wavefunctions. Btw, anybody knows what happens

with the position observables in light cone coordinates? Do the

problems with them become worse or better?

Best regards,

Squark

------------------------------------------------------------------

Write to me using the following e-mail:

Skvark_N...@excite.exe

(just spell the particle name correctly and change the

extension in the obvious way)

Nov 4, 2003, 7:42:45â€¯AM11/4/03

to

"Squark" <fii...@yahoo.com> schrieb im Newsbeitrag

news:939044f.03110...@posting.google.com...

>

> "Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote in message

news:<3fa2f0b3$1...@news.sentex.net>...

> > Without a manifold it makes no longer sense to have the A_mu(x) be

functions

> > of coordinates. Therefore let's assume, being very naive, that the

> > connection is _independent_ of any coordinates.

> >

> > As we learn from LQG, functions (observables) on the space of

connections

> > are spanned by generalized Wilson lines ("networks", graphs - I'll avoid

the

> > word "spin" for the moment).

>

> The spin networks are wavefunctionals (states),

> not observables.

news:939044f.03110...@posting.google.com...

>

> "Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote in message

news:<3fa2f0b3$1...@news.sentex.net>...

> > Without a manifold it makes no longer sense to have the A_mu(x) be

functions

> > of coordinates. Therefore let's assume, being very naive, that the

> > connection is _independent_ of any coordinates.

> >

> > As we learn from LQG, functions (observables) on the space of

connections

> > are spanned by generalized Wilson lines ("networks", graphs - I'll avoid

the

> > word "spin" for the moment).

>

> The spin networks are wavefunctionals (states),

> not observables.

Oops, right, I did not mean to say "observables", I meant to say "states".

> > In ordinary LQG these are embedded into a

> > manifold. But since we have just done away with this manifold we now

have to

> > evaluate our connections on abstract networks that are not embedded into

any

> > a priori structure. The natural way to do that is to equip the network

that

> > comes with a given state (function on the space of connections) with a

> > D-tuple valued (piecewise defined) 1-form k and compute the holonomy of

> >

> > Sum_mu k^mu A_mu

> >

> > along the edges of the network, intertwining at the vertices as desired

and

> > finally tracing over the result.

>

> This is somewhat ad hoc, as one needs

> far less states to span the function

> space of a single triple of Lie algebra

> elements.

I suppose that you are here thinking of mu taking values in {1,2,3}and A

taking values in a finite dimensional Lie algebra. But with hindsight I

would like to generalize this to higher dimensions and large Lie algebras.

For very large (possibly infinite) Lie algebras I expect that the states I

describe are the natural analogues of the usual spin network states.

But you are right of course that this is ad hoc. I am just playing around

with the notions appearing and LQG and in the IKKT model, vaguely

indicating that these are quite similar.

Of course this similarity as such is no big surprise, because it is

essentially a special case of the general relation between Wilson loop

description of gauge theories and "string" theories, as for instance

described by John Baez in hep-th/9309067, where it says:

"The loop representation of quantum gravity has many formal resemblances to

a background-free string theory." (abstract)

"The resemblance of these states [spin network states] to wavefunctions of a

string field theory is striking. It is natural, therefore, to ask whether

the loop representation of quantum gravity might be a string theory in

disguise - or vice versa" (p.2)

But what seems to be surprising in the IKKT model is that the Wilson loop

description of that totally dimensionally reduced SYM theory does not just

give _a_ theory of strings, but actually _the_ theory of strings.

> > Instead of using the action of B^F theory on the

> > space of connections A we'd rather use the simplest action quadratic in

the

> > curvature:

> >

> > S = Tr F^2 = Tr [A,A]^2 .

>

> Is this supposed to be the action or

> the Hamiltonian? Because if you modify

> the action you modify the kinematics.

> If it is the Hamiltonian, how do you

> solve the time problem? Do you expect

> it be a constraint?

This is supposed to be the action. It is just the action of totally

dimensionally reduced YM theory, where all the derivatives drop out.

In which sense does this modify the kinematics?

> How background independant is it, though? The "fields" still

> carry a flat space index.

That's an illusion. :-) If I hand you ten constant matrices and a way to sum

up the trace over squares of their commutators you would not think that I

had defined a Minkowski space background. And indeed Minkwoski space emerges

in this model (locally) only as an approximation to the eigenvalue

distribution of these matrices (or something like that, see the IKKT

papers).

Nov 4, 2003, 1:08:13â€¯PM11/4/03

to

"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote in message news:<3fa79ec5$1...@news.sentex.net>...

> But what seems to be surprising in the IKKT model is that the Wilson loop

> description of that totally dimensionally reduced SYM theory does not just

> give _a_ theory of strings, but actually _the_ theory of strings.

> But what seems to be surprising in the IKKT model is that the Wilson loop

> description of that totally dimensionally reduced SYM theory does not just

> give _a_ theory of strings, but actually _the_ theory of strings.

This is not so clear to me. What I understand of matrix

theory is that a certain n+1-dimensional SYM theory

defines M-theory with n compactified spatial directions.

One of the main problems with that is the SYM being ill

defined (non-renormalizable?) for n > 3, so in these

cases it describes at best a certain limit of the

theory.

In particular the BFFS model (n = 0) defines M theory on

uncompactified 11-dimensional spacetime.

How does the IKKT model enter in this scheme is a

mystery to me.

> > > Instead of using the action of B^F theory on the

> > > space of connections A we'd rather use the simplest action quadratic in

> the

> > > curvature:

> > >

> > > S = Tr F^2 = Tr [A,A]^2 .

> ...

> In which sense does this modify the kinematics?

What you constructed using spin networks is

cetain functions of A. Why do you think these

define states in the IKKT model? In fact, I

don't expect states in the IKKT model to be

functions of A at all, as otherwise all of

the A components commute in the observable

algebra and the system is classical rather

than quantum.

> > How background independant is it, though? The "fields" still

> > carry a flat space index.

>

> That's an illusion. :-) If I hand you ten constant matrices and a way to sum

> up the trace over squares of their commutators you would not think that I

> had defined a Minkowski space background.

I certainly would.

> And indeed Minkwoski space emerges

> in this model (locally) only as an approximation to the eigenvalue

> distribution of these matrices (or something like that, see the IKKT

> papers).

It emerges from the eigenvaludes in the limit in which the

matrices commute and thus can be diagonalized simultaneously. In

this limit what I get is a collection of N instantons in the

Minkowsky space. Btw, the dimension of that space is not clear to

me, is it 12?! Since in general it's (10-n)+1-dimensional (see

above notation for n).

However, that's a long way from claiming background independance.

To do you would have to show curved space emerges as easily

(meaning non-perturbatively).

Nov 4, 2003, 1:24:06â€¯PM11/4/03

to

"Squark" <fii...@yahoo.com> schrieb im Newsbeitrag

news:939044f.03110...@posting.google.com...

news:939044f.03110...@posting.google.com...

> What you constructed using spin networks is

> cetain functions of A. Why do you think these

> define states in the IKKT model? In fact, I

> don't expect states in the IKKT model to be

> functions of A at all, as otherwise all of

> the A components commute in the observable

> algebra and the system is classical rather

> than quantum.

I don't understand what you are saying here. A (kinematical) state is a

function on config space, which here is the space of connections.

> It emerges from the eigenvaludes in the limit in which the

> matrices commute and thus can be diagonalized simultaneously. In

> this limit what I get is a collection of N instantons in the

> Minkowsky space. Btw, the dimension of that space is not clear to

> me, is it 12?! Since in general it's (10-n)+1-dimensional (see

> above notation for n).

It is 10, since there are 10 matrices and also because the whole thing

becomes IIB string theory.

> However, that's a long way from claiming background independance.

> To do you would have to show curved space emerges as easily

> (meaning non-perturbatively).

Yup. And that's precisely what the authors of that model claim to have done.

In particular they claimed to see an automatic compactification to 4 large

dimensions. Now this particular calculation may be correct or not, but in

any case the general idea is quite interesting, I think.

Nov 5, 2003, 7:08:29â€¯PM11/5/03

to

"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote in message news:<3fa7eec6$1...@news.sentex.net>...

> I don't understand what you are saying here. A (kinematical) state is a

> function on config space, which here is the space of connections.

No. The space of connections is the space

of histories, not the configuration space.

It's like saying the collection of all

imaginable trajectories x(t) is the

configuration space for particle mechanics.

The quantum wavefunctions aren't functionals

of x(t) as the x(t) observables don't commute

for different t in the quantum theory.

> > It emerges from the eigenvaludes in the limit in which the

> > matrices commute and thus can be diagonalized simultaneously. In

> > this limit what I get is a collection of N instantons in the

> > Minkowsky space. Btw, the dimension of that space is not clear to

> > me, is it 12?! Since in general it's (10-n)+1-dimensional (see

> > above notation for n).

>

> It is 10, since there are 10 matrices and also because the whole thing

> becomes IIB string theory.

Something crazy and in the same time very cool just occured to me.

As Lubos explained in "The Big Picture of Superstring Theory"

thread, the IKKT model is related to T-duality along time, the

meaning of which is not clear. Consider the following though:

Usual matrix models result by compactifying and T-dualizing 11d

M-theory along spatial directions. What happens though if you

try to do it along time? Well, the time t isn't a spatial

direction, but imaginary time it* _is_, from the point of view of

the metric. So, maybe we're actually compactifying and

T-dualizing along it*! But what does compactifying it mean? It

means passing to finite temperature! So: the IKKT model describes

the thermal sector(s) of 11d M-theory! More generally I'd expect

Euclidean SYM models which would describe the thermal sector of

compactified M-theory.

> Yup. And that's precisely what the authors of that model claim to have done.

I didn't read the article (probably I'll have to now :-) ) but I'm

very sceptical about it.

*This is not the English word "it", it's the imaginary unit i

times the time coordinate t :-)

Nov 6, 2003, 6:42:14â€¯AM11/6/03

to

> "Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote in message

news:<3fa7eec6$1...@news.sentex.net>...

> > I don't understand what you are saying here. A (kinematical) state is a

> > function on config space, which here is the space of connections.

>

> No. The space of connections is the space

> of histories, not the configuration space.

news:<3fa7eec6$1...@news.sentex.net>...

> > I don't understand what you are saying here. A (kinematical) state is a

> > function on config space, which here is the space of connections.

>

> No. The space of connections is the space

> of histories, not the configuration space.

You are right. I managed to confuse my terminology by sticking too much to

LQG analogies instead of spin foam analogies.

> Something crazy and in the same time very cool just occured to me.

> As Lubos explained in "The Big Picture of Superstring Theory"

> thread, the IKKT model is related to T-duality along time, the

> meaning of which is not clear. Consider the following though:

> Usual matrix models result by compactifying and T-dualizing 11d

> M-theory along spatial directions. What happens though if you

> try to do it along time? Well, the time t isn't a spatial

> direction, but imaginary time it* _is_, from the point of view of

> the metric. So, maybe we're actually compactifying and

> T-dualizing along it*! But what does compactifying it mean? It

> means passing to finite temperature! So: the IKKT model describes

> the thermal sector(s) of 11d M-theory! More generally I'd expect

> Euclidean SYM models which would describe the thermal sector of

> compactified M-theory.

Actually this addresses a point that I wondered about when reading

hep-th/9908038: The authors are actually using the euclidean path integral

throughout but apparently get the string field equations from (2.2) and

(2.3) without ever making an analytic continuation.

Nov 8, 2003, 5:21:54â€¯AM11/8/03

to

On 4 Nov 2003, Squark wrote:

> This is not so clear to me. What I understand of matrix

> theory is that a certain n+1-dimensional SYM theory

> defines M-theory with n compactified spatial directions.

> One of the main problems with that is the SYM being ill

> defined (non-renormalizable?) for n > 3, so in these

> cases it describes at best a certain limit of the

> theory.

If you make all the steps of the Seiberg/Sen derivation properly, you

automatically obtain the correct and consistent UV (short-distance)

completion of the gauge theory. For example, for T^4 where one naively

expects a non-renormalizable 4+1-dimensional gauge theory, you actually

derive that the matrix model results from the low-energy dynamics of

D4-branes at *strong* coupling, which is equivalent to M5-branes in

M-theory.

The matrix model for M-theory on T^4 is actually the 5+1-dimensional (2,0)

theory describing the M5-branes, compactified on T^5. It is a local field

theory in six dimensions whose compactification on S^1 gives the

4+1-dimensional maximally supersymmetric gauge theory.

In a similar fashion, you may consider the matrix model for M-theory on

T^5. One naively expects the 5+1-dimensional gauge theory that is again

non-renormalizable. One can derive the matrix description involving the

D5-branes of type IIB at strong coupling, which is by S-duality equivalent

to NS5-branes of type IIB string theory at weak coupling. One can see that

the energy limit is precisely the limit that defines the (1,1) little

string theory in 6 dimensions, that arises from type IIB NS5-branes. This

little string theory is the correct ultraviolet completion of the

6-dimensional gauge theory.

These theories - UV-completed gauge theories - may be defined in various

ways, for example using the tools of deconstruction.

http://arxiv.org/abs/hep-th/0110146

Matrix theory for M-theory on T^6 and higher - where one would expect the

exceptional symmetries and other interesting stuff - can't be defined as a

non-gravitational theory.

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