This Week's Finds in Mathematical Physics (Week 198)
John Baez
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
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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
David Hillman
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".
George Cox
>
> ...
> 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]
Uncle Al
> 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]
Lubos Motl
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.
John Baez
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.
Lubos Motl
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.
John Baez
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.
Urs Schreiber
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.
Squark
"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)
Urs Schreiber
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).
Squark
> 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).
Urs Schreiber
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.
Squark
"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 :-)
Urs Schreiber
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.
Lubos Motl
> 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.