This Week's Finds in Mathematical Physics (Week 168)

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John Baez

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May 30, 2001, 10:43:38 PM5/30/01
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Also available at http://math.ucr.edu/home/baez/week168.html

May 29, 2001
This Week's Finds in Mathematical Physics (Week 168)
John Baez

It's been about two months since the last issue of This Week's Finds,
and I apologize for this. I've been very busy, and my limited writing
energy has all gone into finishing up a review article on the octonions.
I'm dying to talk about that... but first things first!

When I left off I was at Penn State, learning about the latest
developments in quantum gravity. I told you how Martin Bojowald was
using loop quantum gravity to study what came before the big bang...
but I didn't mention that he'd written a nice little book on the subject:

1) Martin Bojowald, Quantum Geometry and Symmetry, Shaker Verlag,
Aachen, 2000. Available at
http://www.shaker.de/Online-Gesamtkatalog/Details.asp?ISBN=3-8265-7741-8

This does not cover his most recent work, in which his program is really
starting to pay off... but it will certainly help you *understand* his
recent work. He's doing lots of great stuff these days. In fact, he
just came out with a paper yesterday:

2) Martin Bojowald, The semiclassical limit of loop quantum
cosmology, available at gr-qc/0105113.

This explains how his new approach to quantum cosmology is related to
the old "minisuperspace" approach. In the old approach, you just take
some limited class of cosmologies satisfying the equations of general
relativity and think of this class as a classical mechanics problem with
finitely many degrees of freedom: for example, the size of the universe
together with various numbers describing its shape. Then you quantize
this classical system.

In this approach, you don't see any hint of spacetime discreteness on
the Planck scale. But in Bojowald's approach, you do! What gives? He
still starts with a limited class of cosmologies and quantizes that, but
he does so using ideas taken from loop quantum gravity. This makes all
the difference: now areas and volumes have discrete spectra of
eigenvalues, and this saves us from the horrors of the singularity at
the big bang. In fact, we can go back *before* the big bang, and find a
time-reversed expanding universe on the other side!

But what's the relation between this new approach and the old one,
exactly? Well, in loop quantum gravity, space is described using "spin
networks", and area is quantized. Each edge of a spin network is
labelled by some spin j = 0, 1/2, 1, ..., and when a spin-j edge
punctures a surface, it gives that surface an area equal to

8 pi gamma sqrt(j(j+1))

times the Planck length squared. Here gamma is a constant called the
"Immirzi parameter" - see "week112" and "week148" for more about that.
Bojowald shows that you can recover the old approach to quantum
cosmology from his new one by taking a limit in which the Immirzi
parameter approaches zero while the spins labelling spin network edges
go to infinity. In this limit, the spacings between the above areas go
to zero - so the discrete spectrum of the "area operator" becomes
continuous! Thus we lose the discrete geometry which is typical of loop
quantum gravity.

I'm also excited by what's going on with spin foams lately. For one, my
friend Dan Christensen is starting to do numerical calculations with the
Riemannian Barrett-Crane model. I've discussed this model in "week113",
"week120", and "week128", so I won't bore you with the details yet
again. For now, let me just say that it's a theory of quantum gravity
in which spacetime is a triangulated 4-dimensional manifold. There is
also a Lorentzian version of this model, which is more physical, but
it's trickier to compute with, so Dan has wisely decided to start by
tackling the Riemannian version.

As you probably know, in quantum field theory, as in statistical
mechanics, the partition function is king. So Dan Christensen is
starting out by using a supercomputer to numerically calculate the
partition function of a triangulated 4-sphere. He has some students
helping him, and he's also gotten some help from Greg Egan....

Anyway: this partition function is a sum over all ways of labelling
triangles by spins - but it's not obvious that the sum converges! For
this reason Dan has begun by imposing a "cutoff", that is, an upper
bound on the allowed spins. Physically this would be called an
"infrared cutoff", since big spins mean big triangles. The question
is: what happens as you let this cutoff approach infinity? Does the
partition function converge or not?

Now, what's cool is that in November of last year, a fellow named
Alejandro Perez claimed to have proven that it *does* converge:

3) Alejandro Perez, Finiteness of a spin foam model for euclidean
quantum general relativity, Nucl. Phys. B599 (2001) 427-434.
Also available at gr-qc/0011058.

I say "claimed", not because I doubt his proof, but because I still
haven't checked it, and I should. But the great thing is: now we have
both numerical and analytic ways of studying this spin foam model, and
we can play them off against each other! This helps a lot when you're
trying to understand a complicated problem.

Of course, the skeptics among you will say "Fine, but this is just
Riemannian quantum gravity, not the Lorentzian theory. We're still not
talking about the real world." And you'd be right! But luckily, there
has also been a lot of progress on the Lorentzian Barrett-Crane model.

This version of the Barrett-Crane model is based on the Lorentz group
instead of the rotation group. Because the representations of the
Lorentz group are parametrized in a continuous rather than discrete way,
in this version one computes the partition function as as an *integral*
over ways of labelling the triangles by nonnegative real numbers. These
numbers represent areas, so it seems that area is not quantized in this
theory - but I should warn you, this is a hotly debated issue! We need
to better understand how this model relates to loop quantum gravity,
where area is quantized.

Anyway, when Barrett and Crane proposed the Lorentzian version of their
model, it wasn't obvious that this integral for the partition function
converged. Even worse, it wasn't clear that the integrand was
well-defined! The basic ingredient in the integrand is the so-called
"Lorentzian 10j symbol", which describes the amplitude for an individual
4-simplex to have a certain geometry, as specified by the areas of its
10 triangular faces. Barrett and Crane wrote down an explicit integral
for the Lorentzian 10j symbol, but they didn't show this integral
converges.

Last summer, in a fun-filled week of intense calculation, John Barrett
and I showed that the integral defining the Lorentzian 10j symbols
*does* in fact converge:

4) John Baez and John W. Barrett, Integrability for relativistic
spin networks, available at gr-qc/0101107.

It took us until this January to write up those calculations. By April,
Louis Crane, Carlo Rovelli, and Alejandro Perez had written a paper
extending our methods to show that the partition function converges:

5) Louis Crane, Alejandro Perez, Carlo Rovelli, A finiteness proof for
the Lorentzian state sum spin foam model for quantum general relativity,
available as gr-qc/0104057.

So now we have a well-defined quantum gravity theory for a 4-dimensional
spacetime with a fixed triangulation, and we can start studying it! The
big question is whether it mimics general relativity at distance scales
much larger than the Planck scale.

Now, I want to say more about what I did after visiting Penn State -
among other things, I've been talking to a bunch of folks who work on
gravitational wave detection - but I think I'll wait and talk about that
next time.

So: octonions!

I've finally finished writing a survey of the octonions and their
connections to Clifford algebras and spinors, Bott periodicity,
projective and Lorentzian geometry, Jordan algebras, the exceptional
Lie groups, quantum logic, special relativity and supersymmetry:

6) John Baez, The octonions,
http://math.ucr.edu/home/baez/Octonions/octonions.html
Also available at math.RA/0105155.

Let me just sketch some of the main themes. For details and precise
statements, read the paper!

Octonions arise naturally from the interaction between vectors and
spinors in 8-dimensional Euclidean space, but in superstring theory and
other physics applications, what matters most is their relation to
10-dimensional Lorentzian spacetime. This is part of a pattern:

1) spinors in 1d Euclidean space are real numbers (R).
2) spinors in 2d Euclidean space are complex numbers (C).
3) spinors in 4d Euclidean space are quaternions (H).
4) spinors in 8d Euclidean space are octonions (O).

(These numbers are just the dimensions of R, C, H and O.)

Also:

1) points in 3d Minkowski spacetime are 2x2 hermitian real matrices
2) points in 4d Minkowski spacetime are 2x2 hermitian complex matrices
3) points in 6d Minkowski spacetime are 2x2 hermitian quaternionic matrices
4) points in 10d Minkowski spacetime are 2x2 hermitian octonionic matrices

(These numbers are 2 more than the dimensions of R, C, H and O.)

The octonions are also what lie behind the 5 exceptional simple Lie
groups. The exceptional group G2 is just the symmetry group of the
octonions. The other four exceptional groups, called F4, E6, E7
and E8, are symmetry groups of "projective planes" over:

1) the octonions, O
2) the complexified octonions or "bioctonions", C tensor O
3) the quaternionified octonions or "quateroctonions", H tensor O
4) the octonionified octonions or "octooctonions", O tensor O

respectively.

Warning: I put the phrase "projective planes" in quotes here because the
last two spaces are not projective planes in the usual axiomatic sense
(see "week145"). This makes the subject a bit tricky.

Now, it is no coincidence that:

1) spinors in 9-dimensional Euclidean space are pairs of octonions.
2) spinors in 10-dimensional Euclidean space are pairs of bioctonions.
3) spinors in 12-dimensional Euclidean space are pairs of quateroctonions.
4) spinors in 16-dimensional Euclidean space are pairs of octooctonions.

(These numbers are 8 more than the dimensions of R, C, H and O.)

This sets up a relation between spinors in these various dimensions
and the projective planes over O, C tensor O, H tensor O and O tensor O.
The upshot is that we get a nice description of F4, E6, E7 and E8 in
terms of the Lie algebras so(n) and their spinor representations where
n = 9, 10, 12, 16, respectively.

It's all so tightly interlocked - I can't believe it's not trying to
tell us something about physics! Just to whet your appetite for more,
let me show you 7 quateroctonionic descriptions of the Lie algebra of
E7:

e7 = isom((H tensor O)P^2)

= der(h_3(O)) + h_3(O)^3

= der(O) + der(h_3(H)) + (Im(O) tensor sh_3(H))

= der(H) + der(h_3(O)) + (Im(H) tensor sh_3(O))

= der(O) + der(H) + sa_3(H tensor O)

= so(O + H) + Im(H) + (H tensor O)^2

= so(O) + so(H) + Im(H) + (H tensor O)^3

I explain why these are true in the paper, but for now, let me
just say what all this stuff means:

"isom" means the Lie algebra of the isometry group,

(H tensor O)P^2 means the quateroctonionic projective plane
with its god-given Riemannian metric,

"der" means the Lie algebra of derivations,

h_3(O) is the exceptional Jordan algebra, consisting of 3x3
hermitian octonionic matrices,

h_3(H) is the Jordan algebra of 3x3 hermitian quaternionic
matrices,

Im(O) is the 7-dimensional space of imaginary octonions,

Im(H) is the 3-dimensional space of imaginary quaternions,

sh_3(O) is the traceless 3x3 hermitian octonionic matrices,

sh_3(H) is the traceless 3x3 hermitian quaternionic matrices,

so(V) is the rotation group Lie algebra associated to the
real inner product space V.

It is fun to compute the dimension of E7 using each of these
7 formulas and see that you get 133 each time!

I also give 6 bioctonionic descriptions of E6. Alas, I could
not find 8 octooctonionic descriptions of E8, probably because
this group is more symmetrical and in a curious sense simpler
than the others.

Time for dinner.

-----------------------------------------------------------------------
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If you just want the latest issue, go to

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Mark William Hopkins

unread,
Jun 2, 2001, 1:19:52 AM6/2/01
to
In article <9f4b4q$bpv$1...@glue.ucr.edu> ba...@math.ucr.edu (John Baez) writes:
>This explains how his new approach to quantum cosmology is related to
>the old "minisuperspace" approach. In the old approach, you just take
>some limited class of cosmologies satisfying the equations of general
>relativity and think of this class as a classical mechanics problem with
>finitely many degrees of freedom: for example, the size of the universe
>together with various numbers describing its shape. Then you quantize
>this classical system.

Just as a diversion ... what's the complication that keeps one from
doing quantum gravity by treating Lorentzian 4-space as being just a
4-surface in a flat 10-space and then quantizing the surface?

This is really the ultimate parametrization of solutions to Einstein's
equations -- the surface, itself, is the parameter.

Urs Schreiber

unread,
Jun 3, 2001, 10:28:49 PM6/3/01
to
John Baez wrote:

>2) Martin Bojowald, The semiclassical limit of loop quantum
>cosmology, available at gr-qc/0105113.
>
>This explains how his new approach to quantum cosmology is related to
>the old "minisuperspace" approach. In the old approach, you just take
>some limited class of cosmologies satisfying the equations of general
>relativity and think of this class as a classical mechanics problem with
>finitely many degrees of freedom: for example, the size of the universe
>together with various numbers describing its shape. Then you quantize
>this classical system.
>
>In this approach, you don't see any hint of spacetime discreteness on
>the Planck scale. But in Bojowald's approach, you do! What gives? He
>still starts with a limited class of cosmologies and quantizes that, but
>he does so using ideas taken from loop quantum gravity. This makes all
>the difference: now areas and volumes have discrete spectra of
>eigenvalues, and this saves us from the horrors of the singularity at
>the big bang. In fact, we can go back *before* the big bang, and find a
>time-reversed expanding universe on the other side!

[...]

There are attempts to extend the "old approach" to quantum cosmology
by replacing the ordinary quantization of minisuperspace cosmology models
by supersymmetric quantization. Basically, one looks for a function
which serves as a superpotential for the ordinary "potential" term in
the Hamiltonian cosmology model. By the standard methods of SUSY QM the
original Hamiltonian is made supersymmetric and thus acquires Fermionic
degrees of freedom which vanish in the hbar --> 0 limit. (I have been told
that this is expected to be the Hamiltonian-cosmology-version of
taking account of the gravitino.)

With regard to Bojowald's "bottom-up" approach to discrete quantum
cosmology this made me wonder if SUSY might provide a "top-down" access
to the problem:
In view of the fact that supersymmetry is closely related to geometry
and to non-commutative geometry in particular, and that non-commutative
geometry is about exotic geometries, could it be possible that
SUSY-Hamiltonian-cosmology somehow does allow or at least help to
"see hints of spacetime discreteness" in "old" quantum cosmology?

I do not know enough about the relation between SUSY and
non-commutative-geometry to make that any more precise. I'd just like
to know if my free association

SUSY cosmology
SUSY -> non commutative geometry ----------------> discrete spacetime

looks suggestive to anybody.


Urs Schreiber

P.S. I know that I am being very vague and speculative again, it's
just that such speculations help me decide where to concentrate my
non-speculative efforts on. A fascinating speculation in the back of
my head helps me and motivates me to navigate through the vast amounts
of knowlegde out there.

--
Urs.Sc...@uni-essen.de

Toby Bartels

unread,
Jun 4, 2001, 9:45:14 PM6/4/01
to
John Baez wrote in small part:

>e7 = ...


> = der(O) + der(H) + sa_3(H tensor O)

> = ...

>I explain why these are true in the paper, but for now, let me
>just say what all this stuff means:

>...


>"der" means the Lie algebra of derivations,

>...

But John neglects to define sa_3(H tensor O).
This is the traceless 3x3 antiHermitian matrices over H tensor O.
(I told him to say "u" instead of "a", but he didn't '_`.)


-- Toby
to...@math.ucr.edu

John Baez

unread,
Jun 6, 2001, 12:14:30 AM6/6/01
to
In article <9f9t1o$lu9$1...@uwm.edu>,

Mark William Hopkins <whop...@alpha2.csd.uwm.edu> wrote:

>Just as a diversion ... what's the complication that keeps one from
>doing quantum gravity by treating Lorentzian 4-space as being just a
>4-surface in a flat 10-space and then quantizing the surface?

First of all, you're being a bit optimistic if you think any Lorentzian
4-manifold can be isometrically embedded in 10d Minkowski spacetime.
Any Riemannian 4-manifold can be isometrically embedded in 10d Euclidean
spacetime, but you need a lot more extra dimensions to handle the Lorentzian
case.

The last time we talked about this, Robert Low mentioned this paper:

Clarke, C. J. S., "On the global isometric embedding of
pseudo-Riemannian manifolds," Proc. Roy. Soc. A314 (1970) 417-428

which apparently proves that any Lorentzian 4-manifold can be embedded
in 91-dimensional flat spacetime with two time dimensions. For more,
see:

http://www.lns.cornell.edu/spr/2001-01/msg0030684.html

Anyway, this is just nitpicking. On to your real question....

A better question might be: what makes you think this approach would make
quantum gravity any easier? If you write down the action for general
relativity in terms of your embedding you'll get a fairly nasty formula,
and also a bunch of gauge symmetries coming from the fact that different
embeddings can describe the same Lorentzian 4-geometry. These gauge
symmetries will include the usual diffeomorphism-invariance of general
relativity, but also extra symmetries as well - e.g., translating the
the whole manifold as embedded in R^{91}. As usual, these gauge symmetries
have to be taken into account when one quantizes. So it'll be a big
mess unless you have some clever ideas.

The most similar thing around is the usual theory of bosonic strings
and membranes. Here the action is simpler: just the area of the worldsheet,
or volume of the "worldvolume". The gauge symmetries are still there
and still tricky. In the case of the string, one can only consistently
handle them in 26 dimensions - otherwise one gets anomalies! In the case
of the membrane, nobody knows what to do. This is why people gave up on
membrane theory back in the old days. Now people think there's something
to it, thanks to other clues... but they're still unable to quantize
the theory in any straightforward way.

By the way, there's no need for the term "maxi-superspace": we're
basically talking about good old "superspace" here.


Lubos Motl

unread,
Jun 6, 2001, 9:50:43 PM6/6/01
to
John Baez wrote:

> First of all, you're being a bit optimistic if you think
> any Lorentzian 4-manifold can be isometrically embedded in 10d
> Minkowski spacetime.

Finally, a Lorentzian manifold can have closed time-like curves but the
10d Minkowski space has no time-like curves. So we need at least two
times, Prof. Itzhak Bars could be happy about that. :-) It is funny that
we need a 89+2 dimensional spacetime, less dimensions are really not
enough? :-) Have they proved that 91 is the minimum?

> membrane theory back in the old days. Now people think there's something
> to it, thanks to other clues... but they're still unable to quantize
> the theory in any straightforward way.

It's better (or worse) than you think. People already know since 1996 why
it is *nonsense* to quantize membranes in the straightforward way, in the
same way as we quantize strings. For example, the 11-dimensional
supermembrane had been known to have continuous spectrum (let us work in
the light cone gauge now), unlike strings whose excitations have
mass^2 = n/alpha' for integer "n".

Since 1996 we know the reason: while people in the old times (80s) thought
that they were quantizing a single membrane (DeWitt, Nicolai and the third
one - sorry), they were in fact quantizing a system that contains an
arbitrary number of membranes whose number and topology can change, as
well as gravitons: the continuous spectrum corresponds to decay of the
membrane into two or more independent membranes or clusters with arbitrary
velocities. In fact the discretized, matrix description of a single
membrane in 11-dimensional spacetime (quantized in the light cone gauge) -
this model is a U(N) matrix quantum mechanics with 16 supercharges - was
shown to describe whole physics of M-theory in 11 asymptotically flat
dimensions, at least in the large N limit. See hep-th/9610043 (Matrix
theory) and its more than 1000 citations (for instance the recent review
of Taylor from 01/01).

This is an example of the surprising way how the new developments can
often answer old questions: the breakthroughs often show us that we were
trying to find an answer to something that does not make sense because we
made wrong assumptions.

Examples: The atoms could not be made stable because people assumed that
the position and momentum commuted and the energy can change continuously
etc. QM solved that in a different way. People did not know how to
"quantize the Fermi's interaction in a straightforward way". But it was
shown that the divergences represent our ignorance about the short
distance physics (and W,Z bosons were included and - of course - finally
observed). Similarly it was not known how to quantize Einstein's
equations: the divergences in fact show new (stringy) physics at
ultrashort distances which was ignored by Einstein (as well as LQG
people). It was not known how to quantize a single membrane. But it was
shown that there is no coupling constant (unlike the case of strings) that
can be chosen small so that the membranes do not interact - and therefore
the correct quantization of a single membrane automatically includes also
multimembrane and multiparticle states.

In all those cases, people had to be open-minded and able to understand
that they were asking questions in a wrong way and some of their questions
assumed something incorrect (usually something that people do not talk
about at all). Finally, disproving wrong prejudices is one of the most
important and exciting manifestations of progress in physics, I think.

Best wishes
Lubos
______________________________________________________________________________
E-mail: lu...@matfyz.cz Web: http://www.matfyz.cz/lumo tel.+1-805/893-5025
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Superstring/M-theory is the language in which God wrote the world.


Phil Gibbs

unread,
Jun 8, 2001, 1:27:25 AM6/8/01
to
In article <9f9t1o$lu9$1...@uwm.edu>,
Mark William Hopkins <whop...@alpha2.csd.uwm.edu> wrote:

>Just as a diversion ... what's the complication that keeps one from
>doing quantum gravity by treating Lorentzian 4-space as being just a
>4-surface in a flat 10-space and then quantizing the surface?

In article <Pine.SOL.4.10.101060...@strings.rutgers.ed
u>, Lubos Motl <mo...@physics.rutgers.edu> writes


>John Baez wrote:
>
>> First of all, you're being a bit optimistic if you think
>> any Lorentzian 4-manifold can be isometrically embedded in 10d
>> Minkowski spacetime.
>
>Finally, a Lorentzian manifold can have closed time-like curves but the
>10d Minkowski space has no time-like curves. So we need at least two
>times, Prof. Itzhak Bars could be happy about that. :-) It is funny that
>we need a 89+2 dimensional spacetime, less dimensions are really not
>enough? :-) Have they proved that 91 is the minimum?
>

You are both being too hard on Mark because you are being far too
theoretical. You should be starting with the experimental side.
Why should Mark care that not all 4D manifolds can be embedded in
a 10D manifold? He only needs to care about the real one which
is our space-time. If a theory excludes space-times with closed
timelike curves it is far from ruled out by that.

Suppose I propose a theory of gravity in which space-time is embedded
in 9+1 Minkowski space and minimises the Einstein-Hilbert action
under this constraint. Can that be ruled out by observation? What
about the more general question for embeddings in s+t Minkowski space.
For which (s,t) can this be ruled out by observation? Let's start
with (5,1) and work our way upwards.


John Baez

unread,
Jun 8, 2001, 6:58:37 PM6/8/01
to
In article <Pine.SOL.4.10.101060...@strings.rutgers.edu>,
Lubos Motl <mo...@physics.rutgers.edu> wrote:

>People already know since 1996 why
>it is *nonsense* to quantize membranes in the straightforward way, in the
>same way as we quantize strings. For example, the 11-dimensional
>supermembrane had been known to have continuous spectrum (let us work in
>the light cone gauge now), unlike strings whose excitations have
>mass^2 = n/alpha' for integer "n".

I've heard that a lot of times, but I never understood what's
so terrible about a continuous spectrum, and people never seem
to say. Is it just that particle physicists like particles with
discrete masses? Or was there some actual reason why a continuous
spectrum led to some mathematical inconsistency?

>Since 1996 we know the reason: while people in the old times (80s) thought
>that they were quantizing a single membrane (DeWitt, Nicolai and the third
>one - sorry), they were in fact quantizing a system that contains an
>arbitrary number of membranes whose number and topology can change, as
>well as gravitons: the continuous spectrum corresponds to decay of the
>membrane into two or more independent membranes or clusters with arbitrary
>velocities. In fact the discretized, matrix description of a single
>membrane in 11-dimensional spacetime (quantized in the light cone gauge) -
>this model is a U(N) matrix quantum mechanics with 16 supercharges - was
>shown to describe whole physics of M-theory in 11 asymptotically flat
>dimensions, at least in the large N limit. See hep-th/9610043 (Matrix
>theory) and its more than 1000 citations (for instance the recent review
>of Taylor from 01/01).
>
>This is an example of the surprising way how the new developments can
>often answer old questions: the breakthroughs often show us that we were
>trying to find an answer to something that does not make sense because we
>made wrong assumptions.

I've seen some of this stuff, and it suggests that this continuous
spectrum was not so terrible after all.

>[...] disproving wrong prejudices is one of the most


>important and exciting manifestations of progress in physics, I think.

I agree!

Robert Low

unread,
Jun 8, 2001, 8:44:49 PM6/8/01
to
Lubos Motl <mo...@physics.rutgers.edu> wrote:
>John Baez wrote:
>
>> First of all, you're being a bit optimistic if you think
>> any Lorentzian 4-manifold can be isometrically embedded in 10d
>> Minkowski spacetime.
>
>Finally, a Lorentzian manifold can have closed time-like curves but the
>10d Minkowski space has no time-like curves. So we need at least two
>times, Prof. Itzhak Bars could be happy about that. :-) It is funny that
>we need a 89+2 dimensional spacetime, less dimensions are really not
>enough? :-) Have they proved that 91 is the minimum?

Oh no. It Clarke's paper shows that any Lorentz (1+3) manifold, M, can
be embedded in E^{2,48} if M is compact and E^{2,89} if M is not
compact. If M is globally hyperbolic (and a forteriori not compact)
then you can get away with E^{1,88}.

It is not known how tight those estimates are, but the last time
I asked Clarke about it, he suspected that they were very
generous. I don't think anybody has done much about improving
the dimensions.

--
Rob. http://www.mis.coventry.ac.uk/~mtx014/

John Baez

unread,
Jun 8, 2001, 10:05:39 PM6/8/01
to
In article <1mJYyKA9...@194.222.186.18>,
Phil Gibbs <philip...@weburbia.com> wrote:

>You are both being too hard on Mark because you are being far too
>theoretical. You should be starting with the experimental side.

Ah yes, experimental results on embedding 4d manifolds in 10d
spacetime. Let's see...

... hmm, there aren't any! Okay, on to the theoretical stuff. :-)

>Why should Mark care that not all 4D manifolds can be embedded in
>a 10D manifold? He only needs to care about the real one which
>is our space-time.

I think we were talking about theories of quantum gravity, and I
think Mark was proposing to express the path integral over all
Lorentzian metrics as a path integral over all embeddings of 4d
spacetime in some higher-dimensional flat spacetime. In this
context it was not too nasty to point out that, as far as we know,
one only gets *all* Lorentzian 4-manifolds as submanifolds of a
91-dimensional flat spacetime with 2 time dimensions. The reason,
of course, is that in a path integral, one wants to integrate over
*all* metrics.

>If a theory excludes space-times with closed
>timelike curves it is far from ruled out by that.

The usual way to exclude closed timelike curves and such
pathologies is to restrict attention to globally hyperbolic
spacetimes. Chris Clarke showed that we can get all globally
hyperbolic 4-dimensional Lorentzian manifolds as submanifolds
of 89-dimensional Minkowski spacetime. So, we can get rid of
two space dimensions and one time dimension this way.

Of course, nobody seems to claim that Clarke's results are
optimal, so these funny large numbers should not be taken too
seriously.

>Suppose I propose a theory of gravity in which space-time is embedded
>in 9+1 Minkowski space and minimises the Einstein-Hilbert action
>under this constraint. Can that be ruled out by observation?

Not if we're talking about classical gravity. If we're talking
about quantum gravity, you'll need to propose the theory before
we can rule it out. :-)


Ralph E. Frost

unread,
Jun 10, 2001, 9:07:57 AM6/10/01
to

John Baez <ba...@galaxy.ucr.edu> wrote in message
news:9fs09j$eda$1...@glue.ucr.edu...

> In article <1mJYyKA9...@194.222.186.18>,
> Phil Gibbs <philip...@weburbia.com> wrote:

> >You are both being too hard on Mark because you are being far too
> >theoretical. You should be starting with the experimental side.
>
> Ah yes, experimental results on embedding 4d manifolds in 10d
> spacetime. Let's see...
>
> ... hmm, there aren't any! Okay, on to the theoretical stuff. :-)

You miss the point EVERY TIME you jump to this conclusion, _particularly_
when it comes to discussing quantum gravity.

The fact that you don't say, "that sounds good but I don't know how to do
it" prevents serves to prevent that leg of the discussion and exploration
from beginning.

IMO...

> Not if we're talking about classical gravity. If we're talking
> about quantum gravity, you'll need to propose the theory before
> we can rule it out. :-)

Yeah, well, like I said, you always find your keys in the last place you
bother to look.

Try, please, to open up a mediation channel on the experimental side.

Squark

unread,
Jun 9, 2001, 7:23:07 AM6/9/01
to
On Sat, 9 Jun 2001 02:05:39 +0000 (UTC), John Baez wrote (in
<9fs09j$eda$1...@glue.ucr.edu>):

>>Suppose I propose a theory of gravity in which space-time is embedded
>>in 9+1 Minkowski space and minimises the Einstein-Hilbert action
>>under this constraint. Can that be ruled out by observation?

>Not if we're talking about classical gravity.

So, any Lorentzian 4-manifold satisfying the Einstein equations in vacuo,
and having no close time-like curves, I suppose, can be embedded in 10d
Minkowski space? This is interesting, though evidently fails when we add
matter...

Best regards,
Squark.

--------------------------------------------------------------------------------
Write to me at:
[Note: the fourth letter of the English alphabet is used in the later
exclusively as anti-spam]
dSdqudarkd_...@excite.com

John Baez

unread,
Jun 11, 2001, 2:16:53 AM6/11/01
to
In article <vGnU6.1890$pb1....@www.newsranger.com>,
Squark <dSdqudarkd_...@excite.com> wrote:

>On Sat, 9 Jun 2001 02:05:39 +0000 (UTC), John Baez wrote (in
><9fs09j$eda$1...@glue.ucr.edu>):

>> Phil Gibbs wrote:

>>Suppose I propose a theory of gravity in which space-time is embedded
>>>in 9+1 Minkowski space and minimises the Einstein-Hilbert action
>>>under this constraint. Can that be ruled out by observation?

>>Not if we're talking about classical gravity.

>So, any Lorentzian 4-manifold satisfying the Einstein equations in vacuo,
>and having no close time-like curves, I suppose, can be embedded in 10d
>Minkowski space?

No, I didn't mean this - I have no idea whether or not *this* is true!

Since Phil Gibbs asked about "observation", I was simply claiming
that *our observed universe*, not "any Lorentzian manifold satisfying
the Einstein equations in vacuo, and having no time-like curves",


can be isometrically embedded in 10d Minkowski spacetime.

But in retrospect, I shouldn't have even made *that* claim! I was
thinking about something like a Robertson-Friedman-Walker universe.
I have no idea whether our actual universe, with all its glorious
wiggles and bumps, can be isometrically embedded in 10d Minkowski
spacetime. So I retract my claim.

In short: I have no idea what the answer to Phil Gibbs' question is.

Phil Gibbs

unread,
Jun 11, 2001, 5:08:20 PM6/11/01
to
In article <9fs09j$eda$1...@glue.ucr.edu>, John Baez <ba...@galaxy.ucr.edu>
writes

>>Why should Mark care that not all 4D manifolds can be embedded in
>>a 10D manifold? He only needs to care about the real one which
>>is our space-time.
>
>I think we were talking about theories of quantum gravity,

Yes, but before quantising the theory it is reasonable to
consider whether or not it works classically. I would be surprised
if we cannot explain all observations of gravitational effects
without going beyond embeddings in 10D Minkowski. In fact, embeddings
in 5D Minkowski would be sufficient to explain the precession of
Mercury and bending of starlight, because the Schwarzchild solution
embeds in 5D Minkowski (Fronsdal 1959) and I would not be surprised if
even the decay of orbits for binary pulsars could be explained
with 6D or 7D embeddings. So to say that we need more than 10D is
probably not even true classically let alone for quantum gravity.

Since there are very few diffeomorphism invariant alternatives to
GR it surprises me that lower dimensional embedding have not been
considered, if only as alternatives to test against. Now back to
Mark's question.

>and I
>think Mark was proposing to express the path integral over all
>Lorentzian metrics as a path integral over all embeddings of 4d
>spacetime in some higher-dimensional flat spacetime.

I don't see the word "all" used in his question :-) He was merely
asking what is the complication when trying to quantise gravity
using the parameters of an embedding in flat 10D space-time. I don't
think that the correct answer is that more than 10 would be required.

Actually, without meaning to drive the knife in further, I found your
other answers unconvincing too!

In article <GEHqG...@world.std.com>, John Baez <ba...@galaxy.ucr.edu>
writes


>A better question might be: what makes you think this approach would make
>quantum gravity any easier? If you write down the action for general
>relativity in terms of your embedding you'll get a fairly nasty formula,

I suppose it depends what you mean by nasty. The expression for the
curvature tensor in terms of embedding co-ordinates is quite clean.
I think it is less complicated than the expression for the curvature
tensor in terms of the metric. It is nasty in the sense that it involves
higher derivatives than you might like. Perhaps this is the real
complication when quantising. Is that what you meant?

>and also a bunch of gauge symmetries coming from the fact that different
>embeddings can describe the same Lorentzian 4-geometry.

Gauge symmetries may well make the maths more complex, but they never
stopped anyone quantising Yang Mills theory. Are you saying that these
ones are more difficult to deal with?

I wonder if anyone has ever attempted it.


Phil Gibbs

unread,
Jun 11, 2001, 10:19:19 PM6/11/01
to
In article <GEHqG...@world.std.com>, John Baez <ba...@galaxy.ucr.edu>
writes
>A better question might be: what makes you think this approach would make
>quantum gravity any easier? If you write down the action for general
>relativity in terms of your embedding you'll get a fairly nasty formula,
>and also a bunch of gauge symmetries coming from the fact that different
>embeddings can describe the same Lorentzian 4-geometry. These gauge
>symmetries will include the usual diffeomorphism-invariance of general
>relativity, but also extra symmetries as well - e.g., translating the
>the whole manifold as embedded in R^{91}. As usual, these gauge symmetries
>have to be taken into account when one quantizes. So it'll be a big
>mess unless you have some clever ideas.

This has got to be one case where at least the choice of gauge is
easy. You can just set the four space-time co-ordinates to coincide
with four of the co-ordinates of the embedding space. That then just
leaves a global rotation and translation symmetry over the remaining
co-ordinates. For the quantum theory you will have to work out the
contribution for integrating out the gauge symmetry as usual. I have
no idea if that is doable.

I can't imagine that the result is renormalisable and perhaps that is
the real answer as to why the approach fails, but it would be
interesting to know if anyone has ever got as far as proving it.

Robert C. Helling

unread,
Jun 11, 2001, 10:29:47 PM6/11/01
to
On 8 Jun 2001 22:58:37 GMT, John Baez <ba...@galaxy.ucr.edu> wrote:
>In article <Pine.SOL.4.10.101060...@strings.rutgers.edu>,
>Lubos Motl <mo...@physics.rutgers.edu> wrote:
>
>>People already know since 1996 why
>>it is *nonsense* to quantize membranes in the straightforward way, in the
>>same way as we quantize strings. For example, the 11-dimensional
>>supermembrane had been known to have continuous spectrum (let us work in
>>the light cone gauge now), unlike strings whose excitations have
>>mass^2 = n/alpha' for integer "n".
>
>I've heard that a lot of times, but I never understood what's
>so terrible about a continuous spectrum, and people never seem
>to say. Is it just that particle physicists like particles with
>discrete masses? Or was there some actual reason why a continuous
>spectrum led to some mathematical inconsistency?
>

My understanding is that it is the former. For the string, once you've
gotten rid of the tachyon (if you don't like it) you are left with
the massless fields, the gauge fields and the metric (among others).
The higher spin fields are made arbitraily unimportant by sending the
string tension to infinity (or some large value motivated by the value
of the Planck mass).

If you want asymptotic particle states a continuous spectrum appears unphysical.
The point is just that with supermembranes (super is important here),
there are no asymptotic separated particles. As far as I can see,
at least for fixed N, there are no mathematical inconsistencies, it's
just ordinary quantum mechanics. But there are open questions regarding
the N->infinity limit, such as renormalizability and Lorentz invariance.

For a review, see for example Hermann Nicolai's Trieste lecture notes
hep-th/9809103.

Robert

PS: The third guy is Jens Hoppe (for: Quantum mechanics of Supermembranes)
or Matrin L\"uscher (The Supermembrane is Unstable). And B. de Wit always
likes to point out that he is not a relative of the de Witts (Cecille and
Bryce)


--
.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Institut fuer Physik
Humboldt-Universitaet zu Berlin
print "Just another Fon +49 30 2093 7964
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling

Squark

unread,
Jun 12, 2001, 5:50:56 PM6/12/01
to
Phil, some things seem to me pretty fishy in this "lower dimensional
embedding" idea. When you restrict yourself to a smaller class of
metrics, namely, those embeddable in n-dimensional Minkowski space-time,
the equations of motion become modified. This is because:
A) You add some sort of constraints, which leads to strengthening of the
equations.
B) The variation of the metric is now limit, what leads to weaking of
the equations.
However, the Einstein equations are the only (lowest order) equations
which are diffeomorphism invaraint and give the Newton equations in the
appropriate approximation. Therefore, I can't see how consistency with
those two basic requirements is retained (I don't think adding higher
order terms would do, as it would probabely still allow most of the
metrics). Of course, you can try to throw one out of the window but:
- Throwing out diffeomorphism invariance is unpleasant, as it is one of
the basic principle of GR.
- Throwing out the Newtonian approximation will lead to inconsistencies
with experiment.
On the other hand, why wouldn't you go for 89+1d space-time? A priori, I
don't see how might it make things worse. Of course, the classification of
90-manifolds is more than non-existent, but you only want good old flat
Minkowski, no? For the matter, you can even try doing something in 91+2d
(was it 91+2?)...

Phil Gibbs

unread,
Jun 13, 2001, 2:03:55 PM6/13/01
to
In article <perV6.5352$pb1.2...@www.newsranger.com>, Squark <dSdqudark
d_Nucdl...@excite.com> writes

>Phil, some things seem to me pretty fishy in this "lower dimensional
>embedding" idea. When you restrict yourself to a smaller class of
>metrics, namely, those embeddable in n-dimensional Minkowski space-time,
>the equations of motion become modified. This is because:
>A) You add some sort of constraints, which leads to strengthening of the
>equations.
>B) The variation of the metric is now limit, what leads to weaking of
>the equations.
>However, the Einstein equations are the only (lowest order) equations
>which are diffeomorphism invaraint and give the Newton equations in the
>appropriate approximation. Therefore, I can't see how consistency with
>those two basic requirements is retained (I don't think adding higher
>order terms would do, as it would probabely still allow most of the
>metrics).

The co-ordinates of the higher dimensional space in which space-time
is embedded become the field variables of the theory. The metric is
expressed in terms of the first derivatives of these fields, so I
suppose this works out as theory which uses higher derivatives and
so it gets round your no-go theorem.

Somehow I think that Brans-Dicke theory might evade your no-go theorem
too. Perhaps you need to state the theorem more carefully with all
its assumptions so we can see what is going on.

>Of course, you can try to throw one out of the window but:
>- Throwing out diffeomorphism invariance is unpleasant, as it is one of
>the basic principle of GR.

It does not throw out diffeomorphism invariance. You can keep general
co-ordinate transformations on the space-time co-ordinates, or you
can fix the gauge by setting the space-time co-ordinates to coincide
with some of the co-ordinates of the space in which you are embedding.
This is just gauge fixing, not a further dynamic constraint.

>- Throwing out the Newtonian approximation will lead to inconsistencies
>with experiment.

Even 5D embedding include the Schwarzchild metric so I dont think we
are throwing out the Newtonian approximation

>On the other hand, why wouldn't you go for 89+1d space-time? A priori, I
>don't see how might it make things worse. Of course, the classification of
>90-manifolds is more than non-existent, but you only want good old flat
>Minkowski, no? For the matter, you can even try doing something in 91+2d
>(was it 91+2?)...
>

There is nothing wrong in principle with using as many dimensions as you
like, but if you have more variables than equations the solution is
indeterminate. For example, if I can embed a space-time in 26 dimensions
then I can embed it in 27 with the 27th co-ordinate being an arbitrary
function of the 26th co-ordinate. I don't want arbitrary functions in my
solutions beyond the gauge symmetries in the dynamic equations, so I
want to use the minimum number, at least classically. By limiting
myself in this way I will be introducing topological constraints, but
not dynamic constraints. The topological constraints would make a
difference in the quantum theory. They might disallow virtual wormholes
for example. This might be a bad thing or it might be a good thing.

Anyway, I like the idea of seeing what happens if you use less than 10
dimensions so that you have dynamic constraints. I am curious to know
if and how this conflicts with observation classically. I am also
curious to know what happens when you try to quantise. Like Mark who
asked the original question, I am curious to know at what point this
becomes difficult.

By the way. I found some old notes of mine which give some of the
equations so in case anyone is interested, Let X^A be the co-ordinates
of the higher dimensional embedding space, A = 1,...,N where N is
e.g. 5,10 or 91. Let x^a be the space-time co-ordinates, a = 1,..,4
Let G_AB be the Minkowski metric and g_ab be the Einstein metric.
Then

g_ab = G_AB X^A,a X^B,b

There are implied sums over A and B and the comma means partial
derivative.

The connection is Gamma_abc = G_AB X^A,bc X^B,a

And the curvature tensor is

R_abcd = G_AB X^A;ac X^B;bd - G_AB X^A;ad X^B;bc

Where the semi-colons are covariant derivatives using the connection.
Notice that there are third derivatives of the X co-ordinates in there.

I think this shows that the equations involved are not very
complicated. The third derivatives may be nasty, but there is no
reason in principle why you cant try to quantise with them there.


Squark

unread,
Jun 14, 2001, 12:17:59 PM6/14/01
to
On 13 Jun 2001 18:03:55 GMT, Phil Gibbs wrote (in
<9g89ub$lnb$1...@news.state.mn.us>):

>The co-ordinates of the higher dimensional space in which space-time
>is embedded become the field variables of the theory. The metric is
>expressed in terms of the first derivatives of these fields, so I
>suppose this works out as theory which uses higher derivatives and
>so it gets round your no-go theorem.

It is difficult for me to see how can you have a metric field theory
which is:

A) Local
B) Diffeomorphism-invariant
C) Gives Newtonian physics in the appropriate approximation.
D) Disallows a large part of the metrics (probabely)

Btw, I don't know about Brans-Dicke theory - is it the same order as GTR?
Maybe it inserts small corrections into the Newtonian approximation?
Also, the mere fact the Schwarzchild metric is in doesn't ensure the
Newtonian approximation is right. Complicated matter distributions may
mess things up (you don't have a superposition principle :-) ).

>There is nothing wrong in principle with using as many dimensions as you
>like, but if you have more variables than equations the solution is
>indeterminate. For example, if I can embed a space-time in 26 dimensions
>then I can embed it in 27 with the 27th co-ordinate being an arbitrary
>function of the 26th co-ordinate.

Why so?? It seems to me this would change your metric considerably...

>I don't want arbitrary functions in my solutions beyond the gauge
>symmetries in the dynamic equations, so I want to use the minimum
>number, at least classically.

I dunno, you might as well try dealing with all of the gauge theory. Time
for simplification comes when you fail to deal with the big thing, if you
ask me.

>...


>I think this shows that the equations involved are not very
>complicated.

The real question is what happens when we use the variational prinicple
(delta)S_EH = 0, where S_EH is the Einstein-Hilbert action, and delta
stands for variations of the worldvolume.

John Baez

unread,
Jun 14, 2001, 7:52:06 PM6/14/01
to
In article <9g3c04$b8s$1...@news.state.mn.us>,
Phil Gibbs <philip...@weburbia.com> wrote:

>So to say that we need more than 10D is
>probably not even true classically let alone for quantum gravity.

As I admitted in a post subsequent to those you were replying to
here, I haven't the foggiest idea if this is true or not. I don't
know where you get your intuition that the metric people are using
to model binary pulsars can be isometrically embedded in 6- or 7-
dimensional Minkowski spacetime... but anyway, I have no intuitions
about this.

>In article <9fs09j$eda$1...@glue.ucr.edu>, John Baez <ba...@galaxy.ucr.edu>
>writes

>>[there are] also a bunch of gauge symmetries coming from the fact that

>>different embeddings can describe the same Lorentzian 4-geometry.

>Gauge symmetries may well make the maths more complex, but they never
>stopped anyone quantising Yang Mills theory. Are you saying that these
>ones are more difficult to deal with?

They seem a lot less well-understood, at least to me. For example:
if we are trying to describe the metric on the unit sphere by embedding
it in flat Euclidean 3-space, the group of gauge symmetries is
just the 3d Euclidean group. In other words, any two isometric
embeddings of the round unit 2-sphere in flat 3-space differ by
a rotation/translation/reflection.

The same is true if we work with just the top hemisphere of the unit
2-sphere. The same is true if we flatten that top hemisphere out a bit.
But if we're trying to describe a perfectly flat 2-disk this way, the
group of gauge symmetries suddenly becomes infinite-dimensional! The
reason is that one can "flex" the embeddded 2-disk without changing its
induced metric.

This "jumping" of the size of the gauge symmetries has to be taken
into account when quantizing. This general type of thing happens
in many other contexts, e.g. Yang-Mills theory and general relativity,
but not in such a drastic fashion. Also, in those other contexts
it's well-understood, thanks to the work of people like Fischer and
Moncrief and Arms. This situation seems scarier to me.

>I wonder if anyone has ever attempted it.

Dunno!


Squark

unread,
Jun 15, 2001, 12:43:00 AM6/15/01
to
On 13 Jun 2001 18:03:55 GMT, Phil Gibbs wrote (in
<9g89ub$lnb$1...@news.state.mn.us>):
>The co-ordinates of the higher dimensional space in which space-time
>is embedded become the field variables of the theory. The metric is
>expressed in terms of the first derivatives of these fields, so I
>suppose this works out as theory which uses higher derivatives and
>so it gets round your no-go theorem.

It is difficult for me to see how can you have a metric field theory
which is:

A) Local
B) Diffeomorphism-invariant
C) Gives Newtonian physics in the appropriate approximation.
D) Disallows a large part of the metrics (probabely)

Btw, I don't know about Brans-Dicke theory - is it the same order as GTR?
Maybe it inserts small corrections into the Newtonian approximation?
Also, the mere fact the Schwarzchild metric is in doesn't ensure the
Newtonian approximation is right. Complicated matter distributions may
mess things up (you don't have a superposition principle :-) ).

>There is nothing wrong in principle with using as many dimensions as you


>like, but if you have more variables than equations the solution is
>indeterminate. For example, if I can embed a space-time in 26 dimensions
>then I can embed it in 27 with the 27th co-ordinate being an arbitrary
>function of the 26th co-ordinate.

Why so?? It seems to me this would change your metric considerably...

>I don't want arbitrary functions in my solutions beyond the gauge


>symmetries in the dynamic equations, so I want to use the minimum
>number, at least classically.

I dunno, you might as well try dealing with all of the gauge theory. Time


for simplification comes when you fail to deal with the big thing, if you
ask me.

>...


>I think this shows that the equations involved are not very
>complicated.

The real question is what happens when we use the variational prinicple

Squark

unread,
Jun 18, 2001, 8:40:12 PM6/18/01
to
On Thu, 14 Jun 2001 23:52:06 +0000 (UTC), John Baez wrote (in
<9gbin6$g7t$1...@glue.ucr.edu>):

>This "jumping" of the size of the gauge symmetries has to be taken
>into account when quantizing. This general type of thing happens
>in many other contexts, e.g. Yang-Mills theory and general relativity,
>but not in such a drastic fashion.

Hmm, this seems to be calling for the use of stacks... Has anyone tried
it?

>Also, in those other contexts it's well-understood, thanks to the work
>of people like Fischer and Moncrief and Arms.

Really? Where can I read about this?

Best regards,
Squark.

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