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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.

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

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

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

obtained at

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

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

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

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

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

If you just want the latest issue, go to

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.

>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.

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.

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

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:

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.

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.

Jun 8, 2001, 1:27:25 AM6/8/01

to

>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.

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:

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!

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?

>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.

Jun 8, 2001, 10:05:39 PM6/8/01

to

In article <1mJYyKA9...@194.222.186.18>,

Phil Gibbs <philip...@weburbia.com> wrote:

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. :-)

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.

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>):

<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

Jun 11, 2001, 2:16:53 AM6/11/01

to

In article <vGnU6.1890$pb1....@www.newsranger.com>,

Squark <dSdqudarkd_...@excite.com> wrote:

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.

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,

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.

Jun 11, 2001, 10:19:19 PM6/11/01

to

>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.

>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.

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?

>

>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

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?)...

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?)...

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).

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.

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.

<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.

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:

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!

Jun 15, 2001, 12:43:00 AM6/15/01

to

>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.

>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

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.

<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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