# 4 11 26 dimensions?

#### Life|Time

Hi
I'm reading this book at the moment called "Parallel Worlds" By Michio Kaku. It's really interesting and has lots of ideas about different quantum processes and stuff, but there is one point I really don't understand.
He sometimes talks about 5, 10 or 11 dimensions (and even at one point 26!)...I really can't see or imagine these dimensions. I mean, I get the first four but whats after that?

I just can't "picture" it.

Any explanations would be really helpful

thanks

Related Beyond the Standard Model News on Phys.org

#### DaveC426913

Gold Member
Many of the higher dimensions are hypothesized to be curled up very small.

Imagine ants living on a garden hose. The hose is infinitely long and infinitely thin - they live in a one-dimensional universe where they are free to move back and forth along its length. They call this dimension "LENGTH". They know of no other freedoms of movement in their universe, and must crawl over each other to get back and forth.

Now, one of the ant scientists says "Hey, did you know our universe is NOT merely one dimensional, it's 2 dimensional! The 2nd dimension is just so tiny and curled up that we don't perceive it! I call this dimension CIRCUMFERENCE."

And so the ant perfects a device that can expand this tiny dimension up to visible scales. They discover they're living on a garden hose with a measurable circumference. Now the ants can travel, not simply along their universes' length, but also around its circumference.

So with our 3D (or 4D) world, many of these higher dimensions are all around us, but curled up so small that we have no freedom of movement through them. Or more accurately, we are moving through them all the time, but they are so short that they don't slow us down in our passage through the familiar 4.

BTW, I highly, highly recommend Brian Greene's 'The Elegant Universe' - it expanded my mind about higher dimensions.

Last edited:

#### Life|Time

Right! That makes sense!
But how does Michio Kaku sometimes know that we need 11 dimensions for something? I suppose it would be more helpful if I had an example...
I couldn't imagine what the other dimensions maybe. lol
I might have to have a look round the Internet.

Thanks

#### Kalimaa23

Some insights :

If you naievely start out with a relativistic string, and then quantify it, you see that your theory is inconsistent exept when the dimenion of your space-time equals 26.This is the so called bosonic string theory. Because it does not include fermions, it is not a good theory for the natural world.

The same phenomenon manifests itself when considering superstrings, i.e. a string theory with supersymmetry on the world sheet. Exept that in this case, you get D=10.

The eleven dimensional case comes from an older theory called supergravity. Here you start out with Einstein's general relativity, and than add supersymmetry. The spacetime of this theory is not fixed, but a quick calculation shows that the spacetime dimension may not exceed 11.

It has been conjectured that certain 10D string theories are dual to 11D supergravity, but this has not been proven rigorously yet. Hope this helps.

#### Life|Time

Yeah, that does help!
Does anyone know of any books that go over this topic and include the maths? Unfortunately, mine doesn't so I think it would be good if I could see where all these numbers are coming from!

#### Parlyne

Life|Time said:
Yeah, that does help!
Does anyone know of any books that go over this topic and include the maths? Unfortunately, mine doesn't so I think it would be good if I could see where all these numbers are coming from!
If you have some background knowledge about physics (an undergraduate major, for example), you might find Zwiebach's book illuminating:

https://www.amazon.com/gp/product/0521831431/sr=8-1/qid=1150667188/ref=pd_bbs_1/002-5677392-7486461?%5Fencoding=UTF8

Without background in physics, I'm not sure the math would mean much to you.

Also, from personal experience, the particular calculation you're talking about is really long (it took me 8 pages, and I write pretty small) and doesn't really lead to much understanding of the issue.

You might just be happier taking our word that the critical dimension arises from requiring that the string theory obey special relativity.

#### Farsight

Life | Time :

IMHO you should read up on "Flatlanders". These are hypothetical two-dimensional creatures whose whole Universe is a surface. They nothing about the third dimension. It's like they're drawings on a balloon, or on a piece of paper. If their surface is rumpled like scrunched-up paper, they can't see it, because they're rumpled along with their whole world. But they can feel a "force" kind of pulling them back when they try to climb a rumple. It seems like something magical, something that's action-at-a-distance with no discernible cause. But all it is, is a hidden dimension.

But also look up dimension. A dimension doesn't have to be an extra spatial direction. It really means "fundamental parameter". So for a Flatlander we could be talking about the "roughness" of their surface. This sort of thing might also translate into something that they can only experience indirectly.

#### DaveC426913

Gold Member
Shoot, I once read an explanation that tied 4, 6, 10 and 26 dimensions together.

Our familiar 4: 3 space and 1 time,
- the extra 6 that split off and shrunk out of sight following the Big Bang, making 10.
- one of those extra 6 is also a time dimension, meaing there are actually 8 spatial dimensions and 2 time dimensions,
- someone postulated that those 8 spatial dimensions are each three spatial dimensions, making a total of 24, add the 2 time dimensions and you have 26.

It's a lousy explanation but it sort of ties together the numbers.

#### Cexy

DaveC426913 said:
Shoot, I once read an explanation that tied 4, 6, 10 and 26 dimensions together.

Our familiar 4: 3 space and 1 time,
- the extra 6 that split off and shrunk out of sight following the Big Bang, making 10.
- one of those extra 6 is also a time dimension, meaing there are actually 8 spatial dimensions and 2 time dimensions,
- someone postulated that those 8 spatial dimensions are each three spatial dimensions, making a total of 24, add the 2 time dimensions and you have 26.

It's a lousy explanation but it sort of ties together the numbers.
I think that you would do much better to completely ignore this, as as far as I can tell it's junky numerology - and not even very convincing numerology at that.

#### DaveC426913

Gold Member
Cexy said:
I think that you would do much better to completely ignore this, as as far as I can tell it's junky numerology - and not even very convincing numerology at that.
Well then you can ignore Brian Greene's books on the subject.

#### SdogV

Heim-Droscher theory?

Anyone aware of this new direction by a German cripple?

Staff Emeritus
Gold Member
Dearly Missed
Last edited by a moderator:

#### Kalimaa23

DaveC426913 said:
Shoot, I once read an explanation that tied 4, 6, 10 and 26 dimensions together.

Our familiar 4: 3 space and 1 time,
- the extra 6 that split off and shrunk out of sight following the Big Bang, making 10.
- one of those extra 6 is also a time dimension, meaing there are actually 8 spatial dimensions and 2 time dimensions,
- someone postulated that those 8 spatial dimensions are each three spatial dimensions, making a total of 24, add the 2 time dimensions and you have 26.

It's a lousy explanation but it sort of ties together the numbers.
Codswallop.

All the extra dimensions are space-like, and compactified on a certain geometrical shape (torus, orientifold, Calabi-Yau...)

#### coalquay404

Dimitri Terryn said:
Codswallop.

All the extra dimensions are space-like, and compactified on a certain geometrical shape (torus, orientifold, Calabi-Yau...)
Precisely. Regardless of how one feels about the justification of these compactifications, they all share the property that the compact directions are spacelike.

#### nitin

DaveC426913 said:
Shoot, I once read an explanation that tied 4, 6, 10 and 26 dimensions together.

Our familiar 4: 3 space and 1 time,
- the extra 6 that split off and shrunk out of sight following the Big Bang, making 10.
- one of those extra 6 is also a time dimension, meaing there are actually 8 spatial dimensions and 2 time dimensions,
- someone postulated that those 8 spatial dimensions are each three spatial dimensions, making a total of 24, add the 2 time dimensions and you have 26.

It's a lousy explanation but it sort of ties together the numbers.
I'm pretty sure this is not the way to think about it. J. Baez has a nice http://math.ucr.edu/home/baez/week126.html" [Broken] on Week 126 as to why the classical bosonic string theory requires a 25+1-dimensional space-time. The Riemann zeta function is for something. As Baez points out, one can also find this in Green-Schwarz-Witten Vol. I, pp 95-96 (this might also figure in Zwiebach's, but I am not sure). To tell you truly, though it mathematically makes sense, it fails to convince me totally.

Last edited by a moderator:

#### nitin

Dimitri Terryn said:
Some insights :

If you naievely start out with a relativistic string, and then quantify it, you see that your theory is inconsistent exept when the dimenion of your space-time equals 26.This is the so called bosonic string theory. Because it does not include fermions, it is not a good theory for the natural world.

The same phenomenon manifests itself when considering superstrings, i.e. a string theory with supersymmetry on the world sheet. Exept that in this case, you get D=10.

The eleven dimensional case comes from an older theory called supergravity. Here you start out with Einstein's general relativity, and than add supersymmetry. The spacetime of this theory is not fixed, but a quick calculation shows that the spacetime dimension may not exceed 11.

It has been conjectured that certain 10D string theories are dual to 11D supergravity, but this has not been proven rigorously yet. Hope this helps.
Dimitri has got it, nice and simple. Still, one has to answer "why", which would take one into pages of algebra here.

#### Kalimaa23

I'll probably have an explanation on my blog in a few days, so let me just give the main points here :

One of the strengths of string theory is that the field theory you define on the world sheet is a conformal field theory, i.e. it is invariant under arbitrary local rescalings of the metric. This extra symmetry along with reparametrisation invariance of the two worldsheet coordinates makes sure that the worldsheet metric does not have any dynamics of it's own, and that it can always be chosen to be "conformally flat". This is called the conformal gauge.

The quantum theory exhibits an anomaly, the conformal symmetry is broken on the quantum level. This manifests itself in the fact that the Virasoro-algebra (that contains both the diffeomorphism invariance and the conformal symmetry generators) gets what is known as a central charge, an extra term in the commutation relations.

On a more physical level, it comes from the fact that one of the Virasoro operators (the one with the central term) is the String Hamiltonian. When promoted to a quantum operator it gets an ordering ambiguity, that can absorbed in a constant that is to be identified with the vacuum Casimir energy of an infinite amount of harmonic oscillators. Of course formally this energy is infite, because each oscillator gives an amount of energy hbar/2.

To total energy is then given by

$$\epsilon_{0} = \frac{d-2}{2} \sum_{m=1}^{\infty} m$$

The factor d-2 comes from the fact that only the transverse oscillator modes contribute to the energy, similar to the situation in QED when one imposes the Gubta-Bleuler conditions. The infinite sum can be represented by the Riemann Zeta-function

$$\zeta(z)= \sum_{m=1}^{\infty} \frac{1}{n^z}$$

This can be analytically continued to it's value in -1, which is what we need. It turns out that this is equal to -1/12.

$$\epsilon_{0} = \frac{d-2}{2} \zeta(-1)=-\frac{d-2}{24}$$

Later, when you check the particle content of your theory, you note that Lorentz invariance (amongst other reasons) demands that this factor is equal to -1. So this fixes d=26. A similar computation can be done in superstring theories, but this gives d=10.

#### colibri

sorry just in french CERN LHC see www.cern.ch

Dans le cas d'une pyramide Il y a 4 faces et un centre de gravité 21 éléments la compose (haut bas gauche droite longueur largeur et l'épaisseur donc 4x7= 21 éléments. (La recherche avance et nous trouverons sûrement ces 21 éléments 20 sont déjà trouvés.

conclusion: IL y a bien 11 dimensions (6 extérieures 5 Intérieures) dans l’univers.
le CERN qui recherche (boson de Higgs , des quark avec alice)

. La règle électrique de la main droite ou gauche : champ chemin courant et un point de référence ca fait déjà 4 dimensions certaines.

Imaginez vous la forme d'une pyramide l’univers n’est pas carré ! L’espace est ouvert… sans sont socle comme si elle l’évitait…
Il y a 3 faces visibles la hauteur de la pyramide et la base (base ou sont posé les pierres qui constituent la pyramide).

Si ont considère que nous humains ne voyons que le haut, le bas, la gauche, la droite, la longueur, la largeur et I’épaisseur d’un objet, la théorie des cordes prends alors sont vraie sens 6 dimensions (exemple un cube a 6 faces)

5 environnements(coté gauche droit le fond et les 2 autres cotés si ont démonte le cube) >>> 5 IEME DIMENSION

15 composants pour que cette pyramide tienne debout : 3 faces multipliées par sa hauteur longueur largeur épaisseur centre de gravité) 3X5=15

Donc 26 dimensions ! 6+5+15 = 26
conclusion we have 26 dimensions and find 20 elements

#### MTd2

Gold Member
Since you know both languages, you could at least translate! :S

#### suprised

sorry just in french CERN LHC see www.cern.ch

Dans le cas d'une pyramide Il y a 4 faces et un centre de gravité 21 éléments la compose (haut bas gauche droite longueur largeur et l'épaisseur donc 4x7= 21 éléments. (La recherche avance et nous trouverons sûrement ces 21 éléments 20 sont déjà trouvés.

conclusion: IL y a bien 11 dimensions (6 extérieures 5 Intérieures) dans l’univers.
le CERN qui recherche (boson de Higgs , des quark avec alice)

. La règle électrique de la main droite ou gauche : champ chemin courant et un point de référence ca fait déjà 4 dimensions certaines.

Imaginez vous la forme d'une pyramide l’univers n’est pas carré ! L’espace est ouvert… sans sont socle comme si elle l’évitait…
Il y a 3 faces visibles la hauteur de la pyramide et la base (base ou sont posé les pierres qui constituent la pyramide).

Si ont considère que nous humains ne voyons que le haut, le bas, la gauche, la droite, la longueur, la largeur et I’épaisseur d’un objet, la théorie des cordes prends alors sont vraie sens 6 dimensions (exemple un cube a 6 faces)

5 environnements(coté gauche droit le fond et les 2 autres cotés si ont démonte le cube) >>> 5 IEME DIMENSION

15 composants pour que cette pyramide tienne debout : 3 faces multipliées par sa hauteur longueur largeur épaisseur centre de gravité) 3X5=15

Donc 26 dimensions ! 6+5+15 = 26
conclusion we have 26 dimensions and find 20 elements
This is complete bull*** ... where is this supposed to be on our web site?

#### haushofer

I'll probably have an explanation on my blog in a few days, so let me just give the main points here :

One of the strengths of string theory is that the field theory you define on the world sheet is a conformal field theory, i.e. it is invariant under arbitrary local rescalings of the metric. This extra symmetry along with reparametrisation invariance of the two worldsheet coordinates makes sure that the worldsheet metric does not have any dynamics of it's own, and that it can always be chosen to be "conformally flat". This is called the conformal gauge.

The quantum theory exhibits an anomaly, the conformal symmetry is broken on the quantum level. This manifests itself in the fact that the Virasoro-algebra (that contains both the diffeomorphism invariance and the conformal symmetry generators) gets what is known as a central charge, an extra term in the commutation relations.

On a more physical level, it comes from the fact that one of the Virasoro operators (the one with the central term) is the String Hamiltonian. When promoted to a quantum operator it gets an ordering ambiguity, that can absorbed in a constant that is to be identified with the vacuum Casimir energy of an infinite amount of harmonic oscillators. Of course formally this energy is infite, because each oscillator gives an amount of energy hbar/2.

To total energy is then given by

$$\epsilon_{0} = \frac{d-2}{2} \sum_{m=1}^{\infty} m$$

The factor d-2 comes from the fact that only the transverse oscillator modes contribute to the energy, similar to the situation in QED when one imposes the Gubta-Bleuler conditions. The infinite sum can be represented by the Riemann Zeta-function

$$\zeta(z)= \sum_{m=1}^{\infty} \frac{1}{n^z}$$

This can be analytically continued to it's value in -1, which is what we need. It turns out that this is equal to -1/12.

$$\epsilon_{0} = \frac{d-2}{2} \zeta(-1)=-\frac{d-2}{24}$$

Later, when you check the particle content of your theory, you note that Lorentz invariance (amongst other reasons) demands that this factor is equal to -1. So this fixes d=26. A similar computation can be done in superstring theories, but this gives d=10.
I'm always a little disturbed by this explanation. Sure, analytic continuation gives the right answer, and later you can reproduce the answer from looking at the 2-dimensional CFT without "funny arguments". In textbooks and notes this explanation is always like "surprisingly analytic continuation of the zeta function gives the right regularization".

But no one explains why it gives the right answer! Ofcourse, analytic continuation is unique, so naively it should give the answer, but I've never seen a rigorous explanation of why it works in the first place.

#### atyy

I'm always a little disturbed by this explanation. Sure, analytic continuation gives the right answer, and later you can reproduce the answer from looking at the 2-dimensional CFT without "funny arguments". In textbooks and notes this explanation is always like "surprisingly analytic continuation of the zeta function gives the right regularization".

But no one explains why it gives the right answer! Ofcourse, analytic continuation is unique, so naively it should give the answer, but I've never seen a rigorous explanation of why it works in the first place.
This one isn't rigourous, but it's really cute! http://math.ucr.edu/home/baez/numbers/24.pdf

#### kai0

Well when einstein concluded in theory of relativity that gravity is a consequence of distorsion of space time fabric it became clear that gravity is a geometric force in a way inspired from this idea and in an attempt to unifie and explain electromagnetism with gravity kaluza concluded that there is a fifth dimension responsible for EM force it was subjected to laugh as there was no proof to support this idea other than maths it remained dormant for few years untill klien reevoked this conclusion by assuming that these dimension are very small and very curled up somewhat near to planck length and these dimension may exists and so kaluza-klein dimensions came into existence later on string theorists included this concept in there their attempt of Grand unification theory the string theory.it is still unsure how many dimension are there but 11 dimensional model is dominating nowadays.i hope that within this lifetime we will see a Unified theory.

#### BruceG

I've been getting into this numberology lately: what you find is that it leads you into a fascinating mathematical world which is actually more fascinating than string theory and I stop caring whether string theory has any physical relevence.

The issues, as John Baez has attempted several articles on, all lead to the magical number 24. I would also refer you to Terry Gannon's book "Moonshine Beyond the Monster".

The first point to understand is why 24 not 26?
I think the best explanation here is that the key dimension is the number of polarization states of a massless vector. In 4 dimensions this is 4-2=2 (so a photon has 2 polarizations). So in bosonic string theory you have 26-2=24 and in superstring theory you have 10-2=8.
For this analysis is turn out that bosonic string theory is more fundemental than superstring theory so it is 24 we must understand.

24 then appears in the mathematics underlying string theory, that is conformal field theory, via the Reimann zeta function and modular forms. But this is all complex stuff.

What is the key mathematic property of 24 underlying this? Here we have different opinions which I have yet to resolve. John Baez gives his answer (see ref in an earlier post).

The one I got hooked on I found from Terry Gannon. The quickest way to explain it is to show you these formula:

5 x 5 = 1 x 24 + 1
7 x 7 = 2 x 24 + 1
11x11 = 5 x 24 + 1
13x13 = 7 x 24 + 1
17x17 =12 x 24 + 1
19x19 =15 x 24 + 1
23x23 =22 x 24 + 1

(5,7,11,13,17,19,23) are all the numbers coprime to 24. 24 is the largest number that does this. As I say, I have yet to follow the thread from here all the way back to string theory, but I am assured there is one.