What Equation Predicts 10 or 11 Dimensions?

[AstrophysicsX]

How many dimensions are there. I can describe the first 3 spatial dimensions plus 1 of time, but all the others are a mystery. I also read that an equation predicted that there would be 10 or 11 dimensions, if this is true, what equation is that?

 

[fzero]

How many dimensions are there. I can describe the first 3 spatial dimensions plus 1 of time, but all the others are a mystery. I also read that an equation predicted that there would be 10 or 11 dimensions, if this is true, what equation is that?

No more than 4 dimensions of spacetime have been observed. Higher dimensions (for a total of 10) are the most obvious interpretation of extra degrees of freedom predicted by string theory. Certain string theories have a limit in which an 11th spacetime dimension seems to open up. This is often called M-theory. The wikipedia entries below should be decent and provide references for further study.

http://en.wikipedia.org/wiki/String_theory
http://en.wikipedia.org/wiki/M-theory

 

[MathematicalPhysicist]

$$\infty$$

 

[kcajrenreb]

Infinite spatial dimensions right?

 

[AstrophysicsX]

Infinite dimensions? I thought it was limited?

 

[mitchell porter]

How many dimensions are there. I can describe the first 3 spatial dimensions plus 1 of time, but all the others are a mystery. I also read that an equation predicted that there would be 10 or 11 dimensions, if this is true, what equation is that?

The equation is "conformal anomaly should be zero".

In writing down equations for string theory, at some point you introduce a one-dimensional coordinate system along the length of the string, so you can say "location x on the string moves like this". But the actual number you use for x is just a label, and it must be possible to label the points differently and still get the same predictions. What I mean by re-labeling: Suppose you have a string which is a line segment, and you have x=0 at one end and x=1 at the other end. There will be points in between which are x=1/4, 1/2, 3/4. But you could "rescale" these labels to a new quantity, x', so that x=1/4 goes to x'=1/2, x=1/2 goes to x'=3/4, and x=3/4 goes to x'=7/8 (for example). You're squeezing the x coordinate at one end and stretching it at the other end, when you go from x to x'.

Since these are just labels for the purpose of description and calculation, the final predictions of the theory have to be independent of the labeling scheme. There's also a label for time along the history of the string, for which the same consideration must apply. "Independence of predictions from labels" is expressed mathematically by saying that the equations of motion for the string have a symmetry; they don't change when the labels are changed.

But a quantum theory can fail to possesses in reality a symmetry which it appears to have in its equations. A quantum theory can usually be understood as a classical theory with some extra fluctuations, and sometimes the solutions to the classical theory have the symmetry but the fluctuations don't. This failure of a symmetry at the quantum level is called an anomaly. Some anomalies are just features of reality - e.g. the matter-antimatter imbalance of the universe may have been caused by anomalous asymmetry of interactions in the early universe - but for the "change of labels" symmetry along the string that I was talking about, you don't want it to break down, because those labels really shouldn't have any intrinsic meaning. The "change of labels" symmetry is part of something called conformal symmetry, which includes the squeezing and stretching operations that I mentioned.

So what you do is you write down equations for a string in D dimensions, you calculate the conformal anomaly, and you find that it will only be zero (as required) for a particular value of D. If you look this up in a string theory textbook, you will find that D=26 for the "bosonic string" and D=10 for the superstring. The bosonic string is the original form of string theory, that doesn't include fermions. The superstring has fermions, and fermions moderate the size of the anomaly, so the number of dimensions you need is less.

D=11 shows up when you discover that superstrings are actually 2-dimensional "supermembranes". The argument here is different. There are "supergravity" theories which describe the gravitational part of string theory, and if you go to more dimensions than 11, you will get "spin 5/2" particles which no-one can write an equation for. So D=11 supergravity was considered the maximal supergravity, but it wasn't a limit of superstring theory, which only exists in D=10. Except that one day, Edward Witten discovered that Type IIA string theory under certain conditions was behaving as if it had an 11th dimension. That was the beginning of M theory in 11 dimensions. ("M" mostly stands for membrane.)

There's probably a way to get D=11 by considering the conformal anomaly for a supermembrane, but I can't find any such calculation and it would necessarily be difficult; membranes were ignored for a long time in favor of strings, because it seemed impossible to calculate with them. However, they are now a standard part of string theory.

Almost certainly there is a deeper reason for why the special dimension is 10 or 11. Right now, it imposes itself on us mathematically, as a necessary condition to have a consistent theory with the desired property of re-labeling (reparametrization) invariance. But it's a common phenomenon in mathematics, that you can prove something without really understanding it - just doing the calculation shows you what the answer is, but not "why". I believe the real reason for D=10,11 remains to be discovered.

 

[MathematicalPhysicist]

I was just kidding.

Though in one of Michio Kaku's pop book he describes one gradaute student saying there should be infinite dimensions in some conference when asked how many extra dimensions there should be?

I read it in HyperSpace.