What are the ten dimensions of string theory?

[Wizardsblade]

Hey, I have not taken any string theory courses yet, but I had heard that String theory says there are probably 8? dimensions. I was wondering what the other 4 dimensions are and how they would be described. I know of:
0d - Spatial Dimension (a dot)
1d - Spatial Dimension (Can be viewed as infinite 0d's stacked, a line)
2d - Spatial Dimension (Can be viewed as infinite 1d's stacked, a square)
3d - Spatial Dimension (Can be viewed as infinite 2d's stacked, a cube)
4d - Temporal Dimension (Time)
5d - ?
6d - ?
ect..

 

[selfAdjoint]

Hey, I have not taken any string theory courses yet, but I had heard that String theory says there are probably 8? dimensions. I was wondering what the other 4 dimensions are and how they would be described. I know of:
0d - Spatial Dimension (a dot)
1d - Spatial Dimension (Can be viewed as infinite 0d's stacked, a line)
2d - Spatial Dimension (Can be viewed as infinite 1d's stacked, a square)
3d - Spatial Dimension (Can be viewed as infinite 2d's stacked, a cube)
4d - Temporal Dimension (Time)
5d - ?
6d - ?
ect..

Superstring theory has 10 dimensions. One of them is time and the other nine are spacelike. Three of the nine are our visible three and the other six are hidden from us, perhaps by being compacted into tiny sizes, much smaller than a subatomic particle.

Although on the surface of Earth it's easy to distinguish height from the other two visible dimensions, in zero gravity that's not so, and anyone of them can be transformed into any other by a rotation. So it's really meaningless to contemplate naming all nine, when even the three traditional names don't work in general.

 

[rtharbaugh1]

This question surfaces again and again. We need a good standard answer. For a start, what is a dimension? Then, what is a basis? Then, is the 3-space 1-time basis sufficient to explain quantum effects? If not, then by Occam, what is the least basis which can explain quantum effects?

String theory has proposed, to my uncertain knowlege, five different solutions to the question. M theory seeks to unite them into a single answer. This approach is criticized for having too many answers, or, in physbuzz, infinite vacua. It is as if we were trying to find a self-consistent algebra in which 3+1 = 2, 3, 4, or 5.

I would like to see the members of this forum start by finding consensus on the first question, implied by the OP: What is a dimension?

My provisionary answer:

A dimension is a straight line on which can be displayed a gradient of values.

Corrections, I hope?

R

 

[rtharbaugh1]

Superstring theory has 10 dimensions. One of them is time and the other nine are spacelike. Three of the nine are our visible three and the other six are hidden from us, perhaps by being compacted into tiny sizes, much smaller than a subatomic particle.
Although on the surface of Earth it's easy to distinguish height from the other two visible dimensions, in zero gravity that's not so, and anyone of them can be transformed into any other by a rotation. So it's really meaningless to contemplate naming all nine, when even the three traditional names don't work in general.

In the Zen-like interest of meaningless contemplation, I offer the following.

Front-back
Right-left
up-down
future-past
in-out


We see that the idea of dimension involves a seperation. This separation can be made into a gradient by choosing a discrete unit and applying it along a line according to some rule for summation. The easiest rule for summation is that all units are equal, no units can overlap, and there is no defined space between units. This corresponds to Euclidean geometry.

SelfAdjoint is correct in that these separations are not universal, but depend upon the imposition of an observer with an implied pre-existing basis. What does right-left mean to a jellyfish? What is up-down in freefall?

Classical spacetime therefore relies on the idea that the basis is orthagonal, that is, that there is a ninety degree rotation between basis lines. We then find that the spatial dimensions are limited to x, y, and z. There also must be a time dimension, and as Wizardblade has shown, the common idea is that the time dimension is added to the others, making x, y, z and t, four in total.

Mathematicians prefer to use more general terms which can be manipulated algebraically. So they say x_1, x_2, and x_3 instead of x, y, and z, where the notation x_1 would be read "x sub one". This leaves them free to generalize further to x_n where n takes on the values in the set (1,2,3). Then why not go further and talk about the set n=(1,2,3...), where the triple period means that the numbers go on and on, presumably to infinity.

But that is math and not physics. Not yet, anyway.

Observable, measurable, experimental results seem to defy explanation under the classical spacetime model, which is that there are three spacelike dimensions and one timelike dimension. We can explain these results by invoking additional dimensions. You can get a good introduction to this idea by reading the book, "Flatland". Basically, in two dimensions, as on a sheet of paper, you can draw a circle and label the inside of the circle A and the outside of the circle B. Logically, the circle separates A from B. There is no point in A which is also in B.

If you draw the circle on a sphere, the problem changes. Again, label the inside of the circle with some mark, A if you like. Now expand the circle to the equator of the sphere and beyond. Make the circle smaller again as it enters the opposite hemisphere from where you marked the A. Now you see that your mark, A, is no longer inside the circle, but has gone to the outside. What happened? In three dimensions, the two dimensional notion of inside and outside has been lost. The mark, A, is not inside the circle or outside the circle, but it is both.

So we find that electrons can be contained inside a quantum well. The quantum well is a device which has walls that the electron cannot penetrate, nor does the electron have enough energy to leap up over the top to escape the well. Yet it can be demonstrated that an electron trapped inside a quantum well can and will escape, no matter how tall or thick the walls. If the well is three dimensional, and the electron escapes, we can explain this behavior by invoking a fourth dimension. As in the circle on the sphere, the A, or electron, is neither inside nor outside the well, so it can escape through the next higher dimension.

But does this solution have any physical meaning? Can we find a rational explanation for the quantum behavior of the electron without invoking a fourth spatial dimension? I don't know. String and M use ten dimensions to explain observable parameters of physical behaviors of particles. Is there a simpler explanation? I don't know.

It does seem to me that Wizardblade and many others have made an error in counting time as the fourth dimension. The rules of measurement demand that time be present in any separation. Einstein pointed out that space and time are the same thing. We need to overcome the idea that time should be counted apart from space. It is clear from first principles that time and space overlap. We cannot therefore make a rational mathematics, or physics, by simply adding time to space. We have to address the question of spacetime equivalence if we are to make any progress.

R.

 

[Wizardsblade]

Actually my thoughts on dimensions, from my current level of education, I built started from the lowest dimension and building up. I start by wondering what makes a dimension and I came to the conclusion that a dimension is a way to measure something with more then one outcome. So first I thought about a point. Is a point a dimension? Well a point can be filled or empty. So, in a since yes, but is it a spatial or temporal dimension? Then I thought about 1d, a line. It is used to measure a distance away and has an infinite set of points. The next dimension would be a circle. I gather that 2d is a circle not a square because if you drop a 2d object to a point it approaches a circle at diameter = 0 (if I remember correctly). Next is a sphere for the same give reason as before. Lastly I thought about time and how it can exist with 0d-3d. So I came to the conclusion that temporal dimensions and spatial dimensions are not of the same cloth, but are 2 separate forms of dimensions. With that reasoning QM's belief that a particle can be in 2 places at once is just simply a belief that there are 3 spatial dimensions and at least 2 temporal dimensions, assuming that temporal dimensions work like spatial dimensions and are not infinite stacks, but rather deviations of the base temporal dimension. And lastly the question of is a point temporal or spatial? I believe the answer is both. A point is present in all spatial and temporal dimensions, it is the way spatial and temporal dimensions are linked.

 

[selfAdjoint]

Actually my thoughts on dimensions, from my current level of education, I built started from the lowest dimension and building up. I start by wondering what makes a dimension and I came to the conclusion that a dimension is a way to measure something with more then one outcome. So first I thought about a point. Is a point a dimension? Well a point can be filled or empty. So, in a since yes, but is it a spatial or temporal dimension? Then I thought about 1d, a line. It is used to measure a distance away and has an infinite set of points. The next dimension would be a circle. I gather that 2d is a circle not a square because if you drop a 2d object to a point it approaches a circle at diameter = 0 (if I remember correctly). Next is a sphere for the same give reason as before. Lastly I thought about time and how it can exist with 0d-3d. So I came to the conclusion that temporal dimensions and spatial dimensions are not of the same cloth, but are 2 separate forms of dimensions. With that reasoning QM's belief that a particle can be in 2 places at once is just simply a belief that there are 3 spatial dimensions and at least 2 temporal dimensions, assuming that temporal dimensions work like spatial dimensions and are not infinite stacks, but rather deviations of the base temporal dimension. And lastly the question of is a point temporal or spatial? I believe the answer is both. A point is present in all spatial and temporal dimensions, it is the way spatial and temporal dimensions are linked.

These thoughts, whie interesting, do not capture the sense of dimension used in physics and mathematics. First of all, every since Minkowski, time and space dimensions HAVE been on the same footing, because what one observer makes of another's time and length turns out to be a linear combination of what the other makes of his own time and length. So if I look at you (in motion) I have

1. my measurement of your time = A*your measurement of your time + B*your measurement of your length
2. my measurement of your length = -B*your measurement of your time + A*your measurement of your length.

Where A and B are two constants depending on our relative speed. So the time and length aren't seperatally determined, but jointly. They cannot be considered truly independent.

Second the number of dimensions can be identified with the number of different measurements it takes to pin something down. On the surface of the Earth two will do (except at the poles where it only takes one) latitude and longitude. In space it takes time and three others, but which of the three is which is just a decision; up can be down or sidewise or at a slant. In string theory it takes nine, plus time, and the physics doesn't care which is which.

 

[rtharbaugh1]

Dimension n, [Latin, dimensio, a measuring, from dimensus, pp. of dimetiri, to measure off; dis-, off, from, and metiri, to measure.]

1. extension in a single line or direction, as length, breadth, and thickness or depth; as, a line has one dimension, length; a plane has two dimensions, length and breadth; and a solid has three dimensions, length, breadth, and thickness or depth.
6. in algebra, the sum of the exponents in a term; as, xy^3z and a^2b^2cd are terms of five and six dimensions, respectively.
7. in physics, a fundamental quantity, as mass, length, or time, in terms of which all other physical quantities, as those of area, velocity, power, etc. are measured; as, the dimensions of density are mass divided by the cube of length.

Webster’s New Twentieth Century Dictionary, unabridged second edition 1983

I notice that in the first definition the staff at Webster’s has avoided the problem of Earth-based reference by using terms that can be applied even in free-fall. Length, breadth, and thickness are ordinary English terms that do not require a gravitational field, as on the surface of Earth, to make sense. The orthogonal basis of rectilinear rotations is implied.

I skip over the next four definitions as not being relevant to this discussion.

In the sixth definition we see how the idea of dimension is used in mathematics. In the seventh, the physical definition is given.

The dimension question is usually stated as some variant of the following: What are the ten dimensions of string theory? We can now see this as confusion among the three quoted contexts. Which of these three contexts applies to string theory?

Is there any reason to require that string theorists name their dimensions, as in the first definition? If not, should we advise those who pose the question to abandon their use of the common dimensional terms? If again not, how best to bridge the gap between the first definition and the correct string theorist sense of the word?

Now we must consider what string theorists mean by their use of ten dimensions. I am not up to the maths, so maybe someone else here can be more helpful, but it seems to me now that the string theorist sense is found in the sixth definition. Is there any reason to try to apply the seventh, physical definition to the ten dimensions of string theory?

Well string theory has physical pretensions, I suppose. Do the ten dimensions of string theory relate in some way to fundamental physical quantities as mass, length, or time? I don’t know but I would suppose not. Ten fundamental physical quantities seems excessive. Most physics is done with only three or four. Time and length can be reduced to each other by setting the speed of light as a unit, so that a length can be expressed as the time it takes light to move some distance, or a time can be expressed as the length light moves in that amount of time.

Personally, I respect selfAdjoint’s interpretation that the ten dimensions are intended to represent nine space-like and one time-like dimension. Brian Greene in his books has explained that the nine space-like dimensions are the usual three, plus six compact dimensions. The compact dimensions are called Calabi-Yau manifolds. Greene gives the model of an ant crawling on a jungle gym. From a distance, the pipe the ant is crawling on may seem to be a one dimensional line. But up close, the ant may find itself on what seems to be a two dimensional surface.

I think the jungle gym model is a bit of a cheat, since the difference is only a matter of scale. It seems unfair to me to count “up” for the ant as a different dimension than “up” for the observer. But there are mysteries of scale that may save this model. Is there a smallest meaningful length, as Planck suggests? Then perhaps it is reasonable to consider anything smaller to be taken on its own. Perhaps ‘up’ for the observer really is a different ‘up’ for the ant, if the ant is smaller than the Planck length. Again, I do not know.


Hope this helps.

Richard Harbaugh