# What is each dimension in string theory?

1. Dec 4, 2005

### thegreensquall

i'm currently reading Hyperspace by michio kaku and it says that there are 10 dimensions in superstring theory, what are each of the dimensions?

2. Dec 5, 2005

### Cexy

There is one time dimension and three spatial dimensions, just as you find in Newtonian physics or general relativity.

There are also six extra 'compact' dimensions, which together form what's called a six-dimensional Calabi-Yau space. How can I explain this..?

A compact dimension is one which is curled up - unlike an extended dimension, which goes on to infinity. For example a circle is a compact 1-dimensional space, whereas a line is an extended 1-dimensional space.

You can get various combinations of these - for example, the surface of a sphere and the surface of a torus (donut) are both compact two-dimensional spaces. The surface of an infinitely long pipe is a two-dimensional space composed of one extended dimension, and one compact dimension in the shape of a circle.

With six dimensions, you can get much more complicated shapes - shapes that are impossible for us to envision, which is why we are forced to talk about them using mathematics. In one dimension, the only compact space is a circle. In two dimensions, you can have a sphere, a torus, a 2-holed torus, a 3-holed torus, etc. There are many possibilities. When you move up to three dimensions, there are more still. By the time you get to six, the number of possible shapes is bewildering!

So that's it - the extra six dimensions aren't space dimensions, or time dimensions. They're just 'extra' dimensions that are necessary for superstring theory to work.

3. Dec 5, 2005

### rtharbaugh1

Hi thegreensquall and Cexy

Good standard explanation, Cexy. In the other thread (Anyone want to explain a few things?) I give a spatial model which generates waves in any required number of dimensions. It is decidedly not standard. However it does show how compressed dimensions might evolve. And it does so without invoking a bewildering number of shapes. Not that it does not involve a bewildering number of shapes. It just doesn't have to invoke them.

It does have six compressed dimensions, in two sets. Wouldn't it be neat if they turned out to be the standard model, the Fermi scale, and the Planck scale?

R.

4. Dec 6, 2005

### straycat

Here's a crazy idea that I had some time ago.

The world of our "everyday experience" has 4 dimensions, and string theory predicts 10 dimensions. Suppose, then, that "the world of our everyday experience" had, not 4, but rather n dimensions. Would string theory have to be modified in some way to have not 10, but rather some different number m dimensions? If so, what would m be (as a function of n)?

Here's my idea. According to GR, the metric g_ij has 16 = 4 x 4 components; but since the metric is symmetric, you can actually specify the metric using only 10 components. Is it conceivable that the number of components of the metric equals the number of dimensions necessary for string theory to "work"? Or is this just a coincidence?

For example, a string theorist who lived in Flatland would think that there are only 3 spacetime dimensions. The metric, in Flatland, would therefore require only 6 components. Here are some more (n, m) pairs:

n m
2 3
3 6
4 10
5 15
6 21

Now my understanding of string theory is minimal, to say the least ... so I have no idea in what manner string theory would have to be modified to accomodate an arbitrary number n of "everyday experience" dimensions. But maybe someone here might have some ideas. Anyone? ...

David

5. Dec 6, 2005

### Chronos

The simple explanation [at least to me] is that each extra dimension represents an extra degree of freedom any particle [or force] is free to move about. In our everyday experience, 4 dimensions are sufficient to completely describe what we perceive with our senses. Mathematically, however, there is no limit to the number of dimensions available. String theorists talk about 11 [or 10, 26, etc.] because it is the minimum number of degrees of freedom necessary to mathematically describe what we perceive. I usually resort to this analogy to visualize extra dimensions:

Imagine a sheet of paper as representing 2 dimensions [an x - y table]. You can expand it to 3 dimensions by overlaying additional sheets of paper. Put these in a binder and call them a book. You can add a 4th dimension by adding more books, and a 5th dimension by creating collections of books.... and so forth.

6. Dec 6, 2005

### rtharbaugh1

Hi Chronos

Your analogy looks like a catagory type approach. It works for me. One could go on to talk about libraries, bibliographies, even grimoires. Or national acadamies, perhaps.

When trying to apply catagory theory to real situations, I always think of the time I bought a box full of cans and jars full of small parts at a farm auction. There were various kinds of screws, nails, bolts, nuts, washers, rivets, paper clips, tacks, staples, ball bearings, pins, and whatever other small bits of metal (mostly metal, but some rubber, glass, ceramic) the farmer had thrown into the box over a lifetime of wrestling with stubborn machinery. Farmers hereabouts don't throw such things away, or at least they didn't used to when there were still small freeholds. You never know when you might need a whatsis. Most farmers usually had one, if they could find it.

Now the problem is to find some rational way to divide the stuff into cans, jars, and boxes in such a way that you can find what you want by going to some subset of the collection, rather than having to sort through all of it. You can see how this would save a lot of time looking for small parts. However it costs a lot of time to do the sorting. The problem is to take some limited number of containers, and divide the (essentially infinite) reality of the collection into some kind of IMAGINARY order. The order is not inherent in the collection, it has to be imposed at some arbitrary level of catagorization. One must of practical necessity limit the number of catagories. Less catagories is more usefull. To see this, you only need imagine putting each part in its very own container. Doing so would make it no easier to find the nut you need at any given moment.

We can make some improvements by imposing a heirarchy of catagories. For example, one might make a box full of cans of different kinds of screws, another box full of cans of different kinds of nails, another box full of cans of different kinds of ball bearings and so on. If the shed gets too full of boxes, one might build a second shed, and put all the boxes containing nails and screws in one shed, all the boxes containing bolts and nuts and everything else in the other shed. I make this division purely on practical terms, since I have found that nails and screws of various kinds usually make up about half of any random collection of small fundamental items in a farmer's junk shed.

My point here is that in any large random collection of small items, order can be imposed "from the top," but all such schemes, useful as they may be, are necessarily imaginary, part of the system of the observer, not an inherent order in the random collection. Catagory schemes operate by a process of selection. You pick out all the bolts, for example, and put them in a box. Then you get a bunch of cans and jars, and you put the bolts in different cans and jars based on another selection scheme. Maybe you decide to divide them up by diameter, then by length, then by thread type. The farmer across the road decides to divide his collection up by another scheme. Maybe he starts out seperating all the clean bolts from the ones that have rust, and all the rusty ones into ones that have only a little rust and ones that have a lot of rust, and then all the ones that have a lot of rust into ones that .... well you get the idea. The two farmers will not end up with any two equivalent cans of bolts, even if they start with the exact same random collection of bolts.

Last edited: Dec 6, 2005
7. Dec 6, 2005

### hossi

As far as I understand, string theory does not care about our everyday experience. Meaning, as long as there are in total 10 dimensions, it works. You only have to get rid of 6 of them to reproduced something that we see. What we see is a problem. You can 'solve' it by compactification. However, when you forget what our world looks like, you might as well only compactify 4 or 3 dimensions and leave the other ones large.

(Maybe String Gas Cosmology has something to say about that).

Anyway, when our everyday experience would have been more than 10 dimensions, now THAT would have been a challenge for string theory!

Actually, there was some paper about that by JoAnne Hewett, something with 'testing critical string theory' or so... ahem... wait... let me look... here it is

http://xxx.lanl.gov/abs/hep-ph/0503178"

About the degrees of freedom of the metric tensor: you can take some sub-sectors of the matrix and interpret them as fields on their own. That was about the original idea by Kaluza and Klein. Still you have to explain where the dependence on the extra coordinates goes (Kaluza called it
the 'cylinder condition').

Last edited by a moderator: Apr 21, 2017
8. Dec 7, 2005

### Chronos

Your point is well taken, Richard. Category theory does impose arbitrary definitions to your choices. But I also think it impossible to avoid such constraints under the of rules of quantum physics.... Any comments, Kea? She is an expert on this stuff. I admit my concept is simplistic.

Last edited: Dec 7, 2005
9. Dec 7, 2005

### Kea

Goodness, I'm no expert on Category Theory: just a struggling physicist. But I can tell you that Category Theory isn't just about putting things in boxes. If it was I can't see that it would be much use. However, the analogy is quite good in that it shows that everything we look at has a context. This is not an inconvenience, it is a physical law.

10. Dec 8, 2005

### rtharbaugh1

Hi Chronos and Kea, and all
I wish I knew more about catagory theory. Kea, could you identify the context as it relates to my nuts and bolts analogy?

I was trying to go on to talk about different kinds of catagories of nuts and bolts and of ways to compare the utility of various schemes of catagorization, but my batteries died. I am in town today, partly to charge them up again.

Anyway I was probably getting distracted again. Really the O.P. wanted to know about dimensions. I got on catagories because of the library analogy Chronos provided. I suppose one could talk about catagories of dimensions.

But Chronos was building higher dimensionality by starting with a flat sheet of paper, which we could agree has two important dimensions. The third dimension, thickness of the paper, becomes evident when you bind the pages into a book, but there is more to what Chronos is saying than that, I think. The amount of information that can be fitted into a two dimensional region can be increased by stacking the pages in a third dimension. Of course it could also be increased by making the information smaller, as used to be done in microfilm applications. Or it could be increased by making the size of the page bigger. The main thing is that the book analogy does show how two dimensions can be built up into a third dimension.

Usually we get into trouble when we try to go to four dimensional analogies. People seem to have trouble envisioning a four dimensional system, which, imho, is why the mathematics becomes so complicated. Mathematicians and physicists have been telling students not to even try to envision four dimensions, but it seems to me it would make the math and physics immensely easier if we could do so.

Is there really any reason why we cannot envision more than three or maybe four dimensions? I was taught in education classes about how children learn to see dimensionality. Small children are puzzled by how one thing can dissappear behind another, and yet come back out again. Physicists are puzzled about how a partical, supposedly a fundamental object, can transform into two or more objects, or interact with other fundamental objects. Mostly the physicists collect information about the transformations, and then try to apply one or another classification scheme, rather as if they were zoologists classifying species. The term "partical zoo" was even popular for a while, before the standard model of partical physics became, well, standard.

Nowadays we have the standard model, a crowning achievement, but there is the nagging doubt, the leftover can of unclassifiable bits and pieces, the night-time dreams of advanced thinkers who talk about preons and axions and instantons and so on. (I rather miss an old internet friend who used to go on about ignorons, ftl particals that physicists prefer to ignore.) So how long do we continue to divide the zoo into smaller and smaller boxes? Is it really useful to do so, or had we better think outside the box, and find, maybe, a higher dimensional geometry to account for the apparent differences between particals?

The presense of families of particals is highly suggestive. Could there be some higher dimensional sense that an electron and its neutrino could be the same thing, only looked at from different dimensional perspectives? And then might a muon and its neutrino be another kind of thing, when looked at the same way? And maybe there is a way to look at muons and tauons and electrons as a single sort of thing, and so on. Perhaps there is really only one kind of thing, and whatever it is vibrates in different dimensions to produce the various kinds of particals. So the promise of string theory.

And so the recurrent question about what the dimensions are, and do they have names, and if so, what are they called?

Adding more dimensions is like adding more cans to divide up your collection. If you have as many cans as you have parts, the process becomes a bit silly, but if you have a big can of nails, it might behoove one to divide it into large nails and small nails or something. Bent nails and straight nails, I don't know. Certainly adding more catagories makes the leftover pile smaller, but I always worry about that leftover pile.

Well as Cexy said, they are all the same so they don't really need names. If you have a box of identical nails, what use would there be in naming them? Bill or Susan or Rexroth's daughter, you just take any one and pound it in to fix the Planck.

I think I will start a new thread to talk about envisioning higher dimensions. Maybe later. My coffee has gone cold.

Be well,

Richard

Last edited: Dec 8, 2005
11. Dec 9, 2005

### Chronos

Agreed Kea, no boxes. I'm thinking intersections. By drawing one dimensional lines through all the intersections, you create the illusion of 3 dimensional space. When you rotate that along any axis, you create 4 dimensional spacetime. Which is to say you need a time coordinate to describe the apparent position of any intersection relative to all other intersections. Bear in mind you can rotate this coordinate system on multiple axes without creating a paradox.

12. Dec 9, 2005

### rtharbaugh1

A summary thus far:

1. Thegreensquall asks: what are each of the 10 dimensions in superstring theory?

2.Cexy replies with Newtonian 3-space 1-time plus six compact Calabi-Yau dimensions, and notes the Calabi-Yau are not space dimensions or time dimensions, but just "extra."

4.Straycat notes that "according to GR, the metric g_ij has 16 = 4 x 4 components; but since the metric is symmetric, you can actually specify the metric using only 10 components", and asks "is it conceivable that the number of components of the metric equals the number of dimensions necessary for string theory to "work"?"

5.Chronos relates dimensions to degrees of freedom and gives a model building higher dimensions by stacking lower ones like stacking pages of a book.

6.rth mentions catagory theory, then talks about dividing up a large assorted collection of small parts like nuts and bolts into boxes, notes that there are many ways to do the division, and suggests that an order of heirarchies of divisions, while useful, is not inherent to the collection of parts, but is imposed by the observer.

7.Hossi replies to straycat noting that string theory isn't about our everyday experience, says you could as well compact two or three dimensions as six, mentions 'string gas cosmology', gives a link to a JoAnne Hewit paper about finding blackholes in many dimensions at the LHC, notes that you can take subsectors of the metric tensor and treat them as fields, as Kaluza-Klein did, and points to a problem with this approach, called 'the cylander connection'.

8.Chronos asks Kea to comment on catagory theory

9.Kea replies that Catagory Theory is not about putting things in boxes, and says the nuts-and-bolts analogy does show that "everything we look at has a context"

10.rth talks about trying to envision geometry in higher dimensions and suggests that doing so would simplify the mathematics.

11.Chronos replies to Kea agreeing that it is not about boxes, then suggests intersections of lines, and rotations into higher dimensions.

I apologise if I have misunderstood or misrepresented the gist of anyone's post here. For me, at least, Hossi's post was the one that was the most dense with new ideas. Corrections would be welcome.

rth

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Last edited: Dec 9, 2005
13. Dec 9, 2005

### rtharbaugh1

Returning to the original post, what are the ten dimensions of string theory?

First, one should note that string theory is a mathematical theory, not a physical one. Many very promising attempts have been made toward showing a physical interpretation of string theory, but none of them afaik has been widely accepted in the general physics community. For this reason, we must look for a mathematical answer to the question. It may not be useful after all to begin by examining the meaning and function of dimensionality in physical theory.

So, a proper answer to the question should probably be deferred to the string theory mathematicians. I am a rather poor student of maths, interested in the popular idea of string theory, but hardly qualified to give a proper answer, so I should probably just shut up about it and let my betters have a go. Unfortunately I do not have the wisdom to do so, and I rather enjoy playing the fool.

What is a dimension, mathematically? Height width depth and duration are examples of physical dimensions with which we are all familiar, but are they really useful as mathematical examples? What other dimensions are we familiar with, that may be more in line with mathematical analysis?

Wolfram Research has Mathworld by Eric Weisstein, found at

http://mathworld.wolfram.com/index.html

where a search on the term Dimension results in about 280 hits. There are listed entries on Hausdorff-Besicovitch or fractal dimension, Krull dimension, correlation dimension, information dimension, q-dimension, Lebesque Covering Dimension, and Lyapunov Dimension, all on the first page.

The first entry takes an approach from topology, and may after all be the one we want.

"The dimension of an object is a topological measure of the size of its covering properties. Roughly speaking, it is the number of coordinates needed to specify a point on the object. For example, a rectangle is two-dimensional, while a cube is three-dimensional. The dimension of an object is sometimes also called its "dimensionality." "

Eric W. Weisstein. "Dimension." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Dimension.html

One is tempted to quote the entire article.

"Finite collections of objects (e.g., points in space) are considered 0-dimensional." This is, I think, the shorthand way of saying that dimensionality is imposed "from the top" by some organizational scheme of the observer. Real collections of objects, such as the particals in the standard model, are zero dimensional, but we choose to impose a dimensional order upon them, rather as the farmer sorts his nuts and bolts into boxes, for his own convenience. We are, I think, most familiar with the 3-space 1-time system of order, but it is not something inherent to the finite collection of objects we find in our universe, rather it is a system we impose upon the universe for our own purposes.

I like this. If the system of dimensionality is imposed by the observer, then the observer is free to impose any convenient system of dimensions. There is nothing really special about the physical system of 3-space 1-time, we are just used to thinking that way. We could just as well think 1-space 1-time, and in fact we often choose to do so, as in the graphs of ballistic curves showing how objects fall in a gravitational field. We show them in 1-space 1-time for the simple reason that our graph paper is two dimensional, so it is most convenient to show only one spatial and one time dimension, even though we know there are two more common spatial dimensions. We just leave the other two spatial dimensions out of the picture, which is quite all right, because in simple cases anyway there is not much sideways motion, such as might be caused by wind shear.

Then they go on to talk about how we drag one dimensional objects to impose a second dimension, drag two dimensional objects to impose a third dimension, and so forth. "Dragging" and "adding pages" (from the book analogy) are the same thing.

When a farmer takes an assorted collection of small parts and makes a seperation, she is in a sense dragging the collection....perhaps she pulls out everything that is a nail from the collection. You might imagine that she has a large canvas upon which she spreads out the entire collection in a layer one item deep. Then she "drags" everything which is a nail from the collection to another part of the canvas. Now she has two collections, one of nails and one of everything else, and the collection as a whole has become one dimensional. (Notice there is no mention of boxes!)

Now she may take the subset which is all the nails, and drag again. This time, she may choose to make the separation of maybe large nails from small nails. An arbitrary distinction is imposed, perhaps based on the size of a certain nail. Anything larger is dragged into a new pile, leaving behind the smaller ones. Is the nail pile two dimensional? No, still one dimension of nails. So how do we advance the scheme into a second dimension?

Well, we can take our zero dimensional collection of nails, divide it into large and small, that is one dimension. Then we can take our one dimensional collection of large-small nails, and divide it again, this time into common nails and every-other-kind of nail. The result is four distinct groups of nails, large common and small common, large not-common and small not-common.

You see we can display this collection still in two dimensions, by dragging all the small nails to a new part of the canvas, and then by dragging all the common nails to yet another part of the canvas, preserving as we do so (the second drag) the distinction between large and small. The number of dimensions of the canvas (2) IS NOT THE SAME as the number of dimensions of the set of nails, even though we have chosen to display the two dimensional nail collection upon the two dimensional canvas.

Lets now proceed to use our nails to make a three dimensional set. To do so, we might drag all the steel wire nails out of our collection, leaving behind the brass nails, galvanized nails, square nails, roofing nails, finish nails and so on. As we do so, once again, we preserve the separations that have already been made, so that we have now eight groups of nails. The divisions have been:

Large-small
Common-not common
Steel-not steel.

Each time we divided in two (that is, we added one dimension to our scheme) so we now have 2x2x2=8 seperate groupings. Again, we can display this set of eight groups on a single canvas laid out on the barn floor. It is a three dimensional set, displayed on a two dimensional surface. It could as well be displayed in three dimensions. We could, for example, make our steel-not steel division by dragging all the steel nails upstairs into the hayloft and displaying them on another canvas, preserving the separation into large-small and common-not common.

However I feel that our poor farmer is becoming exhausted by all this dragging stuff around. I suggest the farmer should retire to the kitchen to have a cup of coffee while we intellectual types decide what we have learned about dimensionality.

(241 views at 0512091510)

Well that is a review of how we evolve the lower dimensions. In math, we think of dragging a set from one place to another. Dragging is the same as seperating. In math, you are just seperating space, but in physics you are presumably seperating physical objects. Nails works as an analogy. We are mostly interested in the standard model of partical physics here, of course. It is notable that the dimensionality of an object is not the same thing as the dimensionality of the display of that object. We can and commonly do display a three dimensional object in a two dimensional space.

break

Last edited: Dec 9, 2005
14. Dec 9, 2005

Staff Emeritus
Very good post Richard. Following that idea, any point on a string can be dragged in ten indpendent directions, including duration in time as one of them. Ten comes out of, well yes, the math, but the math is based on the kind of math used to describe existing kinds of things since the 19th century, and especially since Einstein introduced relativity 100 years ago this year.

Recall that a point moving in spacetime (introduced by mathematician Minkowski, as you know) has a worldline[/i] consisting of the successive events (points in space and moments in time) that it has occupied. Now following Lagrange we set up a mathematical expression for the action and derive its physics from that. And for this relativistic particle the action turns out to be proportional to the differential of proper length $$d\tau$$ along the worldline.

So the physicists (not mathematicians) who created string theory modeled their action on this. And the great, too much unsung physicist Nambu proposed the string action to be proportional to the differential of area $$d\sigma d\tau$$ on the worldsheet, by two-dimensional analogy with the worldline action. This led to many wonders - the worldsheet with this action has all sorts of nice symmetries that particle physics doesn't. But it also required very directly and mathematically that the dimension of spacetime (string theory is all relativistic) be 26 for strings without supersymmetry and 10 for strings with it.

Four of the 10 are visible, time and three space ones, and the other six space ones are hidden, which it is proposed to do by compacting them on tiny manifolds. Then by choosing those manifolds carefully you can make some of the things known in the standard model happen with your strings. Admittedly this last is, umm, rather epicyclic, if you catch my drift, but the stuff leading up to it is not epicycles at all, it all falls out of that very plausible choice for the string action.

15. Dec 9, 2005

### rtharbaugh1

I was just thinking of how to extend this into hyperspace, and here you come along with world lines and world sheets! Most excellent.

So, I guess we take a zero dimensional point (it could be any set, really, as we saw above from Mathworld) and we drag it to a new space, which makes one dimension, or division. Of course in order to drag anything, we must already include the idea of time, not so? It was in one space. We drag it to another space, and observe that the trace of the movement is a sort of one dimensional line. So we see that time and space are already inextricable even in the first dimension.

In fact, even in the zero dimensional set object, or point, there is already the insistance upon duration. What is an object that has no duration? For anything to be called an object, it must have duration. So the spacetime equivalence principle starts at the very beginning and goes all the way through our discussion, even though up until now I have not considered the effect of time.

Since I have spent so much time on developing a three dimensional set of nails, let me try to go on with that analogy. In the beginning the nails were all mixed together in a common space, and from Newton we know they would have stayed that way, more or less, until someone or something came along to make them behave differently. The farmer is our operator. She split the nails into two sets by dragging all the large nails to another part of the canvas. We may imagine this occurance as it flows through time as well as space. The path of each nail as it is moved is a kind of trace, and we can imagine a line that connects the space where it was to the space where it is after being moved. That is the world line of the nail during the operation of being moved, correct? We see that the world line is both in time and in space at once, and that in fact the time part and the space part cannot be thought of as seperate from each other.

Then the world line is a kind of object, and it too has duration, so we get a world sheet as the world line has duration and therefore sweeps across time.

Now we are getting to the new stuff, or so I imagine, being largely uneducated. I am from this point on making stuff up as I go along. In fact, I have tried to make these connections before, with little success and much criticism. Maybe this time I can achieve some clarity.

There are two kinds of time in the world sheet, or so it seems to me. First there is the trace of the movement of the set along its world line. Then there is the operation of separation. (Drat that word. Have I been misspelling it all along? Well it happens that google has it spelled both ways. But now that I think of it, the spelling with an 'a' looks better to me. Oh well.)

Anyway, we see that in order to accomplish the separation, an acceleration had to be applied to the set. That would be length per time per time, or as we say, length per time squared. Yet it is usual in physics and math circles to specify that there is only one dimension of time. I am not sure why we make this distinction between space dimensions and time dimensions, even though we are assured by the equivalence principle that they are the same.

Anyway we take our three dimensional set of objects, and we extend it in time, which is to say it has duration, and that is its time line. We move the set in space over a time, which is an acceleration, and then we see that the time line itself has duration, and so it becomes a world sheet. Now a clever man can see that the worldsheet itself has duration and may be moved in time, and so we have the possibility of another direction in time. Length over time cubed is called jerk. You see it is the case in which the rate of acceleration itself is undergoing a change.

Now we could go further and further this way, talking about time dimensions of even higher orders, but it happens that the difference, as far as we are concerned, becomes very small, in fact almost invisible, in fact, rather compact. I know this is not what the string theorists say about their compact dimensions, but really, since space and time are equivalent, isn't this it? The changes in higher dimensions of time are so short that we cannot record them as changes.

But if you and others insist on refusing to see it this way, I am content. Only I want to go on trying to build dimensions in hyperspace, then, without considering the effect of time at all. It seems to me that if we are going to mix time elements with space elements at all, we must think as I have outlined above. If this line of reasoning is unacceptable for some reason, then it must also be unacceptable to introduce even one dimension of time when talking of the ten (or is it nine now?) dimensions of string theory.

If that line of reasoning holds, then I am inclined to go forward with an attempt to derive nine dimensions of space as a reasonable way of dividing up the universe. I think I see how to do it. Yes. I think I see it clearly. But I should wait and see what the others will have to say about this. If we are going to count one dimension of time, how do we justify ignoring the other dimensions of time, especially since we use them freely in formula such as l/tt, which is acceleration, and l/ttt, which is jerk, and may I add, ll/tt, which is force?

Furthermore, I should like to point to the fact that the highest order term I have seen in the physical equations is c, the speed of light, raised to the fifth power. In my notation, that becomes lllll/ttttt. Five dimensions of space, five dimensions of time, ten dimensions. Is this a coincidence?

Happy regards,

Richard T. Harbaugh

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Last edited: Dec 9, 2005
16. Dec 9, 2005

Staff Emeritus
The worldsheet is a two-dimensional surface, and it is given a Riemannian (actually Lorenzian) geometry, just like GR, but only 2-dimensional. One dimension is spacelike and represents length along the string, the other is timelike and represents the time direction (actually it is defined locally, and is the tangent to the worldline of any point on the string at a given moment).

17. Dec 10, 2005

### rtharbaugh1

Yes, I was just getting to that. I was finishing the post when you came in.

Well, of course there can be two kinds of waves in a one dimensional object, transverse and compressional, so perhaps it is not unreasonable to talk about two kinds of time in a single time dimension. And spacetime equivalence should come into this discussion. When you extend a line in space, a time extension is implied.

So I should think that the world sheet, having one timelike dimension and one spacelike dimension, should have plenty of room for two kinds of time. As you can read in the end of my previous post, I am thinking of time and the square of time. In physical terms, that would be velocity, l/t, and acceleration, l/tt.

I would like to know how Nambu came to the twenty seven dimensions idea, and how supersymmetry pared it down to ten. Are you saying then that ten dimensions implies the supersymmetric partners of the standard model particals? I should also like to know more about how that comes about, of course.

Thanks,

Richard

Last edited: Dec 10, 2005
18. Dec 10, 2005

Staff Emeritus
I don't think Nambu had anything to do with the 26 dimensions idea (anyone who knows better is free to correct me of course!). That was a later discovery, due to Schwartz, I think. It does come out of Nambu's choice of action though.

Supersymmetry is usually presented in terms of the Standard Model particle set, but obviously this is not the appropriate context for its use in string theory. I believe the core of introducing SS in string theory is the introduction of Grassmann variables, which square to zero. That is, if g is a Grassman variable then its product with itself gg = 0. This mathematical excercise has various results, but I'm not really up on supersymmetrical string theory so I'll have to leave you there.

19. Dec 10, 2005

### rtharbaugh1

All right. I guess I should have known that, from all the complaints about string theory making no predictions.

But I do find the following statement in WIKI:

"If the Large Hadron Collider and other major particle physics experiments fail to detect supersymmetric partners or evidence of extra dimensions, many version of string theory which had predicted certain low mass superpartners to existing particles may need to be significantly revised."

And much of the article in WIKI seems to go back and forth between superstring theory and the idea of supersymmetric partners for standard model particals.

However I do notice that this page of WIKI has been tagged for improvements. Here is the address:

http://en.wikipedia.org/wiki/Supersymmetry

Maybe this statement is part of the artical that needs revision.

Thanks,

R

Last edited: Dec 11, 2005
20. Dec 11, 2005

### rtharbaugh1

Humor: a google search for Nambu Action turns up an offer to sell us a Nambu Action Figure. I knew that action figures of various super heroes had been made for sale, for example the one of the ex-Governor of Minnesota and World Wide Wrestling Champion, Jesse Ventura. I did not however realize that the collection had been extended so far as to include obscure physicists!

21. Dec 11, 2005

Staff Emeritus

It is perfectly true that in the run-up to the starting of the LHC the research at other experimental sites, Tevatron and so on, is putting tighter and tighter constraints on what hoops the supersymmetry people have to jump through in order to mate their theory with standard model physics. This is, in the narrow view, independent of any string theory, but obviously if supersymmetry should be experimentally falsified (possible but by no means threatened at this point) then superstrings falls with it.

String theory people are already doing proactive things against the day that might come about, and I understand they have some options. Bosonic string theory never used supersymmetry at all, and Sen and Witten's String Field Theory, which is based on bosonic, has recently had a nice new accomplishment, celebrated by Distler and Motl last week.

22. Dec 11, 2005

### rtharbaugh1

I want to bring up another model of compact dimensions which I recall, IIRC, from Brian Greene's popular books about string theory. An ant crawls along a pipe on a childhood playground jungle gym. From a distance, one sees the pipe as a single dimensional line, but to the ant, it is a two dimensional curved surface, a cylander. So by this model the compressed dimensions are a matter of scale. They are really not extra at all, they are in fact the same as our usual three dimensions of space, only seen at a much reduced scale.

I am not confident of this model at all. In fact, I don't like it. It seems a cheat to me to offer up the idea of more than three dimensions, and then to go back and say it is a mere matter of scale. Up and down are up and down both to the ant and to the playground observer, so there is nothing extra about that. In fact there is nothing in the movement of the ant that cannot be described in the three dimensions of the observor, only requiring some very fine numbers. It is as if Dr. Greene were saying that 10^23 is not on the same number line as 10^-23.

No, if it is true that Nambu action requires twentysix dimensions then they must be independent dimensions. We have seen that the number of dimensions is the same as the number of parameters required to specify the position of any point on the world sheet. Not fair if two or more of the parameters change in direct proportion to each other.

Instead, if we require twentysix spacetime dimensions, we should be prepared to give them individual names, like the up-down front-back left-right past-future names we are familiar with from our customary 3-space 1-time view. I cannot accept up-down-big and up-down-small as a proper varient. But I may be willing to accept big-small as an independent direction, or in-out, or perhaps >c-<c. I have to think about it.

356 views as of 0512111219
R.

Last edited: Dec 11, 2005
23. Jan 19, 2006

### stevenvee

2 / 3+1sqrt(5) 13 / 521-233sqrt(5)
/ ---------- - / -------------- = 1
\/ 2 \/ 2

32.272727272.....

Does this equation or number have any significance in this discussion?

Last edited: Jan 19, 2006
24. Jan 20, 2006

### IACOB DUMITRU

The dimension

It is a grave error to assimilate the dimension as a unique spatial direction. The mankind does not know that till now. The line does not have theoretical solution. It is premature to make a general prediction about space and time before to discover the real cause of their existence. Sorry, but for moment you wander with these inefficient study. The Euclidian space is a theoretical nonsense. We always take into account as a loyal host the emptiness. The Math defines this neutral space as an unlimited and different number of points. The Physics makes a big mistake borrowing without discernment this paragon. In reality, the emptiness is an absolute inhomogeneous space.
The theorists do not found the rules of this behavior, and therefore they name it “chaos”. The motion does not exist as a spatial travel into continuous space. If you want know more about this guileful character, I can give you (free) “The space’s harmony” concept. anin1958@yahoo.com

25. Jan 23, 2006

### pavsic

Usually it is assumed that the extra dimensions, necessary for consistency of string theory, have to be compactified.

But there is an alternative approach, known as 'brane world scanario'. The well known Randall-Sundrum model in which the bulk has 5 dimensions, is just one possibility. One can assume that the bulk has 10 dimensions, and that we live on a 3-brane (that sweeps 4-dimensional manifold).

In such scenario there is no need to compactify extra dimensions. I have been pursuing such approach for many years, starting with a paper that appeared in 1985. A complete list of papers can be found on my home page http://www-f1.ijs.si/~pavsic/ .

In 2001 I published a book
"The Landscape of Theoretical Physics: A Global View;
From Point Particles to the Brane World, and Beyond, in Search of a Unifying Principle" (Kluwer, 2001)
in which amongst many other topics (including geometric calculus based on Clifford algebra) I discuss the brane world scenario.

My opinion was that the string theory, by its compactification business, etc., has somehow deviated from the right track, although I still think that string and brane theory has many good features which in one or another way will be employed in constructions of the future "unified" theory.

It is interesting that when I started to write the book the explosion of the brane world scenario papers did not yet occur.

Last edited: Jan 23, 2006