# Size of small dimensions

Gold Member

## Main Question or Discussion Point

I am not sure under which rubric questions about string theory or M-theory should go. Anyway, since this question concerns sizes down to the Planck distance, I suppose it should go here.

The question is two fold: in such a case as the 6 extra spatial dimensions of string theory, how does one precisely define the size of a spatial dimension? Secondly, how does one measure it?

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tom.stoer
This post should be moved to the "beyond ... - forum"

Short answer: there is no mathematical necessity to compactify these dimensions, nor is there a mathematical reason for a specific size. Even the number of compactified dimensions is not determined mathematically.

Instead one looks for solutions which allow for small extra dimensions in order to "hide" them. There is a very large number of solutions (10^x, sometimes you find x=500), e.g. so-called Calaby-Yau spaces (something like generalized tori). Size and shape remain free parameters (so-called moduli).

String theorists found a way to "fix" some of these moduli and "freeze the breathing" of the compactified dimension by wrapping "flux lines" around the "tori". Anyway - the size is defined by physical arguments, e.g. the dimensions must be small enough to be invisible.

The measurement is difficult. Small extra dimensions (~ Planck length) cannot be measured at all. Large extra dimensions can be "measured" as they modify constants and laws well-known in 4 dimensions. That means that e.g. the strength of a force which usually has a 1/r potential (like gravity or electromagnetism) changes its behavior if r is comparable with the size of the extra diemension. So what one does is to look for deviations from predictions known from gravity, high energy physics (e.g. at the LHC). Once one finds those deviations one can try to fit models using large extra dimensions (where the size is comparable to a typical length scale defined e.g. by 1 / energy scale at the LHC).

The reason for the deviation can be understood as follows. By solving the equations of classical electrodynamics one finds for the D dimensional Coulomb potential 1/rD-2 where D counts space dimensions (for D=3 we have D-2 = 1). For a length scale much larger than the size of the small dimension we expect 1/r
For a length scale much larger than the size of the small dimension we expect 1/r
For a length scale much smaller than the size of the small dimension we expect 1/rD-2
For an intermediate length scale it's complicated but at least one expects to identify the scale below which the 1/r law gets modified.

arivero
Gold Member
Short answer: there is no mathematical necessity to compactify these dimensions, nor is there a mathematical reason for a specific size. Even the number of compactified dimensions is not determined mathematically.
This is true for 10 dimensions ("ten into four does not go") but for 11 there exist a special tensor which makes room for a very specific kind of solutions, called "Freund-Rubin solutions", separating two pieces of 7 and 4 dimensions. Stilll, the size of each piece is undefined.

http://www.slac.stanford.edu/spires/find/hep/www?j=PHLTA,B97,233 [Broken]
Freund and Rubin said:
In d-dimensional unified theories that, along with gravity, contain an antisymmetric tensor field of rank s-1, preferential compactification of d-s or of s space-like dimensions is found to occur. This is the case in 11-dimensional supergravity where s = 4.
The tensor is interesting by itself. The 11 D graviton has 44 components, and the total multiplet of supergravity has 128 bosons. So it is needed to fill 84, exactly the number of components of this tensor.

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bcrowell
Staff Emeritus
Gold Member
Although the OP was about string theory, there is also the possibility of large extra dimensions: http://en.wikipedia.org/wiki/Large_extra_dimensions In this case, the scale is the electroweak unification scale, and the true Planck scale in these models is actually the same as the electroweak unification scale -- the apparent Planck scale is actually not the true Planck scale.

Gold Member

Dimensions are measures so they can't have size and they can't be curled up. Measuring a dimension is measuring a measure which doesn't make sense.

tom.stoer
I think you are confusing "meters, kilogram, etc." with "length, width, etc."

Certainly not. All spatial dimensions are necessarily measured in meters, or derivatives, in the metric system. Did you ever hear of any of the 3 spatial dimensions we are familiar with having size? If a dimension has size it would itself have dimensions, which is impossible.

tom.stoer
This is stillnot clear to me.

Think about the usual garden hose example: the two-dim surface of the hose has R*S1 topology.

Measuring length with a fixed meter stick along the hose results in some length x m where x can be arbitrary large and m means meter.
Measuring length with a fixed meter stick around the hose S1 results in some length y m where y is limited by the circumference L.

In that sense the second space dimension, the dimension along the circle S1, has the length L.

What is you wording for these facts?

Measures are of particular objects, the objects have size, the measures do not.

tom.stoer
Measures are of particular objects, the objects have size, the measures do not.
Please read carefully: I am not talking about the measure of of a measure but of space itself. Would you deny that the procedure is reasonable? What is the conceptual problem of measuring the extension of space (except that it may be a practical problem)?

Chronos
Gold Member
A toad hiding from children with a flashlight . . . well done tom! A dimension is the 'freedom' to occupy more than one state. I flip on a light using a switch, which gives it two dimensions - on and off. No 'meter sticks' required. I perceive extra dimensions in ST are more like switches than meter sticks.

tom.stoer
I perceive extra dimensions in ST are more like switches than meter sticks.
I absolutely agree!

Think about the heterotic string: one fermionic sector lives in 26 (!) dimensions where 16 are compactified to a lattice on which an E8*E8 or a SU(32) gauge algebra lives, whereas from the remaining 10 another 6 are compactified on a CY space or something.

Therefore 16 dimensions (which nobody talks about, unfortunately) are internal gauge degrees of freedom, whereas the remaining 10 are spacetime. In a truely unified picture this is nonsense and all "dimensions" are just "means to count degrees of freedom".

Example: look at a gluon field

$$A_\mu^a(x)$$

in QCD with a spacetime index and a color index.

In string theory this disctinction somehow goes away or becomes subject to interpretation depending on the vacuum.

Haelfix
In string theory, there are a many different 'pictures' of how certain features behave (think Heisenberg vs interaction picture in usual quantum mechanics). Indeed that is one of the richness' of the theory, and why field theorists love it, b/c often a confusing statement in gauge theory can be explained rapidly in stringy language....

Regarding extra dimensions, I think one of the great mistakes people make is to view them as a sort of 'real' thing seperate from the formalism or the applicable mathematics.

It's not entirely wrong to view the world (through the lens of perturbation theory and the worldsheet) as 2 dimensional, with the resulting spacetime map as emergent and the other dimensions are simply 'Radion' like fields that ensure certain symmetries. Alternatively you can view it as 11 dimensional in M theory, 10 in heterotic string theory and 4 with compactified dimensions. Since in the appropriate limits these can all be shown to be mathematically equivalent, asking how many dimensions string theory actually predicts, is a slightly loaded question. That's why it can be so frustrating answering critical questions on this board. One really has to specify what particular stringy formalism you are talking about to make sense of the question... (again I make the analogy to Shroedinger vs Heisenberg picture) except that here there are a veritable multitude of different formalisms, approximations and pictures to work in, even before we specify which vacuum or side of a duality we wish to work in..

In the same way, in 4 dimensional supersymmetric theories, its not wrong (in the superfield formation) to think of an additional 4 components of Grassmann coordinates, where you can generalize pretty much the apparatus of regular dimensions into these coordinates. So for instance, you can create boosts in those coordinates, and think of the generators as being kind of like 'momentum' and so forth. These dimensions can be thought of as infinitisemal, and are of course integrated out in the final answer (to get the F or D terms in the lagrangian), however mathematically they are as 'real' as any other.

jal
... how does one precisely define the size of a spatial dimension? Secondly, how does one measure it?
If you are a beginner then the first place to look is at wiki?

http://en.wikipedia.org/wiki/Degrees_of_freedom_(physics_and_chemistry [Broken])
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system. The set of all dimensions of a system is known as a phase space.
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http://en.wikipedia.org/wiki/Dimension
In mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it.
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http://en.wikipedia.org/wiki/Fourth_dimension
Comparatively, 4-dimensional space has an extra coordinate axis, orthogonal to the other three, which is usually labeled w.
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What I did not find in wiki was the physics implication of a fourth or more spatial dimensions. It involves expanding the definition of a closed system to preserve “conservation of energy”. To explain gravity and why it is so weak in 3d, the 4th dimension is considered to be a large sheet/brane that is parallel to our 3d and that the gravity force extends/connects into our 3d by following an exp. curve.

Using more than 3d is a way of hiding a particle/energy from our observation.
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http://en.wikipedia.org/wiki/Subatomic_particle
In particle physics, the conceptual idea of a particle is one of several concepts inherited from classical physics. This describes the world we experience, used ( for example ) to describe how matter and energy behave at the molecular scales of quantum mechanics.
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... how does one precisely define the size of a spatial dimension? Secondly, how does one measure it?
Do an exercise ... use minimum length on dimensions to achieve/realize/permit the degrees of freedom for a particle. You will find that if you assign one unit for the size of a particle then the result will be that the minimum size of the dimension will be 2 units. Any size less than one would not be permitted since that is the definition. If the dimension is equal to the size of the particle then it cannot have any degrees of freedom/movement.

A proton is 1f or 10^-15m.
The experiments being done at CERN will give us a better range on the sizes of particles.
jal

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arivero
Gold Member
I think that we all understand that the use of words in conversation has not the same precision that in mathematics. Most science journalism is about translating between the two kingdoms. In this sense it is allowed to speak of dimensions, radius of a dimension, size of a dimension, etc, to avoid to clutter the conversation with references to geodesics and metrics.

But be careful, it is wrong to use the generic multiple meanings of the word in the arguments, everyone should think the argument in the mathematical setup and only "traslate" to conversational speak after.

arivero
Gold Member
Do an exercise ... use minimum length on dimensions to achieve/realize/permit the degrees of freedom for a particle.
This exercise is misleading, because you are using a double meaning of dimension, and besides you need to introduce a minimum length anyway.

The proper way to grasp the concept in introductory mathematics is to meditate about the [surface of a ] sphere and the [surface of a] torus (or "donut"). How do objects in the surface see the "radius"?

In the sphere, it is simply the "maximum length" of a geodesic curve, which is a global property (you need to circle around). But it is also the "curvature" of the sphere, which is a local property (you do a small triangle with geodesic sides, measure the area, do a even smaller triangle, measure the area, see the variation of the area law respect to the euclidean plane).

In the torus, you have zero curvature and geodesic lines can wrap around, but even there you have a way to find a pair of "max" and "min" circles, and compare the length across these circles without needing to look "outside" to the 3D euclidean space. You can even find the number of holes by noticing the failure of the count of faces, edges and points in a polyhedral figure.

It is convenient to remember, by the way, that in "mathematics of space", dimension is a topological notion, while size is a geometrical notion.

There is no conceptual problem in measuring the extent of space, that is not what were talking about, but there is in assigning size to a measure. And you are confusing different types/ definitions of dimension. There is spatial dimension, the size of a matrix, the number of elements in a vector space basis, the dimension of a manifold, the dimension of a simplex, the power of a fundamental physical quantity, and degrees of freedom. These are all entirely different entities. What string theory is talking about are spatial dimensions, extensions of normal 3-D space, and these cannot have size nor shape.

Shape? Calabi-Yau compactifications? Am I missing something or is the original poster's arguments fallacious.

arivero
Gold Member
There is no conceptual problem in measuring the extent of space, that is not what were talking about, but there is in assigning size to a measure. And you are confusing different types/ definitions of dimension. There is spatial dimension, the size of a matrix, the number of elements in a vector space basis, the dimension of a manifold, the dimension of a simplex, the power of a fundamental physical quantity, and degrees of freedom. These are all entirely different entities. What string theory is talking about are spatial dimensions, extensions of normal 3-D space, and these cannot have size nor shape.
I am not confusing different types of dimension, I am telling that each of them has a precise meaning when in math and a very concrete use, and become badly specified and used in common language. So sometimes in common language they speak of "size" of a "dimension" instead of length of a particular closed geodesic, or similar things.

You are welcome to criticize this way of speaking in common language, and I will agree that it is a poor use. But you can not use it as a way to criticize the theory we are speaking of.

As for a dimension having shape, is just common parlance for the fact that manifolds of equal dimension can have different topologies. I hope that you can recognize this fact, can you? Or I can do it more precise, speaking of homeomorphisms etc. Do we need to do it, every time we speak?

arivero
Gold Member
Or, as tom as asked you, please give your own wording for the fact that the transverse size of a garden hose is a lot smaller that its longitudinal size.

Or you go to the mail office with a package measuring 30x30x900 and then you are informed that you need special package rate. Is it not tempting to use the word "dimension" to state the problem? Is your typical answer to the postman that "it is not a problem with the dimensions, it is a problem with the object, but dimensions are OK, please give me the standard rate"? :-D. Hey, I will try it, next time.

(Aside, it is very funny that fedex calls a 30x30x30 box a "square box". Flatland guys :-DDDD )

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tom.stoer
Or, as tom as asked you, please give your own wording for the fact that the transverse size of a garden hose is a lot smaller that its longitudinal size.

Or you go to the mail office with a package measuring 30x30x900 and then you are informed that you need special package rate. Is it not tempting to use the word "dimension" to state the problem? Is your typical answer to the postman that "it is not a problem with the dimensions, it is a problem with the object, but dimensions are OK, please give me the standard rate"? :-D. Hey, I will try it, next time.
It becomes cheaper if you are able to compactify six of the three dimensions on Planck scale :-)

jal
Do an exercise ... use minimum length on dimensions to achieve/realize/permit the degrees of freedom for a particle. You will find that if you assign one unit for the size of a particle then the result will be that the minimum size of the dimension will be 2 units. Any size less than one would not be permitted since that is the definition. If the dimension is equal to the size of the particle then it cannot have any degrees of freedom/movement.
... you need to introduce a minimum length anyway.
I did ... and CERN will give us a better range.

also you forgot

What I did not find in wiki was the physics implication of a fourth or more spatial dimensions. It involves expanding the definition of a closed system to preserve “conservation of energy”. To explain gravity and why it is so weak in 3d, the 4th dimension is considered to be a large sheet/brane that is parallel to our 3d and that the gravity force extends/connects into our 3d by following an exp. curve.

Using more than 3d is a way of hiding a particle/energy from our observation.
jal