Defining Superstring's Additional Dimensions

[Max™]

Indeed, a mathematical space with n dimensions is not necessarily the same as a physical spacetime with n dimensions.

It isn't that it predicts 10 dimensions as much as it seems to produce a description of reality when formulated in that many dimensions.

 

[tom.stoer]

nevertheless - remarkable coincidence

 

[OB 50]

First off, I really don't see the logic to that argument at all.

Well, good luck then.

When a 3D sphere passes through a 2D plane, it manifests as a point, which expands to a circle and then recedes back to a point. Any 2D observer would conclude that the 3D space was compactified because of the circle's apparent origination from an infinitely small point in space.

I see a direct similarity to the line of reasoning regarding compactified dimensions and the prospect that they might be visible if only we could look at them closely enough.

Secondly, I hadn't noticed you were using the word 'compact' in the English sense -- its use as a mathematical term has pretty much nothing to do with size. You can look up the definition yourself, the bit relevant here is that there are only two kinds one-dimensional manifolds: open intervals and circles. Circles are compact, open intervals are not.

The difference is important: If the extra dimension is a small open interval, then there is absolutely nothing keeping matter from remaining inside that interval, and GR puts no constraints whatsoever upon what happens at the endpoints. If the extra dimension is a small circle, then we don't have any of those problems.

Both uses of the word "compact" would seem to describe this accurately. Compact dimensions, as described to me, are both closed and exceedingly tiny.

As I've said a few times, any extra dimension we encounter would necessarily have to appear compact to us, in exactly the manner you described. All I'm trying to say is that even if we are living an a 3D space that is embedded in 4D space, our interactions with that 4D space would appear to originate from a compactified dimension as well. I fail to see how one could make a distinction between this scenario and that of a literal 4D "pocket dimension" using only the information available to an observer within the 3D space.

Maybe there is literally no difference and each scenario is equivalent.

Interesting fact: GR is 100% consistent with the hypothesis that the universe is nothing more than a ball 3' in diameter surrounding your head, and stuff just appears on the boundary as necessary to maintain the illusion of an extensive universe.

Seriously? Did you just pull out the internet equivalent of jingling your keys in my face?

 

[Max™]

Ya know, we actually have a great example of an extended dimension which appears otherwise.

Time.

The past and future don't stop existing because you aren't looking at them, yet as far as we can see, Now is all that exists of time.

 

[Hurkyl]

When a 3D sphere passes through a 2D plane, it manifests as a point, which expands to a circle and then recedes back to a point. Any 2D observer would conclude that the 3D space was compactified because of the circle's apparent origination from an infinitely small point in space.

Right -- that's exactly the argument that doesn't make any sense.

I can imagine how a 2D observer might be led to speculate they were seeing 2D slices of objects in \mathbb{R}^3.

I can imagine how a 2D observer might be led to speculate they were seeing a "parallel dimension^*" -- another 2D plane completely disjoint from his own, except that wormholes may connect them at times. (Other interesting geometries are possible as well)

But I cannot imagine how a 2D observer might be led to speculate that the the geometry of his space is \mathbb{R}^2 \times S^1. (No matter what the circumference of the loop is)


I notice that you used the phrase "pocket dimension" too -- but not in a way that resembles any usage I've ever seen of that term.


*: Dimension used here in the sense that laypeople use it. e.g. the kind of parallel dimensions you see in star trek, or the alternate planes of reality you see in dungeons & dragons.

Seriously? Did you just pull out the internet equivalent of jingling your keys in my face?

I'm not familiar with that phrase. Yes, I was serious. But it's irrelevant if you weren't hypothesizing that the geometry of space is similar to \mathbb{R}^3 \times (-\epsilon, \epsilon) as I thought you were.

 

[OB 50]

Right -- that's exactly the argument that doesn't make any sense.

I can imagine how a 2D observer might be led to speculate they were seeing 2D slices of objects in \mathbb{R}^3.

I can imagine how a 2D observer might be led to speculate they were seeing a "parallel dimension^*" -- another 2D plane completely disjoint from his own, except that wormholes may connect them at times. (Other interesting geometries are possible as well)

But I cannot imagine how a 2D observer might be led to speculate that the the geometry of his space is \mathbb{R}^2 \times S^1. (No matter what the circumference of the loop is)

I notice that you used the phrase "pocket dimension" too -- but not in a way that resembles any usage I've ever seen of that term.

*: Dimension used here in the sense that laypeople use it. e.g. the kind of parallel dimensions you see in star trek, or the alternate planes of reality you see in dungeons & dragons.

This is going nowhere. I was trying to make a very basic observation which requires no math or geometry beyond a grade school level, and I've somehow ended up sounding like a complete lunatic in the process of attempting to explain it.

Just to help me understand where this disconnect lies, I'm curious as to which of the following scenarios you think more accurately describes a situation where something has been constrained to 1 dimension.

A. An ant is walking on a wire.

B. The data representing an ant is being transmitted through a wire.

 

[Max™]

An ant walking on a wire could walk around it, thus it has 2 degrees of freedom, 1 dimensional would be restricted to forwards or backwards along the wire.

 

[OB 50]

I knew that would come up.

The distinction I'm trying to get to is whether the ant is on the wire or in the wire.

 

[Max™]

I meant that in my post, when I said along the wire I should have said within the wire, though again that is a failure of language to properly represent the condition of being one dimensional. How can you be within a one dimensional object?

Nonetheless, in a hypothetical sense, if the ant were restricted to the single degree of freedom on or in the wire, the effect is the same, in is a better description, but for the conceptual difficulties which arise.

 

[OB 50]

Those conceptual difficulties are the very things I've been trying to point out.

The ant on the wire is still a 3D observer that has been artificially constrained.

The ant contained within the wire has to be reimagined and constructed from the ground up using the principles of a 1D universe.

The two scenarios are profoundly different.

 

[Max™]

Indeed, that's why I'm trying to help talk it out, perhaps we can find a way to explain it that is clearer than either of us tried on our own.

 

[OB 50]

Okay, here goes.

Let's go with the premise that a 1D observer within the line would have to be constructed in an entirely different fashion than that of a 3D observer traveling on the same line.

The ant traveling on the line is aware of its constraints. An ant is a 3D object, and simply limiting its motion in an arbitrary manner says nothing about true degrees of freedom.

The ant within the wire would not be aware of its constraints. The only possible motion within the wire is forward or backward; A or B (the x axis). Every force, particle, and resulting object within this space is limited to motion in A or B. Even interactions with forces or objects that intersect this 1D space from higher dimensions would manifest themselves only in directions A or B. Directions C (y axis) and up (z axis, etc.) do not exist for the 1D observer. You can weave the wire into a sweater, and the 1D observer will always be looking in either direction A or B, never C. 2D space and up can never be directly observed by the 1D observer.

The same holds true for any observer within a given space. It works for 2D, and there is no reason it shouldn't work for 3D.

Does that make any more sense at all?

 

[Max™]

Yeah, that's a good way to describe it.

Interesting idea is if the 1-D space could self intersect, what would the 1-D observer see?Regarding the dimensions passing through, a circle passing around the 1-D observer through his dimension would appear as two points on either side of him, that vanished mysteriously.

Extrapolating to higher dimensions is what you've been trying to do, and that is a good point to an extent.

In another way though, we have a 4 Dimensional object which can be described in terms of passing through a 3 Dimensional space, the Universe when considered correctly regarding Time.

The difference is, that 4 Dimensional shape completely fills the 3 Dimensional space as it passes, so we're aware of a change IN that shape, but we can not say that the shape is not present at this point or that point in 3 Space.

An object in a higher dimensional space could pass through in ways similar to what you're describing, but once you reach 3+ Dimensions, it is hard to pass a large extended space through one with smaller dimensionality unless you're only doing it in pieces, or if it is folded up in various ways such as the various Calabi-Yau manifolds.

 

[Adrian59]

Yeah, that's a good way to describe it.

Interesting idea is if the 1-D space could self intersect, what would the 1-D observer see?


Regarding the dimensions passing through, a circle passing around the 1-D observer through his dimension would appear as two points on either side of him, that vanished mysteriously.

Extrapolating to higher dimensions is what you've been trying to do, and that is a good point to an extent.

In another way though, we have a 4 Dimensional object which can be described in terms of passing through a 3 Dimensional space, the Universe when considered correctly regarding Time.

The difference is, that 4 Dimensional shape completely fills the 3 Dimensional space as it passes, so we're aware of a change IN that shape, but we can not say that the shape is not present at this point or that point in 3 Space.

An object in a higher dimensional space could pass through in ways similar to what you're describing, but once you reach 3+ Dimensions, it is hard to pass a large extended space through one with smaller dimensionality unless you're only doing it in pieces, or if it is folded up in various ways such as the various Calabi-Yau manifolds.

Its quite easy to discuss what these imaginary one dimensional or two dimensional beings will see with various 3D objects passing through their world. If these extra dimensions are purely mathematical then that is also quite easy to explain. Alot of situations can be analysed in a mathematical space: for example economies can be described as points moving in a mathematical space where the dimensions could be GDP, inflation, % employment, average wage and tax revenue and we don't see these as real spaces.

However, it is more difficult to visualise > 4 space-time dimensions which appear counter-intuitive, certainly to me. As such I've waded through what is generally regarded as the definitive text for Superstring theory by Green, Schwarz & Witten and I have even acquired some of the references but a clear exposition of the need for extra space-time dimensions eludes me.

 

[OB 50]

Regarding the dimensions passing through, a circle passing around the 1-D observer through his dimension would appear as two points on either side of him, that vanished mysteriously.

Extrapolating to higher dimensions is what you've been trying to do, and that is a good point to an extent.

Before we start talking about higher dimensions, I want to make sure we're on the same page as far as the 1D scenario goes.

Let's take your example of the circle passing through the line. If we observe this from our 3D perspective, it happens pretty much the way you describe; two points appear and then disappear. However, this event as perceived from within the 1D space appears slightly differently.

Looking at the line from outside, it appears to be made a continuous series of points. Inside the 1D space, the same points are perceived very differently. There would be no sense of being confined to a line, as that direction of confinement does not exist. In order to imagine what it would be like "inside" the 1D space, we have to artificially construct an environment that our 3D brains can understand.

<<DISCLAIMER - The following is a non-technical thought construct intended to illustrate a relationship between concepts, and should not be taken as literal truth.>>

To do this, we have to imagine a space filled entirely with stacked planes. Each point on the line as observed from "outside" corresponds to a plane in the stack. Each plane is infinite, and cannot "slide" relative to other planes; it is the most basic element of geometry within this space. The only motion possible is either toward or away from the next adjacent parallel plane, and every particle and object within this space is the emergent result of this limited motion. Assuming that any object or observer within this space would necessarily have to consist of 2 or more planes, there is no way for such an observer to orient himself in a direction parallel to the surface of any of the stacked planes. It is, by definition, impossible. It also follows that any higher dimensional force or intersecting object (such as the circle) would have to originate from the very direction in which it is impossible to "look", which is parallel to the planes.

What I'm trying to illustrate is that there is no direct one-to-one correspondence. The 1D observer isn't looking around thinking, "Oh no, I'm trapped in a tiny line," because that's what a 3D observer would think. The 1D observer, should one emerge, is completely ignorant of any constraints, and considers his universe to be infinite.

In completely abstract mathematical constructs, maybe a point is a point, no matter the dimension. But, if you actually have to construct something useful like a universe, which is what I assume the whole purpose of this endeavor to be, then it's just not that simple.

 

[Hurkyl]

Let's take your example of the circle passing through the line. If we observe this from our 3D perspective, it happens pretty much the way you describe; two points appear and then disappear. However, this event as perceived from within the 1D space appears slightly differently.

A 1D observer's visual organs would see points mysteriously appear on either side of him, and equally mysteriously vanish. How is that different?

(I'm assuming that vision could perceive distance; e.g. due to a transluscent mist as in flatland. If you don't like that, then let's assume he senses with echolocation)

(assuming, of course, that the circle has some sort of interaction with the 1-D universe)

The 1D observer isn't looking around thinking, "Oh no, I'm trapped in a tiny line,"

Has anyone been suggesting such a thing?




Your stacked plane example is still problematic. Your premise is flawed, I think; you want to create a 3D environment that I would perceive as one-dimensional... but for it to work, I would have to be made out of infinite planes, which I'm not.

I don't see why you think there is anything to be gained by first trying to imagine how a being made out of infinite, translation-symmetric planes would observe a (translation-symmetric) universe, rather than just trying to imagine a one-dimensional being in a one-dimensional universe directly.

It also follows that any higher dimensional force or intersecting object (such as the circle) would have to originate from the very direction in which it is impossible to "look", which is parallel to the planes.

If you're going to change the line into a 3-D space of beings made out of planes, you have to change the circle into a kind of hypercylinder (geometry R²xS¹), and the thing our observer would see is a plane mysteriously appearing on either side of him, and then mysteriously vanishing.

(Assuming the hypercylinder interacts with the 3-D space)


The idea of something "originating" from a direction parallel to the planes breaks the symmetry of your universe; it seems like you've contradicting yourself.

But anyways, if you're going to invent a totally new scenario where we're going to throw an asymmetry at our translation-symmetric beings, then we have to figure out how such beings would react. There are, I believe, only two reasonable cases:

(1) The symmetric beings are incapable of interacting with the asymmetric object in any way
(2) The symmetric beings lose their symmetry

Working through case (2) seems difficult, since (IMHO) the symmetry was the only thing that made thinking of a being made out infinite planes palatable.

(Note that if we assume (1), then the converse applies too: the asymmetric object cannot interact with the symmetric being. Really, we shouldn't be putting both of them in the same universe -- we should be describing it as two disjoint 3D universes)

 

[OB 50]

Your stacked plane example is still problematic. Your premise is flawed, I think; you want to create a 3D environment that I would perceive as one-dimensional... but for it to work, I would have to be made out of infinite planes, which I'm not.

I'm trying to use 3D elements to illustrate how objects within a 1D space would relate to one another; not necessarily to literally construct a 3D environment. And yes, as a 1D observer, you too would be made up entirely of infinite planes.

I don't see why you think there is anything to be gained by first trying to imagine how a being made out of infinite, translation-symmetric planes would observe a (translation-symmetric) universe, rather than just trying to imagine a one-dimensional being in a one-dimensional universe directly.

Because this is the only way I can think of to try and remove any 3D observer bias from the scenario. We can't think any other way, so the only way to illustrate my point is to artificially constrain a 3D environment. None of us are capable of truly thinking in 1D or 2D.

The idea of something "originating" from a direction parallel to the planes breaks the symmetry of your universe; it seems like you've contradicting yourself.

Not at all. This is my main point.

If we observe the line that describes this 1D universe from the outside, we can draw an additional perpendicular line. That perpendicular line would intersect that 1D space in a direction parallel to the surface of the planes. To a 1D observer constructed entirely from planes, this direction does not exist.

The same scenario can easily be constructed for a 2D space, and I have no doubt that the same can be said of 3D space. There are directions that exists in higher dimensional space which simply do not exist for us. Consequently, compactified or not, additional dimensions will never be directly observable through any theoretical means of magnification, probing, or super-colliding.

We will always be looking in the wrong direction.

 

[Hurkyl]

There are directions that exists in higher dimensional space which simply do not exist for us. Consequently, compactified or not, additional dimensions will never be directly observable through any theoretical means of magnification, probing, or super-colliding.

We will always be looking in the wrong direction.

The extra dimension of Kaluza-Klein geometry does exist for us. It is a direction in which we move constantly, and have been observing for centuries. The catch is that we call it "electromagnetism" and not "geometry".


Let's start with a 2D-version of your construction. You have a Euclidean plane which is populated by line particles of identical orientation. (I'm using 2D so that you can make use of your spatial intuition to follow all of the details of this construction)

Now, let's replace all of the lines with infinitely many copies of a point particle, equally spaced and separated by an incredibly small distance.

Surely, you would agree that observers in this universe would still perceive it as one-dimensional? However, the second dimension provides them with additional physical variables -- our "lines" have measurable velocity in the second direction which could have an effect our one-dimensional observers could measure. Also, the relative "phase" of two lines might be measurable, and the separation between points determined as a physical constant.

Now, roll the plane up into a cylinder so that all of the points along our "lines" are coincident. This change has absolutely no effect whatsoever on the physics of this universe. However, it does eliminate the physical implausibility of "lines" being constructed from infinitely many points in a completely perfect translation-periodic fashion. (e.g. an Occham's razor-type thing: "cylinder" is better than "plane where everything has a perfect periodic behavior".


Anyways, reflect upon what we've constructed. We have a two-dimensional space, and it's even isotropic -- all directions behave identically as far as physical laws and (local) geometry is concerned. Furthermore, all objects are created out of familiar point particles without any sort of strange coincidences. The only difference between this universe and what classical Newtonian physics would be in two dimensions is the global geometry of the universe -- it's a cylinder of small circumference rather than a plane. And that makes all the difference, because it means our observers perceive the universe as being spatially one-dimensional.

The above might even resemble what Kaluza and Klein did.

P.S.

None of us are capable of truly thinking in 1D or 2D.

I am not limited by your lack of imagination.

 

[OB 50]

I am not limited by your lack of imagination.

That's an interesting way of putting it.

Congratulations on your blind spot. May it serve you well.

 

[Adrian59]

Before we start talking about higher dimensions, I want to make sure we're on the same page as far as the 1D scenario goes.QUOTE]

Moving on from the one dimensional situational and going back to the original statement of the thread, I’m still struggling to understand the imperative for additional dimensions. The proofs I’ve read have not been cogent. They start with reasonable premises and several pages of maths later a linear equation appears that can only be solved by D (no. of dimensions) being 26 or 10. I thought in my naivety when I first heard about string theory’s requirement for additional dimensions that there would be a vaguely topographical reason for this so I was surprised at the proofs I am presented with.

The main device appears to be to tensor contract the identity tensor which should give 4 but it is left as an open question and assigned the variable D. This gets reinserted into the Virasoro commutator extra term which is D/12(m^3-m). Later, in the bosonic case, the identity a = (D-2)/24 appears and since by now a = 1 (originally a <= 1) , D has to be 26. Further research by me, reveals that the Virasoro Lie group is a two dimensional group and doesn’t contain the parameter D but ĉ.

Notwithstanding this, if a string theorist was brave enough to take a step back from the detailed maths they would see the absurdity of the enterprise. At the beginning a supposedly water tight proof is given that bosons require 26 dimensions. Then, fermions are bolted onto the theory and low and behold the new theory requires 10 dimensions. Someone should by now have realized how wrong this was since nature presumably (unless infinite) has a fixed number of dimensions and remembering a previous discussion in this thread, that the extra dimensions in string theory are real space-time dimensions and not just a mathematical device, having different dimensionalities for different parts of nature appears contrived.

An attempt to solve this anomaly is made by the heterotic string theories which is a technical euphemism for having it both ways. Now the bosons still vibrate in 26 dimensions and the fermions in their 10. Strangely, the added term in the string (action) equation (the one that adds fermionic modes) can now have bosonic modes as well. Then, in the heterotic theories another term is added to the string (action) equation that has a SO(32) or E8 x E8 group symmetry.

My basic criticism, contrary to enthusiastic proponents who when questioned about the need for extra dimensions and supersymmetry and I have been to meetings where I have posed this question, is that the explanation of the allure of string theory is that it makes inevitable the final description of nature as oppose to the fine tinkering needed in the standard model. However, string theory is actually an unwieldy patchwork of ideas, cobbled together by increasingly desperate attempts to make it work.

 

[Adrian59]

Is anyone still reading this thread? I know I was a bit provocative in my last comments but I was trying to stimulate some debate but instead there is silence. Have I broken a taboo. Surely my allusion to some serious maths hasn't put everyone off.