Defining Superstring's Additional Dimensions

[OB 50]

"Never does he directly observe any element of 3D space."

And we've been trying to tell you that's incorrect for 2 pages worth of completely elementary physics. If I'm confined to live on the 2d plane of my computer screen, I still absorb photons from the full 3d geometry around me. I not only indirectly deduce that the world is 3dimensional, I also observe it (whatever that means) and I further can infer that I am simply living on a constrained subspace. In fact, your very eyes (the surface of which is more or less 2 dimensional) does the same exact thing, and its your brain that models and extrapolates the information it receives as 3dimensional.

At this point, I can tell that what you guys think I'm saying, and what I'm actually trying to articulate are two entirely different concepts. Short of being in the same room and drawing pictures, I don't know what else to do.

There have been a lot of examples given, none of which directly refute, or even correctly address the idea I'm actually trying to discuss. I acknowledge that this is mostly due to my lack of proper terminology preventing me from accurately expressing my ideas, but I won't concede that I'm incorrect simply because I've been misunderstood.

Your example of the human eye is an instance of 3D objects arrayed in a 2D configuration. There is a profound difference between this and the idea of a 2D observer who exists within 2D space.

The 2D observer doesn't get to look out into 3D space like General Zod in the Phantom Zone. All of his observations and movement are constrained to the plane of his 2D world. Likewise, do not think of him as a 3D observer squished down to 2D, because that's what it sounds like everyone is assuming. The construction of a 2D universe, and any subsequent observers contained within would be entirely foreign to us.

Perhaps he can construct a vast array of detectors which entirely fill an area of his 2D space in order to detect the presence of photons as they pass through. Fair enough, but what will he actually observe? Likely, he will detect a number of point-like particles popping in and out of existence as the photons pass through the plane. He will not be able to directly measure any aspect of the photon's 3D velocity, but only the deflection of the 2D elements with which they interact while passing through.

He may correctly deduce that these photons originate from a higher dimensional space. He may even deduce that there is an additional degree of freedom afforded by this higher dimension, but the observer himself is never aware of his own constrains. Within his 2D space, he is free to move in all directions that exist.

Now, he sets out to find this extra dimension. He assumes that since the particles appear to originate and terminate as points, the extra dimension must be compact. It must be really really tiny. So, he starts constructing a super powerful microscope as a start. It doesn't matter, because he can only magnify in the directions afforded to him by the constrains of his 2D space. He will never directly "see" the third dimension. What aspect of the third dimension could he possibly find by looking at smaller and smaller pieces of 2D space?

Maybe extra dimensions are compactified; maybe they aren't. I'm not even proposing one over the other. My example attempts to illustrate that even if they are not, they will appear to be to us. Either way, no matter how much we magnify, or how hard we smash particles together, if we exist in 3D space we will always be looking at 3D space.

 

[Hurkyl]

Now, he sets out to find this extra dimension. He assumes that since the particles appear to originate and terminate as points, the extra dimension must be compact. It must be really really tiny.

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

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.

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.

 

[IqbalGomar]

Hello again, all. Glad to see this thread has remained alive, even if it has gone a bit off topic. I've been looking into the Standard Model a bit over the last few days, and I feel like I'm a lot closer to understanding what I was hoping to understand about all this. I've seen the notation SU(3)xSU(2)xU(1) a great many times without ever really knowing what it meant, and now it makes sense to me, so thanks for the insight. If my presumption of understanding has not misled me, the model is something like the following.

D1, D2, D3: Space
D4: Time
D5: U(1), electromagnetic force
D6, D7: SU(2), weak force
D8, D9, D10: SU(3) strong force, corresponding to red, green, blue

SU(3) and the operations of QCD are fairly intuitive to me. The behavior of the strong force seems complex, but its rules are highly logical and easy to understand if you think of the three color-anticolor designations as three mutually-orthogonal continua.

I remain deeply confused about SU(2) though. I've always had trouble forming an intuitive picture of the weak force, since its operations relate to events that are so far removed from everyday life. So far I understand that when a proton changes into a neutron, it emits a weak gauge boson, thereby turning an up quark into a down quark. But the boson quickly decays into an electron and a neutrino, so the weak force is sublimated into electromagnetic force.

It is very confusing to me that the weak force is considered a force unto itself, as it seems more like a mediator between the strong and electromagnetic realms. Why does it require its own doublet symmetry? What are the two parameters of SU(2)? If U(1) has one parameter corresponding to the strength of the electromagnetic field, and SU(3) has three corresponding to the color identities of quarks bound into triplets, what does the (2) in SU(2) signify? Does it have something to do with the chiral symmetry breaking of the force? I would appreciate it if someone could put this into intuitive terms I might understand. Many thanks!

 

[IqbalGomar]

The more I look into this topic, the more it seems that quantum spin and mass must be independent degrees of freedom corresponding to extra dimensions. Am I really incorrect about this? If I am, how are these identities represented within the 10D manifold?

I am also repeatedly finding descriptions of the extra dimensions as spatial or space-like. Is this simply a misleading description designed to make the idea intuitive to laymen?

I don't feel as if the question I'm asking should require that much technical jargon to parse. All I'm looking for is a list of the identities of the extra dimensions posited in string theory and beyond.

1) Length
2) Width
3) Height
4) Time
5) Electromagnetism
6)
7)
8)
9)
10)

Is this really that tall an order?

 

[Adrian59]

Hello again, all. Glad to see this thread has remained alive, even if it has gone a bit off topic. I've been looking into the Standard Model a bit over the last few days, and I feel like I'm a lot closer to understanding what I was hoping to understand about all this. I've seen the notation SU(3)xSU(2)xU(1) a great many times without ever really knowing what it meant, and now it makes sense to me, so thanks for the insight. If my presumption of understanding has not misled me, the model is something like the following.

D1, D2, D3: Space
D4: Time
D5: U(1), electromagnetic force
D6, D7: SU(2), weak force
D8, D9, D10: SU(3) strong force, corresponding to red, green, blue

SU(3) and the operations of QCD are fairly intuitive to me. The behavior of the strong force seems complex, but its rules are highly logical and easy to understand if you think of the three color-anticolor designations as three mutually-orthogonal continua.

I remain deeply confused about SU(2) though. I've always had trouble forming an intuitive picture of the weak force, since its operations relate to events that are so far removed from everyday life. So far I understand that when a proton changes into a neutron, it emits a weak gauge boson, thereby turning an up quark into a down quark. But the boson quickly decays into an electron and a neutrino, so the weak force is sublimated into electromagnetic force.

It is very confusing to me that the weak force is considered a force unto itself, as it seems more like a mediator between the strong and electromagnetic realms. Why does it require its own doublet symmetry? What are the two parameters of SU(2)? If U(1) has one parameter corresponding to the strength of the electromagnetic field, and SU(3) has three corresponding to the color identities of quarks bound into triplets, what does the (2) in SU(2) signify? Does it have something to do with the chiral symmetry breaking of the force? I would appreciate it if someone could put this into intuitive terms I might understand. Many thanks!

I agree with you final count of dimensions. Earlier comments that SU(3) was eight dimensional appeared to be confusing the number of generators with the dimensions of the group. For SU(n) Lie group in n dimensions has n^2-1 generators. You could envisage each member of SU(3) has a point in an eight dimensional space but then you would have to for consistancy regard SU(2) as three dimentional.

However, counting dimensions as: four space-time, one for QED, two for the weak force & three for the colour force; gives ten dimensions for the standard model. As an amateur I am not aware if these suggested ten dimensions correlate with the ten dimensional supersymmetry of string theory, except of course for the ordinary space-time.

Also, I can not find any convincing mathematical derivation of this imperative for all these extra dimensions in string theory. Are they real or just a mathematical device. If they are real are they compactified or macroscopic but outside our awareness like the mythical flat-landers trying to appreciate a third dimension. Finally, if they are compactified space dimensions why doesn't gravity become super strong like the curvature of space at a singularity.

 

[tom.stoer]

I tried to understand string theory for years. I newer saw this idea before!

10 = 4-dim spacetime + dim U(1) + dim SU(2) + dim SU(3)

Are the string guys not aware of it? is it too simple for the Wittens, Vafas, Stromingers etc.? is it completely silly?

I know that J. Baez asks such questions? Anybody here who is in contact with him?

 

[apeiron]

I too was hoping for the right answer from someone more expert.

My understanding was that the three gauge symmetries would be nested. So SU3 would require 6D and decompose (via unstable SU2) to U1. So 6 rather than 3 dimensions to account for SU3. And subset for the others.

Here are a few possibly relevant bits I've clipped from elsewhere.

"Isospin doublets - Isospin doublets look like: (v, e-), (u_R, d_R), (u_G, d_G), and (u_B, d_B). Note particle on the right is always one EM charge unit more negative than the one on the right. This triplet is both colour and anticolour (so giving the nine-ness)."

So we have six quark states and then electron/neutrino would fall out of this symmetry as a lower energy state.

"The two sides of SU(3) matrix – columns give three slots for the colour orientation and then the rows give three anti-slots. So this is why there are nine doublet states (with the ninth self-cancelling to white)."

So again 6D to capture SU3?

And a quote that supports the nesting story I think.

"Symmetry breaking allows the full electroweak U(1)×SU(2) symmetry group to be hidden away at high energy, replaced with the electromagnetic subgroup U(1) at lower energies. This electromagnetic U(1) is not the obvious factor of U(1) given by U(1) × 1. It is another copy, one which **wraps around inside** U(1) × SU(2) in a manner given by the Gell-Mann–Nishijima formula."

So the answer could be thus SU3 takes up three compactified dimensions (with the others each having their own), or six (with the others nested inside). Or none of the above.

I have never seen the answer stated clearly.

 

[Haelfix]

What you guys are trying to do doesn't work for string theory.

The classical Kaluza Klein model is essentially this method, where you get the gauge group from the isometries of the compactified manifold.

Otoh, in String theor(ies), the gauge structure comes from other effects and for consistency reasons, cannot be built up in the exact KK way.

It wasn't a bad guess, but it didn't work out that way.

 

[IqbalGomar]

SU(2) and U(1) as subsets of SU(3)... compelling. If I understand the concept correctly, that does seem to be one way of thinking about it. Since any particle with an identity in SU(3), namely quarks, also necessarily have identity in SU(2) and U(1)... that is to say that quarks also interact via the weak force and possesses electromagnetic charge, but particles that interact via the weak force do not necessarily interact via the strong force, but do necessarily interact via electromagnetic force, and electrons do not have identity in either SU(2) or SU(3), it is as if the dimensions are nested. It's like there are three "tiers" to the dimensionality of the universe; spacetime, which is macroscopic; three dimensions of the electroweak realm, which opens up at a much smaller scale and higher energy; and three dimensions of the strong realm, which opens up at a yet higher energy.

BTW, after doing a bit more research, the two dimensions of SU(2) which I have failed to understand seem to correspond to weak hypercharge and isospin... neither of which I can make sense of intuitively. Are there intuitive descriptions of these properties beyond the maths that describe them, or are they purely mathematical concepts?

So is this more or less correct?

D1) Length
D2) Width
D3) Height
D4) Time
D5) Electromagnetism
D6) Weak hypercharge
D7) Isospin
D8) Red
D9) Green
D10) Blue

Do these ten parameters serve to describe the total state of any particle or system of particles in the universe? Is this a complete list? If not, please enlighten us.

 

[IqbalGomar]

What you guys are trying to do doesn't work for string theory.

The classical Kaluza Klein model is essentially this method, where you get the gauge group from the isometries of the compactified manifold.

Otoh, in String theor(ies), the gauge structure comes from other effects and for consistency reasons, cannot be built up in the exact KK way.

It wasn't a bad guess, but it didn't work out that way.

Please enlighten us then. What do the additional dimensions posited by string theory represent? What are the additional parameters, and why are they necessary?

 

[Max™]

The term Dimensions used in the case of gauge theories should be in reference to parameters available in a mathematical space.

Not specific "direction" type Dimensions in a spatial manifold, which is it's own type of mathematical space.

You could describe your desk surface in terms of dimensions, with each dimension representing some state of the objects on it.

String Theory involves a literally 10 (+1) Dimensional Space, and the amazing discovery was that folding the dimensions up in certain types of shapes, then allowing strings to propagate along them seemed to make Gauge groups "fall out" of the framework as possible ways the strings could interact.

Comparing that to something like an n-dimensional Hilbert space is misleading though, as those dimensions are quantities that can be measured and whatnot, not necessarily spatial degrees of freedom.

 

[apeiron]

String Theory involves a literally 10 (+1) Dimensional Space, and the amazing discovery was that folding the dimensions up in certain types of shapes, then allowing strings to propagate along them seemed to make Gauge groups "fall out" of the framework as possible ways the strings could interact.
.

And is this not the same as saying the three gauge symmetries are/could be nested within the six compactified dimensions?

So the maths was found first. Then a physical explanation in terms of available resonances in compact spatial dimensions followed. Although that intuition has never been cashed in. The way the symmetries could actually be the shape of a 6-space has never been agreed?

 

[IqbalGomar]

And is this not the same as saying the three gauge symmetries are/could be nested within the six compactified dimensions?

So the maths was found first. Then a physical explanation in terms of available resonances in compact spatial dimensions followed. Although that intuition has never been cashed in. The way the symmetries could actually be the shape of a 6-space has never been agreed?

I think it's often assumed that there is no way to intuitively visualize higher dimensions. I can tell anyone who thinks this is the case, from experience, it is not so. I've been thinking about and in multiple dimensions for so long that the idea has become intuitive to me. Whether one relies on the 'cheat' of simply thinking about a higher dimensional manifold in three dimensions at a time, or one is able to actually visualize a system incorporating multiple degrees of freedom simultaneously, the visualization is possible.

I'm simply asking for a straightforward and intuitive description of these dimensions. If I've been wrong in thinking of these dimensions as classically spatial, I have no problem admitting that and reworking the model I have operating in my head. But even if they're not spatial, there must be some analogy that would make these conceptual manifold spaces comprehensible to a layman who does not possesses the mathematical prowess to construct equations in them.

Any help would be appreciated. I'm not asking for the meaning of life here, just a laymanized version of something which should be fairly simple. I <i>get</i> higher dimensions. What I don't get is the identities of these dimensions according to physicists' best understanding of them.

 

[Max™]

If they were, String Theory would pretty much be THE end all of theoretical physics by now, I'd think.

The mathematics of gauge theories works in 3+1 spatial dimensions, with the additional parameters which are often mathematically described as dimensions.

The natural way to produce gauge symmetries is one of the key things keeping String Theory going, it's hard to accept that it can be so naturally "right" about things like that without it being right about everything else.

It's somewhat of a sleight of hand though, the shapes were found which rather naturally produce the mathematical relations of gauge theories, but in a sense the shapes themselves were first thought of as a way to embed gauge structures into spacetime, with Kaluza and Klein's early 5-D Relativity ideas.It's kind of like noticing that you can write out numbers from 1 to 10, and write some multiplication tables for them, then writing out higher multiplication tables which give numbers from 10 to 20.

There's still MUCH work which needs to be done to fully embed String Theory in a predictive physical theory, and it still has that loose parameters problem where you can nearly always get the results you want, by setting it up to give the results you want.

 

[Adrian59]

The term Dimensions used in the case of gauge theories should be in reference to parameters available in a mathematical space.

Not specific "direction" type Dimensions in a spatial manifold, which is it's own type of mathematical space.

You could describe your desk surface in terms of dimensions, with each dimension representing some state of the objects on it.

String Theory involves a literally 10 (+1) Dimensional Space, and the amazing discovery was that folding the dimensions up in certain types of shapes, then allowing strings to propagate along them seemed to make Gauge groups "fall out" of the framework as possible ways the strings could interact.

Comparing that to something like an n-dimensional Hilbert space is misleading though, as those dimensions are quantities that can be measured and whatnot, not necessarily spatial degrees of freedom.

Thanks for your explanation. To summarize how I interpret your answer it would seem that the standard model gauge theories are using a mathematical space not a real space whereas string theory actually predicts 10 (+1 for M theory) space-time dimensions. However, your penultimate paragraph does suggest that these two may be linked since as you say the gauge groups may "fall out" of the compactified extra dimentions.

 

[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.