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Author Topic: How does compactifying dimensions work?  (Read 1807 times)

Offline Bill S

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How does compactifying dimensions work?
« on: 02/02/2014 18:58:30 »
 
I have always had problems with the idea of extra dimensions being “rolled up”. I can accept that if mathematicians need extra dimensions in which to make their ideas work they can invent them, and roll them up I they need to. I can also accept that if these mathematicians are also physicists, they can apply these ideas to a conceptual model of the universe, complete with Compactified dimensions. Of course, when I say I can accept these things, I certainly don’t mean that I can understand them. In practical terms, what does introducing extra dimensions and compactifying them actually mean?

I have seen it illustrated as 3D grid of tubes filling space. What does this represent? If a dimension is rolled into a tube with a cross section of about the Planck length, we will not be able to detect it. However, what seems to be happening is that the same dimension is being rolled into trillions of these tubes. Can that be right?

I become more confused. Each of these tubes is about 1.6X10^-33 cm in cross section, so we cannot see it; but if these tubes fill space, are they packed together? If so, 10^33, side by side form a layer. Then, 10^33 such layers, placed one on top of the other would present an end section that was 1.6X1.6 cm. Why would we not see that?

Another question that comes to mind is: why would these dimensions need to be rolled up in order to be unseen? Some multiverse theories suggest that trillions of “shadow” universes occupy the same space as ours, and these are completely unseen.

Could Calaby-Yau spaces throw some light on my problems? Jim Baggott 9Fairwell to Reality) says: “If I mark an infinitesimally small point on the desk in front of my keyboard, and could somehow zoom in on this point and magnify it so that a distance a billionth of a trillionth of a trillionth of a centimetre becomes visible, then superstring theory says that I should perceive six further spatial dimensions curled up into a Calaby-Yau space.”

Does this mean that in superstring theory space must be granular? Are there six extra dimensions in each Calaby-Yau space? How do they relate to one another?




 

Offline Soul Surfer

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Re: How does compactifying dimensions work?
« Reply #1 on: 03/02/2014 19:55:49 »
I agree that it is a difficult concept and I am not absolutely 100percent sure that I fully understand it.  I see it a bit like the way we use complex numbers to describe oscillations and waves.  After all the "dimensions" that we cannot see are complex numbers in quantum mechanics.

The compactified dimensions that we are talking about here are genuinely very small as to precisely how small they are is still open to question.  The range of some of the short range forces associated with strong and weak interactions may have something to do with their size but they may not.

The properties of particles are defined by not the position but the phase in these recycling dimensions just like where something is in an orbit.

The important result of this is that the universe is very small in these dimensions and only large in our familiar dimensions of space and time.  If you accept this the oddities of a lot of quantum mechanics disappear
 

Offline Bill S

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Re: How does compactifying dimensions work?
« Reply #2 on: 03/02/2014 23:49:47 »
Thanks SS.  It seems you may be saying that the "3D grid" is not a good illustration.
 

Offline yor_on

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Re: How does compactifying dimensions work?
« Reply #3 on: 04/02/2014 12:54:35 »
Well, using a argument in where a dimension is provable, a axiom, then there exist no two dimensional subjects (and objects) inside SpaceTime. That as they should 'disappear', observed from a angle of that third, 'non existing', dimension they lack. If you exchange dimension for degrees of freedom though, then the way something can move is what define its dimensions. Matter can move (in itself, as well as in space) in, or if you like :) call it,  'defining' three room dimensions and one 'time' dimension.
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Defining it from degrees of freedom it becomes perfectly acceptable, to me, defining two dimensional lattices laid upon each other, becoming a three dimensional piece of matter to our three (four) dimensional observer, that is. (but it sure as h** took me quite some time reaching this conclusion:)

Or maybe not? It depends on how they move, doesn't it :) The point may be that one lattice has only two directions which in its constituents can move, but as you join two together, there might become a new direction through which they can interact. At least it solves the question of how to prove a two dimensional lattice for me. It's not as perfect as I would like it though.

You could possibly also relate it to 'Matter tells space how to curve. Space tells matter how to move' which then inevitably will lead us to gravity and geometry those days. Einstein didn't define relativity from geometry though, that is a later concept as I've understood it.
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The most important thing to me is that you can't have it both ways. Either dimensions preexist, to then gets filled with stuff, or it is degrees of freedom that will define what we call dimensions. I prefer the later myself as it fits relativity better (locality and observer dependencies). But you need co-existing frames of reference for the later to become a measurable SpaceTime, as it is those frames of reference interacting that creates our possibility of measurements. Any which way one prefer it one can't have a string 'vibrate' though, without both introducing a arrow and one 'room dimension', or degree of freedom, more than the one defining a string 'at rest'. A tension is slightly different possibly? But not really, from where I look at it. Better to call it a property, like 'spin', if so. You could imagine a string that is stressed in the one dimension it exist in possibly, as compressing and decompressing, but that still need to introduce a arrow in which it can do so. From a degree of freedom it is the same, you need that arrow.
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There is a difference between what we call room dimensions and a arrow. The arrow has one direction in which it takes everything with it. For a room dimensionality, or degree of freedom, it doesn't matter what direction you measure a distance in, a meter is still a meter. So when speaking of time reversibility, it may make theoretical sense, but as we all know we are going to die, it practically doesn't. And btw, as I measure that distance, the arrow I do it in will have a same direction, no matter that 'meters' orientation inside a SpaceTime.
« Last Edit: 04/02/2014 15:06:21 by yor_on »
 

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Re: How does compactifying dimensions work?
« Reply #3 on: 04/02/2014 12:54:35 »

 

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