Quick Thought on Non-locality

[Gopi]

While encountering the concept of non-locality, there is one thing that we are probably skimming over... that non-locality need not necessarily be localized only in the reciprocal aspect, but merely in a higher dimension.

For example, let us take a line... 1 Dimension. In Line-land, the distance between two points is all that can be measured, and let us say it is some value x. Now, if I take the same line, but loop it and form a circle, in 2 Dimensions, then I can make a circle with circumference = x, such that the starting and ending point of the previous line segment coincide. With respect to the 1D situation however, NOTHING has changed, the two are still separated by x. What is non-local in 1D is localized in 2D.

A similar extrapolation is possible, take a piece of paper and draw a square on it. Now, fold it such that one side of the square touches the opposite side of the square, like a cylinder. Then the two lines coincide, but on the paper, the lines have not moved! So they are localized in 3D, and non-local in 2D. This is similar to the idea of wormholes, where one needs to go via hyperspace to get somewhere far off, something that is localized in 4D can be non-local in 3D.

Now, that is all considering space alone. Now what would happen in case of time region? If something is localized in space, but non-localized in time, that would mean that the dimensions of space has to be REDUCED by one in order to represent the correct number of dimensions of time. Similarly, if we are considering the motion in time, and want to represent it in space, we have to increase the dimensionality by one, in other words, when considering 1D motion, it would lead to the Second Power relationship which we commonly use in the case of the time region.

In other words, one dimension of time = "-1" dimension of space.

[bperet]

Quite a fascinating idea you've got here... feels right, but still trying to wrap my mind around it. Some questions:

I'm assuming you are speaking of coordinate dimensions (extension space and time), and that the locality and non-locality of the system is determined by:

1. the projection of scalar dimensions onto the coordinate system?
2. the projection of the inverse/conjugate aspect of one coordinate system upon the other?

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For example, let us take a line... 1 Dimension. In Line-land, the
distance between two points is all that can be measured, and let us say
it is some value x. Now, if I take the same line, but loop it and form a
circle, in 2 Dimensions, then I can make a circle with circumference =
x, such that the starting and ending point of the previous line segment
coincide. With respect to the 1D situation however, NOTHING has changed,
the two are still separated by x. What is non-local in 1D is localized
in 2D.

So you are defining the process of the geometric reciprocal of linear to polar as "localization," and the polar to linear as non-localization? Sort of along the lines of dimensional reduction and expansion, discussed earlier?

I think I've confused myself here... if x was infinite in extent, then the endpoints would both be at infinity--coincident, yet separated by infinite distance, both nonlocal (x distance) and local (endpoints coincide). That condition would be the same for any unbounded object of any dimensions (like the cylinder mentioned below). So the concept of locality and non-locality can only apply to geometry of limited extent (finite)?

Also, why would the length become a circle? If you are starting with a line segment, in 2D wouldn't that just be an arc, with non-coincident endpoints? (Don't get me wrong--not trying to knock holes in your concept, but trying to refine it, as I think you've found the link between bounded and unbounded structures that Nehru discussed years ago... infinite line becoming bounded SHM, and infinite turn becoming rotation. We identified this relationship, but never had a mechanism for it--and I think that's part of what you have here.)

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A similar extrapolation is possible, take a piece of paper and draw a
square on it. Now, fold it such that one side of the square touches the
opposite side of the square, like a cylinder. Then the two lines
coincide, but on the paper, the lines have not moved! So they are
localized in 3D, and non-local in 2D. 

Let me see if I have this right... you aren't considering the area on the paper, but are treating the boundaries (finite extent) as "locations", and because there are multiple locations bounding the area it is 'nonlocal'. When you wrap the paper, the edges become coincident and become local (coincident).

So if I were to draw a circle on a paper and fold--what happens? Obviously, there is only a point of coincidence, not a line. If you follow the contour of the circumference, you end up with a taco, or when distributing the distortion, a football shape. Is that correct?

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This is similar to the idea of
wormholes, where one needs to go via hyperspace to get somewhere far
off, something that is localized in 4D can be non-local in 3D.

So the Babylon 5 Jump Gate is a tool to control dimensional localization? That's an interesting thought, and might actually lead to a mechanism to do it.

I think I'm confused by the way you are using local and non-local. The way I understand it is that "local" is "location"--an identifiable point or plane. Non-local has no directly identifiable source (for the realm you are in) but can influence location, like a force field (electric, magnetic, etc). It seems your use of local and non-local are more along the Latin lines of locus and nexus, where loci are the boundaries and nexi is the stuff bounded (or omitted). In your line example, the endpoints are loci and the distance is the nexus ("nexus" means to tie or bind together, versus "locus", which means point or position, from where we get "location").

I guess the difference in my mind would be that a nexus would be the observable line segment, whereas non-local would be like an invisible line segment--something influencing two points that is not directly observable. The difference between distance and force.

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Now, that is all considering space alone. Now what would happen in case
of time region? If something is localized in space, but non-localized in
time, that would mean that the dimensions of space has to be REDUCED by
one in order to represent the correct number of dimensions of time.

Wouldn't this just be "dimensional reduction" of unit speed? Not understanding how localization/non-localization is fitting in.

If you take a unit, scalar speed, the projection in space would be a bivector and it time would be a birotation. The non-local component of the birotation would be dimensional reduction, causing what would be a linear motion in space to vary in time--SHM. At the same time, the non-local component of the bivector in time would be a complex SHM (RV), dimensional expansion, giving rise to charge (frequency). Am I close?

This would also indicate that the spatial speed of light controls the frequency, so light should travel at different speeds for different frequencies, like QED predicts. But remember the photon is CARRIED by the progression, and the frequency would be in the SAME dimension but has a SHM--the photon would speed up a bit, then slow down a bit, average being the speed of light. A property we would call "heat." Therefore the photon speed IS dependent upon the frequency, but since it manifests as a SHM in the same, linear direction as the progression, it APPEARS independent of frequency.

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Similarly, if we are considering the motion in time, and want to
represent it in space, we have to increase the dimensionality by one, in
other words, when considering 1D motion, it would lead to the Second
Power relationship which we commonly use in the case of the time region.

Would that infer that acceleration, s/t^2, is the expression of each aspect upon the other?

It does make sense. For example, a 1D motion inside the unit speed boundary (time or space region) would have a 0D influence outside--in other words, that electric motion would show up as a point charge, EITHER positive or negative. A 2D motion would show up as 1D -- a N-S magnetic line of force.

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In other words, one dimension of time = "-1" dimension of space.

Interesting... I would look at it as "one dimension of space = +1 dimension of time", because I observe from the POINT of consciousness. Since observation is sensory, that means I "intuit" from the FIELD of consciousness (non-local). Which is about right, since my intuition can tell me what is going on miles away in space, or weeks away in time. That's probably why I cannot understand LM technology--but you might--because being of a social memory complex, they "observe" as a FIELD (non-local) and intuit as a POINT (localized). So keep thinking the way you are thinking--don't want to break the pattern of thought trying to think as I do right now, as you are seeing something from a perspective that I have never been able to.

If you have access to the Stargate SG-1 season 1 episode, Enigma, take a look at what they are talking about around 33 minutes in. That one, little scene always left quite an impression on me, and seems to be saying the same thing you are.

[bperet]

Was thinking further on this concept...

Space cannot be directly represented in time (nor time in space). As you proposed in your earlier papers, space can be represented by an imaginary quantity in time (and time and space). Therefore, when a spatial displacement crosses the boundary into time, a dimensional increase must follow--the "time" dimension + the imaginary axis of the spatial effect on time. So we see "s" (1D) become "t + si" on the temporal side (2D). Reflecting that complex motion back also requires a dimensional increase--(t + si) can transform directly to (s + ti), formerly "s", since what was imaginary in time becomes real in space, and vice versa. Once you reach the complex form in 2 dimensions, you have stability--local (real) and nonlocal (imaginary), the particle-wave duality.

Looking back at Larson's system, his 1D "electric" motion must be a complex quantity, since that is the point of stability. And when you look at electronics, you find that is, indeed, the case... all operations concerning electric fields are expressed using imaginary quantities--the measurable electron disappears from "reality" and becomes and electric field, as it crosses over the unit boundary--the capacitor.

Considering your lineland example... in order to "loop it," it must traverse a rotation of i^4. One at i^0 and the other at i^4, or ends at +/-i^2. The yang line, projected into the yin would do exactly that, so it fits. Take a unit bivector, +/-1 and transform into yin (polar) and you get +/-PI (PI being the unit angular distance). Just like in that Stargate clip--"the two ends of the vine appear distant, until you do this" (touches ends together in a loop). As you said, nothing has changed--2 natural units of yang (linear space) to 2 natural units of yin (polar time). Still 2 units, except from the spatial perspective, what was now distant is coincident.

"In other words, one dimension of time = "-1" dimension of space."

So the dimension is eliminated because it is zeroed-out, like the imaginary axis on Nehru's dimensional reduction? I suppose that begs the question as to whether a dimension "exists" if the quantity on that dimension is equal to its datum? My take would be based on Nehru's dimensional formula--independent dimensions must always exist (3D). Any dimensions created beyond (4+) or beneath (1, 2), being dependent dimensions, would be eliminated--not zeroed.

So to build a Stargate or Jump Gate, each gate would be the endpoints of a line. The mechanism would then convert the yang, spatial distance between them into yin, temporal angle, making the gates spatially coincident, regardless of how far apart they are. Then you step through, and are transported instantly from one place to another. Though without any cool "tunneling" effects!

[Steve]

It is my understanding the tunneling experience does happen with stargate travel. it is not felt as instantaneous. The movie Contact is said to show a reasonably good depiction of both visual and audio.

[bperet]

It is my understanding the tunneling experience does happen with stargate travel. it is not felt as instantaneous.

You are probably right... consciousness, when faced with the task of transforming the angular motion of the connecting "turn" into some kind of linear projection (as it is familiar with), would probably perceive a round tube--though it would not take "clock time", but "clock space" to make the journey, therefore giving the sense of it having some distance to it, though still being instantaneous. (We perceive clock time as duration, and clock space as distance.) You would experience the motion through a tunnel, though your watch would not measure any passage of time for the journey.