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NON-COMMUTATIVE SPACE

May 1, 2012

(What are Space and Time, Really, and Can We Do Without them?)

(From The Elegant Universe, Brian Greene, Vintage Books, p.376-78)

In many of the preceding chapters, we have freely made use of the concepts of space and of spacetime…envisioning the fabric of space and space-time as if it were somewhat like a piece of material out of which the universe is tailored. These images have considerable explanatory power; they are used regularly by physicists as a visual guide in their own technical work. Although [this] gives us a gradual impression of meaning, one can still be left asking, What do we really mean by the fabric of the universe?

This is a profound question that has, in one form or another, been the subject of debate for hundreds of years. Newton declared space and time to be eternal and immutable ingredients in the makeup of the cosmos, pristine structures lying beyond the bound of question and explanation. Leibniz and others disagreed, claiming that space and time are merely bookkeeping devices for conveniently summarizing relationships between objects and events within the universe. The location of an object in space and in time has meaning only in comparison with another. Space and time are the vocabulary of these relationships, but nothing more. Although Newton’s view held sway for more than 200 years, Leibniz’s conception, further developed by Austrian physicist Ernst Mach, is closer to our current picture. As we have seen, Einstein’s special and general theory of relativity firmly did away with the concept of an absolute and universal notion of space and time. But we can still ask whether the geometrical model of space-time that plays such a pivotal role in general relativity and in string theory is simply a convenient shorthand for the spatial and temporal relations between various locations, or whether we should view ourselves as truly imbedded in something when we refer to our immersion within the space-time fabric.

Although we are heading into speculative territory, string theory does suggest an answer to this question. The graviton, the smallest bundle of gravitational force, is one particular pattern of string vibration. And just as an electromagnetic field such as visible light is composed of an enormous number of photons, a gravitational field is composed of an enormous number of gravitons—that is, an enormous number of strings executing the graviton vibrational pattern. Gravitational fields, in turn, are encoded in the warping of the space time fabric, and hence we are led to identify the fabric of space-time itself with a colossal number of strings all undergoing the same, orderly, graviton pattern of vibration. In the language of the field, such an enormous, organized array of similarly vibrating strings is know as a coherent state of strings. It’s a rather poetic image—the strings of string theory as the threads of the space-time fabric—but we should note that its rigorous meaning has yet to be worked out completely.

Nevertheless, describing the space-time fabric in this string-stitched form does lead us to contemplate the following question. An ordinary piece of fabric is the end product of someone having carefully woven together individual threads, the raw material of common textiles.
Similarly, we can ask ourselves whether there is a raw precursor to the fabric of space-time; a configuration of strings of the cosmic fabric in which they have not yet coalesced into the organized form that we recognize as space-time. Notice that it is somewhat inaccurate to picture this state as a jumbled mass of individual vibrating strings that have yet to stitch themselves together into an ordered whole because, in our usual way of thinking, this presupposes a notion of both space and time; the space in which a string vibrates, and the progression of time —that allows us to follow its change in shape from one moment to the next. But in this raw state, before the strings that make up the cosmic fabric engage in the orderly, coherent vibrational dance we are discussing, there is no realization of space or time. Even our language is too coarse to handle these ideas, for, in fact, there is even no notion of before. In a sense, it’s as if individual strings are “shards” of space and time, and only when they appropriately undergo sympathetic vibrations do the conventional notions of space and time emerge.

Imagining such a structureless, primal state of existence, one in which there is no notion of space or time as we know it, pushes most people’s comprehension to their limit (it certainly pushes mine).

The hope is that from this blank slate starting point—possibly in an era that existed before the big bang or the pre-big bang (if we can use such temporal terms, for lack of any other linguistic framework)—the theory will describe a universe that evolves to a form in which a background of coherent string vibrations emerges, yielding the conventional notions of space and time. Such a framework, if realized, would show that space, time, and by association, dimension, are not essential defining elements of the universe. Rather they are convenient notions that emerge from a more basic, atavistic, and primary state.

…whereas strings show us that conventional notions of space and time cease to have relevance below the Planck scale [10-35 m], studies show that ordinary geometry is replaced by something known as non-commutative geometry, an area of mathematics developed in large part by the French mathematician Alain Connes. [note—p412: one way to think of noncommutative geometry is to replace conventional Cartesian coordinates which commute under multiplication, with matrices, which do not]. In this geometrical framework, the conventional notions of space and of distance melt away, leaving us in a vastly different conceptual landscape.

Nevertheless, as we focus our attentions on scales larger than the Planck length, physicists have shown that our conventional notion of space and time does re-emerge.

This idea is fascinating, and to me fits right into the notion that God is not contained in space and oeven time. It thus answers a lot of skeptical questions I have seen, such as how could God be eternal if space wasn’t.
It seems that afterwards, Greene and the other string theorists stopped talking about this non-commutative “primeval” realm, and instead basically replaced it with an 11D “superspace” in their emerging “M theory” (now said to stand for “membrane”). This space is commutatve, as colliding branes are said to be floating around in it, which create “Big Bangs” in them when they collide. This of course doesn’t tell us anything about where this 11D spacetime came from. It looks like they’ve just pushed back the theory of creation of existence to some totally different dimension.

Relating all this to the notion of the “chance” continuum, the chance realm would be non-commutative. In a commutative realm, there is a simple, linear connection between points. You can commute from one point to another (even in time, where you can’t go back to an earlier point). With this chance realm, there is no simple connection like that. It is, from our perspective, hypothetical.
I wonder if I’m even correct in considering it a vector space. You could perhaps create vectors with a logically connected string of different alternative events. Like the different “choices” I mentioned in that discussion, with some being closer than others. I could conceive of at any given pint in time, having on a red shirt, and create a line through chance, where each counterfactual has me wearing a shirt differing in color by one nanometer, and then plot points on this line, where I’m wearing 700nm (red), or 600nm (orange), 500nm (blue-green) or 400nm (deep violet, on the edge of the visible spectrum). Like with any length of time or space, there are all the infinitessimal intermediate points inbetween.

But that’s just one possible parameter on which I can plot a line through chance. It doesn’t have the logical necessity that a line connecting points in space (where matter is located relative to each other) or time (causative) does. However, when we think of the different states simultaneously, as connected by a line, we are sort of converting the realm to a kind of space. The “line” is really only a construct to view relative differences arranged by some sort of logical progression or succession.
So this realm is not really commutative. It probably is a good way to think of the noncommutative realm string theory [used to] consider.

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