25 June 2010

Disordered Locality

One of the most interesting ideas of loop quantum gravity is that the structure of space-time may be disordered and lack the same kind of locality found in macroscopic classical physics. A paper exploring the concept from September 2009 by Chanda Prescod-Weinstein and Lee Smolin is found here.

The basic idea is that fundamentally, space-time may be made up of a bunch of points that are connected to some other points. Points may be two, three, four, or, for example, five hundred points away from each other by the shortest route. But, it might be possible that two points that are by hundreds of other routes, no less than several hundred hops apart from each other are also connected to each other directly.

If macroscopic distance is defined to be the average number of hops that it takes to get from point A to point B by all possible routes, then non-local links are those that involve many fewer links than average. Prescod-Weinstein and Smolin's paper defines the concept in terms of volume rather than length, but the differing expressions express the same basic concept.

In this kind of world, macroscopic distance is only an emergent property of fundamental quantum distance.

A non-local connection in this theory is a bit like the science fiction concept of a wormhole, a direct connection between two macroscopically non-adjacent points. Only unlike the science fiction idea of a wormhole, each one is only big enough to fit a single quantum particle. There can be (and indeed should be) gillions of them, but if they tend to become more rare with macroscopic distance relative to local connections, then they become almost invisible in classical physics.

By analogy, in classical thermodynamics there is some probability that all of the oxygen molecules in a room will end up entirely on the North side of the room turning the South side of the room into a vacuum for a moment. But, the longer the period of time involved, and the more oxygen molecules there are involved, the less likely this is to happen. And when the probability of this happening is less than say 50% in 15 billion years (the age of the Universe) it becomes effectively impossible.

One way to interpret the probability that a photon will travel at above or below the speed of light which is a product of 1/I (in which I is the space-time distance between two points), is that it is a measure of the extent to which space-time is non-local over a distance I (we would expect non-locality to average out over greater distances in any disordered space-time model that appears to be local at the macroscopic level).

This could also help to explain a fundamental cause of the uncertainty principle. Perhaps it is impossible to know location and momentum with more than a certain degree of specificity because the concept of distance becomes increasingly ill defined as one looks closer and closer. Indeed, folks like Stephen Hawkings have used this notion of disorder to explain why one could have a "Big Bang" without having any particular moment that would clearly be the beginning of everything.

This sounds crazy and weird. And it is weird, sort of. But, not that weird.

Have you every heard of the concept "six degrees of separation"? It is a social networking idea. You might know somebody personally, and that is one degree of separation. If you know somebody who knows somebody, you are separated by two degrees of separation from that other person. The theory goes that almost no two people are separated by more than six degrees of separation.

People who are close to each other physically, for example, in the same small town or neighborhood, or who go to the same school or work at the same business, are likely to have few degrees of separation between them. People who live in worlds that are very different geographically and culturally, for example, someone on a Wisconsin farm and someone in a Papua New Guinea village with few ties to the outside world, are likely to have many degrees of separation between them.

You could probably draw a map ordered by degrees of social separation that would look very much like a map of degrees of physical separation. The linkage between social separation and physical separation would have been even stronger in pre-modern times, when travel and long distance communication were much more rare.

By analogy, fundamental distance might be analogous to our degrees of social separation based map, while apparent distance might be analogous to our physical separation map.

So, if locality is disordered, then the fundamental nature of space-time is that if you look really closely, at a point by point level, you can't line the points up neatly on a number line in any dimension of space or in time.

It gets worse. In this kind of disordered locality scenario, it may be that the number of dimensions in space-time itself is not well defined. There may be several different paths you can take to get from point A to point B, but the choices in those paths may not be clearly identified with up and down, left and right, forward and back, future and past. Four directions may eventually provide a pretty good approximation of the network, but you may need a fairly sizeable network of points for those directions to become well defined.

Thus, rather than the idea from string theory that we may have four macroscopic dimensions and six really tiny curled up dimensions, in loop quantum gravity, at small scales, the concept of the number of space-time dimensions may simply become ill defined. Both approaches are pretty weird, but the loop quantum gravity approach is "natural" by comparison; one goes from a very simple notion of space-time as points connected by lines, to a very familiar one (the four dimensional universe), without having to find a way to make six clearly predicted dimensions of space-time become invisible.

This also sounds weird, but the idea of non-integer dimensions which are emergent actually has solid mathematical precedent. It is the fundamental idea behind the concept of a "fractal dimension." The definition of a fractal dimension takes a relationship between two quantities that happen to be "1" on a line, "2" in a flat space, "3" in a three dimensionally flat space, and so on, and finds a way to generalize it so that a form that is squiggly in a certain kind of way at all scales (e.g. a shoreline) can be assigned a fractional dimension that is not an integer. It also happens that fractal dimensions are quite important in explaining a non-obviously related concept called "chaos" which is a kind of ordered near randomness that can arise when a deterministic equation is very sensitive to initial conditions. Chaotic functions tend to produce fractal shapes when plotted.

The definitions of distance and volume in a disordered locality model are quite similar in concept to the definition of dimension used in the mathematics of fractals. So, it turns out that there is already a mathematics well crafted to describing how a network could become emergently four dimensional in a mathematically consistent way.

A system that has disordered locality at a small scale is interestingly mathematically even if it isn't an accurate description of the universe. In the advanced calculus classes called "real analysis" and "complex analysis", one learned that "continuity" of a number line made up of points is a central and key assumption upon which most of advanced mathematics relies. Even discrete mathematicians, who deal with number lines where there are gaps, normally deal with far simpler systems than that approximately continuous at large scales quantum space-time that loop quantum gravity is considering.

And, there are important macroscale differences between the two approaches. For example, one of the major conclusions of the theory of social networks is that a very small number of non-local links dramatically reduce the number of degrees of separation between very large groups of people. Similarly, it doesn't take many wormholes to dramatically reduce the shortest distance between two points in space. Even a small number of non-local links can have a great effect on the behavior of the entire system.

In the paper linked above, for example, it is demonstrated that it would take only one non-local link in every box of 100km on each side to produce the dark energy effect observed in the universe, even though the effect is equivalent to about 70% of the mass-energy in the universe. A few wormholes can have immense effects.

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