A spin network is a generalization of a knot or link: a graph embedded in space, with edges labelled by representations of a Lie group, and vertices labelled by intertwining operators. Such objects play an important role in 3-dimensional topological quantum field theory, functional integration on the space A/G of connections modulo gauge transformations, and the loop representation of quantum gravity. Here, after an introduction to the basic ideas of nonperturbative canonical quantum gravity, we review a rigorous approach to functional integration on A/G in which L^2(A/G) is spanned by states labelled by spin networks. Then we explain the `new variables' for general relativity in 4-dimensional space-time and describe how canonical quantization of gravity in this formalism leads to interesting applications of these spin network states.
- John C. Baez
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If we take a guitar string and fix both its ends before plucking it, we will find that only a certain set of notes can be produced. These harmonics are defined by the geometry of the arrangement, since the only frequencies allowed are those that have anti-nodes (zero-points) at the two fixed ends. We could say that the set of notes has been quantised, that is, reduced to a finite set of discrete tones. The mathematics used when dealing with waves and vibrations is to a large extent that used when describing the structure of reality at the very small, quantum levels. An electron orbiting an atom may be represented as a sum of possible states of vibration using the techniques first developed by Fourier, as mentioned earlier. We find that an electron, like the guitar string, has a discrete set of possible states. One of the discrete variables used to represent the state of a subatomic particle is that known as 'spin,' a peculiar quality which is quantised into one of only two values, 'up' and 'down.' For a closed system --- one not open to external influences --- we know that ''spin is conserved,'' meaning that the total sum of all spins in the system remains constant.
It goes like this. We know that an atom is more than 99% empty space and less than 1% actual matter. (These numbers are oversimplified, albeit greatly, for the sake of an easier read.) The atom looks solid to us because we are larger and perceive at a rate so slow that the atom appears to spin astronomically fast. It is not just the spin which makes the atom take on the properties of a solid block of matter. It is more than that. It is our perception of that spin which makes it seem like matter. Our rate of perception is slower: the spin of the atom is faster. It is this vast difference in the two relative scales of time that enables a mostly empty structure to appear solid to the observer.
If we were able to shrink ourselves and accelerate our speed of perception, the electrons of that atom would appear to slow in their orbits. Eventually, the atom would become more apparent for its empty space than for its solidness. It would thus transform from solidness to emptiness. Accelerate our speed of perception fast enough, and we would find ourselves looking at the components of an atom standing still amid a sea of other atomic components standing still. More than 99% of the matter in the universe, as we formerly knew it, would have disappeared and we would be looking at the less than 1% of matter that remains.
What has just happened is that we eliminated matter when we accelerated the scale of time. We started by measuring the amount of matter in the known universe as it exists within the time scale of, say, a human second. By altering our observation to span the time of merely an astronomical fraction of a second, we find matter has disappeared. This is because matter is a function of time. It takes a certain measurement of time for the components of the atom to complete the amount of orbits necessary to guard their circumference and turn empty space into an impenetrable unit. Accordingly, whether this is solid or empty depends entirely on the amount of time being measured.
A supposition by Tom Burns Bacon October 1999
In Loop Quantum Gravity (LQG), reality is built of loops that interact and combine to form so-called spin networks-- first envisioned by English mathematician Roger Penrose in the 1960s as abstract graphs. Smolin and Rovelli used standard techniques to quantize the equations of general relativity and in doing so discovered Penrose's networks buried in the math. The nodes and edges of these graphs carry discrete units of area and volume, giving rise to three-dimensional quantum space. But because the theorists started with relativity, they were still left with some semblance of a space outside the quantum networks.
Markopoulou Kalamara approached LQG's extraneous space problem by asking, Why not start with Penrose's spin networks (which are not embedded in any preexisting space), mix in some of the results of LQG, and see what comes out? The result was networks that do not live in space and are not made of matter. Rather their very architecture gives rise to space and matter. In this picture, there are no things, only geometric relationships. Space ceases to be a place where objects such as particles bump and jitter and instead becomes a kaleidoscope of ever changing patterns and processes.
Each spin network resembles a snapshot, a frozen moment in the universe. Off paper, the spin networks evolve and change based on simple mathematical rules and become bigger and more complex, eventually developing into the large-scale space we inhabit.
By tracing this evolution, Markopoulou Kalamara can explain the structure of spacetime. In particular, she argues that the abstract loops can produce one of the most distinctive features of Einstein's theory-- light cones, regions of spacetime within which light, or anything else, can reach a particular event. Light cones ensure that cause precedes effect. We can understand this concept by gazing upward and knowing that there are countless stars we cannot see because not enough time has passed since the birth of the universe for their light to shine our way; they are beyond our light cone.
It is not so obvious, though, where light cones fit into the spin networks. Those networks are subject to quantum mechanics. In that wonderland of uncertainty, any network has the potential to evolve into infinite new ones, leaving no trace of a causal history. "We didn't know how, in the language we were working in, to put in the notion of causality" in LQG, Smolin says. Markopoulou Kalamara found that by attaching light cones to the nodes of the networks, their evolution becomes finite and causal structure is preserved.
But a spin network represents the entire universe, and that creates a big problem. According to the standard interpretation of quantum mechanics, things remain in a limbo of probability until an observer perceives them. But no lonely observer can find himself beyond the bounds of the universe staring back. How, then, can the universe exist? "That's a whole sticky thing," Markopoulou Kalamara says. "Who looks at the universe?" For her, the answer is: we do. The universe contains its own observers on the inside, represented as nodes in the network. Her idea is that to paint the big picture, you don't need one painter; many will do. Specifically, she realized that the same light cones she had used to bring causal structure into quantum spacetime could concretely define each observer's perspective.
Because the speed of light is finite, you can see only a limited slice of the universe. Your position in spacetime is unique, so your slice is slightly different from everyone else's. Although there is no external observer who has access to all the information out there, we can still construct a meaningful portrait of the universe based on the partial information we each receive. It's a beautiful thought: we each have our own universe. But there's a lot of overlap. "We mostly see the same thing," Markopoulou Kalamara explains, and that is why we see a smooth universe despite a quantized spacetime. "I actually think theoretical physics is very much like art," concludes Markopoulou Kalamara, the daughter of two sculptors. "Putting these things together is like taking clay and making something out of nothing, and it should work from every side. I like the creative part, but I also like that you can check."
- _Scientific American_ -
November 11, 2002
|spinning top levitron|
Roger's OR is based on the idea that quantum superpositions are separations at the most basic level of the universe at the Planck scale. So you ask yourself, what is this basic level? What is the universe made of? Even mass is not fundamental according to Einstein. Atoms are mostly empty space as is most of the universe. So what is the universe made of? This argument has been going on since the Greeks. Is there a background fabric, or just an empty void? In the last few decades there's been a lot of intense work trying to understand the background pattern of the universe. It turns out that as we go down in scale, well below the size of atoms, things are smooth and featureless until we get to the apparent basement level of the universe known as the Planck scale, some 25 orders of magnitude smaller than atoms. Empty space seems smooth but at the Planck scale things get coarse and irregular, with a vast amount of information and energy. It's kind of like viewing the surface of the ocean from an airplane at 33,000 feet. The ocean seems smooth but if you were on the surface in a small boat you'd be tossed about by waves. How can we describe the Planck scale, basically quantum gravity? String theory has tried, but others, for example Lee Smolin, argue for spin networks, based on Roger Penrose's original idea that at this level everything is spin. The universe is made of spiderwebs of spin which define ultrasmall Planck volumes, or pixels of reality.
- Stuart Hameroff - Morality On The Planck Scale