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Spin Networks
This nOde last updated November 27th, 2004 and is permanently morphing...
(9 Ix (Jaguar) / 17 Keh (Red) - 74/260 - 12.19.11.11.14)
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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
604
track _Heinous Groove
Gland_ MP3 (192k)
by Higher Spin States off of _10 kilors sampler 3_ 10"
on 10 kilo (1998)Ray
Castle |
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Maypole
Spins_ MP3by
Dead
Can Dance off of _Aion_ on 4AD (1990)
by Electronic Home Entertainment
(1998)
by Ozric
Tentacles off of _The Hidden Step_
by Spinback & Windmill off of _Divine Inspiration_ 12" on Legend
#002 (1993)ambient
track _Spinning Spirits_ MP3 (160k)
by Haruomo Hosono off of _NDE_ (1995)
by The Boo Radleys off of _Lazarus (and other songs from Giant Steps)_ promo
CDs
on Creation/Columbia (1992)_Drums
In Friction_ MP3 (192k)
by Ken Ishii off of _Flatspin_
12"x2 on Exceptional #103 (2001)
by Brian
Eno & John Cale off of _Which Way Up_ on Opal (1990)_Solar
Powered_ (2:39) by Binary Star(3:29)
by Large ProfessorContact_
MP3
(3:43) by Deltron 3030Beastie
Boys_Breakdance
- Electric
Boogie_ MP3
by West Street Mob off of 12" on Sugar Hill (1983)Beastie
Boys 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.
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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
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
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