Transcript No Slide Title

Numerical Experiments in Spin
Network Dynamics
Seth Major and Sean McGovern ‘07
Hamilton College Dept. of Physics
INTRODUCTION
Quantum gravity is a theory that attempts to unify Albert Einstein’s general theory of
relativity with quantum mechanics. General relativity works well with large scale
phenomena like planets, galaxies and other large bodies, while quantum mechanics
effectively deals with the subatomic world. When dealing with a system that can be
both massive and tiny, such as a small black hole, neither theory will work alone and
when used together they generate non-physical solutions.
So a theory which incorporates the insights from both theories is needed and loop quantum gravity is one
potential candidate. Our project was to model a version of this theory with a computer program and run
numerical experiments to simulate quantum gravity dynamics. Our goal is to find long range interaction
in our model, which would be an indication of gravity.
We began with a model put forth by Roumen Borissov and Sameer Gupta, who studied regular trivalent
2-dimensional spin networks[1].
Spin networks
Model Dynamics
abc
bc  a
c must
a  be
b even,
and a+b+c
where a,b and c are labels on the edges.
If these rules are upset, then measures must be taken to
restore the invariance. See example to the right.

Critical vertices are ones where one of the inequalities
is saturated. These are of interest because at a critical
triangle, any addition to the biggest edge or subtraction
from one of the smaller two will result in a disruption of
gauge invariance and therefore action to restore it.
Sandpile Model
Numerical Experiment
The model that we considered would ideally exhibit selforganized criticality. This is the idea that following very
simple rules, a simple system can exhibit complex
phenomena.
A nice example of the principle of self-organized criticality is
a sandpile to which sand is being added. Take sand and pour
it slowly on one spot. The pile grows to a certain point, but
then the addition of even one grain of sand can cause an
avalanche of arbitrary size. The avalanches could be very
tiny or they could be system-wide. This system exhibits selforganized criticality since it builds up to a critical point, then
an avalanche sets the system back to before the critical point.
The continued addition of sand will cause the system to reach
criticality again and react with another avalanche. The
computer simulation parallels this scenario in that the edges
are changed to the point where the fraction of critical
triangles is high enough to support an avalanche of arbitrary
size, then one occurs and resets the system and so forth. This
is what we would like to happen, anyway.
When doing a run of the program, there are multiple variables that can
be altered. The size of the lattice being used and the various
probability variables affect the course that the run will take. What we
are looking for is the fraction of critical vertices over total vertices (F)
to remain constant at a non-zero value after a certain number of
iterations of the program. One iteration is one time that a random edge
was chosen and altered. Also the frequency of any given size of
avalanche is of interest. The size of an avalanche means how many
vertices were altered during the course of the avalanche. The
frequency is how many times an avalanche of a given size occurs in
the run. Here are some sample result graphs.
Count
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F
In one approach to quantum gravity, space is represented
as labeled graphs or spin networks. In 2-dimensions, this
takes the form of the dual graph of a lattice of triangles.
Each edge of the spin graph receives a value, called a
color, which represents the length. The three edges that
form each vertex must obey certain rules. Each vertex
must obey the four rules of gauge invariance:
Count
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0.1
Dynamics
One change of a random edge is one iteration of the
program and what we are interested in is the ratio of
critical vertices to total vertices as a function of
iteration number. We are also interested in how
many vertices need to be changed during one
iteration, or the size of the avalanche. If a vertex on
the other side of the lattice has to be changed as a
result of changing the initial vertex, then this is a
long-range interaction. We hope to be able to
observe this.
0
0
2000
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8000
Iteration number
10000
12000
400
General results
Artwork by Elaine Wiesenfeld [3]
We were very close to finding appropriate values for the probability
variables that would give a constant non-zero value for the fraction of
critical triangles(F). The graph shown is an example of the rapidly
decaying F value.
In general, we were unable to find evidence of self-organized criticality.
The plot of the frequency versus size of an avalanche should give a linear
graph, which is characteristic of the power-laws of self-organized
criticality. Instead, we have been finding decaying exponentials for these
graphs, like the one shown.
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300
frequency
In Loop Quantum Gravity, dynamics is roughly
expressed by changes to edge color. So in an effort
to model dynamics, our program is concerned with
changing edges and observing the results.
Starting with a randomly initialized lattice which is
completely gauge invariant, we choose an arbitrary
edge and modify it. If this does not violate gauge
invariance then the iteration is over and another
random edge is picked. If the change did violate the
gauge invariance at neighboring vertices then
according to given probabilities, certain steps are
taken to restore gauge invariance at the vertex. If
this alteration upsets another adjacent vertex, then
that one must be changed and so forth, until either a
change no longer upsets invariance or a vertex on
the edge of the lattice is reached, a dead end.
0.05
References:
[1] R Borissov, S Gupta, Propagating Spin Modes in Canonical Quantum Gravity,
gr-qc/9810024, Phys.Rev. D60 (1999) 024002
[2] Bak, Per. How Nature Works.Oxford University Press, 1997.
[3] http://www.cz3.nus.edu.sg/~chenk/gem2503_3/notes8_3.htm
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avalanche size
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