Numerical Experiments in Spin Network Dynamics

1
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 Einsteins 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 networks1.
Spin networks 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 and
abc must be even, 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.
Model Dynamics
Sandpile Model The model that we considered would
ideally exhibit self-organized 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 self-organized 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.
Numerical Experiment 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.
Dynamics 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. 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.
General results 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.
Artwork by Elaine Wiesenfeld 3
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