Numerical Experiments in Spin Network Dynamics Seth Major and Sean McGovern ‘07 Hamilton College Dept. of Physics 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 a+b+c 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. 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. 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. 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] 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]. Model Dynamics References: [1] R Borissov, S Gupta, Propagating Spin Modes in Canonical Quantum Gravity, gr-qc/ , Phys.Rev. D60 (1999) [2] Bak, Per. How Nature Works.Oxford University Press, [3] INTRODUCTION