Kenneth Ratliff (Hamilton '16)
Spectral Relativity Summer Research (2014) 
In collaboration with Mike, we derived some of the results of special relativity such as length contraction, time dilation, and frequency shifts in the conetxt of the Doubly Special Relativity. We assumed an additional scale, an energy dependent speed of light, and event refraction. Using theoretical experiments, our results show that a contradiction arises from frequency shift and the notion of event refraction leading to multiple times and, ultimately, an inconsistent set of assumptions. 

Mike Verostek (Hamilton '16)
Spectral Relativity Summer Research (2014) 
In collaboration with Kenny, we derived some of the results of special relativity such as length contraction, time dilation, and frequency shifts in the conetxt of the Doubly Special Relativity. We assumed an additional scale, an energy dependent speed of light, and event refraction. Using theoretical experiments, our results show that a contradiction arises from frequency shift and the notion of event refraction leading to multiple times and, ultimately, an inconsistent set of assumptions. 

Chris Lockwood (Hamilton '14)
Constraining Quantum Spin Statistical Mechanics of Black Holes Senior Project (2014) 
A preliminary look at the possibility of using constraints from loop quantum gravity in modeling the spin punctures for a black hole is performed. Spin punctures are treated as quantum particles, so they can be modeled using quantum statical mechanics. Most of the work is set on the foundation built by Ghosh, Noui and Perez. This thesis adds an additional constraint to the system steming from the triangle inequality. The results obtained include the expectation values for both individual spins and expectation values of the horizon's "shape". 

Sunrose Shrestha (Hamilton '14)
Analytic Approaches to Explain Bifurcations in Spin Network Dynamics Senior Project (2014) 
Loop quantum gravity is a theory that attempts to unify classical general relativity with quantum mechanics. In loop quantum gravity, the basis of quantum states are spin networks, which are essentially graphs with labeled edges. We look for selforganized criticality in the dynamics in such graphs, which brings forth the potential to recover the classical continuum in a limit. In particular, we attempt to understand the results of some past numerical experiments performed in spin network dynamics by McGovern, Minella, and Schwedock and explain a particular bifurcation in their frequency data. We present results based on our analysis of the structure of the underlying graphs of these networks, making use of theorems from algebraic graph theory. We find results that suggest that the bifurcation that prevented McGovern et. al.'s from achieving selforganized criticality, was caused by the fundamental bias in the number of even walks versus the number of odd walks in the underlying graph itself. 

Tsion Tesfaye (Hamilton '16)
On the consistency of extensions to special relativity Summer Research (2013) 
While we have no quantum theory of spacetimes as yet, preliminary results indicate a subsutructure on small scales, much like the energy levels of electrons in an atom. If this is correct then high energy photons traveling through such a spacetime would ``see" this substrcuture. This project seeks to show whether an energydependent speed of light can be made compatible with the relativity principle. In other words we will pretend we are Einstein in 1905 but instead of ``the speed of light is equal for all inertial observers" we will use "the speed of light is given by c(E) = c(1  E/E_P), where E_P is the Planck energy, for all inertial observers." 

Grace WilliamsDuHamel (Hamilton '15)
On the consistency of extensions to special relativity Summer Research (2013) 
While we have no quantum theory of spacetimes as yet, preliminary results indicate a subsutructure on small scales, much like the energy levels of electrons in an atom. If this is correct then high energy photons traveling through such a spacetime would ``see" this substrcuture. This project seeks to show whether an energydependent speed of light can be made compatible with the relativity principle. In other words we will pretend we are Einstein in 1905 but instead of ``the speed of light is equal for all inertial observers" we will use "the speed of light is given by c(E) = c(1  E/E_P), where E_P is the Planck energy, for all inertial observers." 

Jeremy Adelman (Hamilton '13)
Towards the Quantization of Plane Gravitational Waves Senior Project (2012) 
Recent work by Hinterleitner and Major on the loop quantization of plane gravitational waves proposed a form for the rightmoving constraint. We show that nontrivial solutions to the constraint are normalizable but generically give infinite expectation values for length, even for the shortest possible lines.In addition, it appears as though compactification fails to ameliorate this issue, suggesting that the constraint and geometric operator must be reformulated before future fruitful work on the quantization of plane gravitational waves can commence. 

Jake Zappala (Hamilton '12) Semiclassical States for an Atom of Geometry Senior Project (2012) 
The successes of quantum field theories over the past halfcentury suggest the idea that general relativity may be unified with the principles of quantum mechanics to create a theory of quantum gravity that would describe spacetime on the microscopic scale. One such theory is the theory of loop quantum gravity (LQG). LQG describes a structure of space via spin networks (networks of angular momentum) discrete on scales on the order of the Planck length. Though the direct effects of this discreteness occur on a scale that is far too small to be observed, it has been shown that there exists a possible lever arm to lift up the effects to a mesoscopic level. This lever arm is associated with asymmetries in angular distributions on the atoms of the geometry. We review the geometric observables of area, angle, and volume on these atoms and the creation of this lever arm through such observables. To test the assumptions used in building the lever arm, we create semiclassical states for the atoms ultimately based on an operational point of view using a scattering thought experiment. We present a simple example of these states and show how their geometry can be used to model the state of space. Said examples reinforce the possibility of a lever arm by yielding nonclassical distributions for angle and squashed geometries on the semiclassical states of the atom. Particular attention is given to ensuring that the information used in these semiclassical states comes from the intrinsic geometry of the atom. 

Abrar Ahmed (Hamilton '14) Modifed Lorentz Transformations Summer (2011) 
A continuation of work by Ileana and Will 

Ileana Becerra (Hamilton '11) Deformed Special Relativity Senior Project (2010) 
Deformed Special Relativity (DSR), a modification of Special Relativity, includes the possibility that the speed of light is energydependent. Recent studies suggest that ultrahigh energy gamma rays may arrive later than their lower energy counterparts. DSR could be the new relativistic theory that explains these phenomenon. The DSR theory discussed here requires a new set of Lorentz transformations and "event refraction" which appears as nonlocality in the classical spacetime framework. 

William Kalbacker (Hamilton '11) Deformed Special Relativity as a Coherent Model for Spacetime Senior Project (201011) 
The subject of Deformed Special Relativity (DSR) is a modification to standard Special Relativity for high energies. It theorizes that the speed of light is not constant in all reference frames, but is dependent upon a propagating photon's energy. In addition, DSR introduces two invariant scales that are constant to all observes: the speed of light c(E) and the Planck energy. This paper explores and investigates the plausibility of DSR as a coherent model for spacetime. Furthermore, it derives modified slip in simultaneity, time dilation, and length contraction assuming the postulates of DSR to be true. Finally, this paper will derive modified Lorentz transformations for DSR in order to transform between reference frames without having to rederive relations on a situation to situation basis. 

Gregory Schwedock (Hamilton '10) SelfOrganized Criticality in a Model of Loop Quantum Gravity Summer Research (2008) 
Continuation of the work of Tim and Sean. 

Walter Schoen (Hamilton '08) Primordial Gravitational Waves with a LQG Correction Term Senior Project (2007) 
Using the FriedmannRobertsonWalker (FRW) metric for a homogeneous and isotrpic universe, we introduce the Laplacian and use it to find the wave equation in a space time with the FRW metric. We claim that gravitational waves exist and they satisfy the wave equation for an expanding space time. We review the calculation by Dodelson to find the power spectrum from gravitational waves generated by the initial inflationary period of expansion of the universe. We investigate what gravitational waves generated during the initial expansion of the universe might look like with a quantum loop theory modification found by Yubo Lu. We describe an approach for analytical solutions to solve the equation for gravitational waves modified by loop quantum gravity effects. 

Timothy Minella (Hamilton '09) SelfOrganized Criticality in a Model of Loop Quantum Gravity Summer Research (2007) 
Loop quantum gravity proposes a physical theory for the smallest possible scales. A key feature of the theory is that space is not continuous, but discrete. A spin network represents this discrete space. In this representation edges labeled with color determine areas while the vertices, where the edges meet, determine volumes. We use a twodimensional triangular spin network and evolve it in time to determine whether it exhibits “selforganized criticality”. Selforganized critical systems “evolve on their own,” without any outside tuning of parameters. Selforganized critical systems have the unique feature that there is a power law relation between the size of the effects and the frequency of their occurrence. Using a numerical simulation, we model this system and evolve it using specific rules that roughly correspond to LQG theory. We find evidence of selforganized criticality, with a power law relation F(s)=a*sb, where F is the frequency of avalanches, s is the size, a is a constant, and b is approximately 0.9. 

Micael Gregg (Hamilton '08) On Modified Dispersion Relations and the Chandrasekhar Mass Limit Summer Research (2006,2007)

Modified dispersion relations from effective field theory are shown to alter the Chandrasekhar mass limit. At exceptionally high densities, the modifications affect the pressure of a degenerate electron gas and can increase or decrease the mass limit, depending on the sign of the modifications. These changes to the mass limit are unlikely to be relevant for the astrophysics of white dwarf or neutron stars due to wellknown dynamical instabilities that occur at lower densities. 

Yubo Lu (Hamilton '07) Quantum Cosmological Effects on Primordial Gravitational Waves Summer Research (2006, 2007) 
Primordial gravitational waves are oscillations of spacetime that were generated during the Big Bang. According to one approach to the theory of quantum gravity, geometry is discrete. In the very early universe, the effects of discrete geometry may be significant and so primordial gravitational waves may be affected. Two different classes of modified gravitational wave equations were derived by quantizing the inverse scale factor based on the Loop Quantum Cosmological model and the HusainWinkler Cosmological model. The ambiguities in the parameterization were also studied. It was shown that the leading order contribution scales as a power of the ratio of the Planck lendgth to the scale factor at the end of inflation.


Julia MacDougal (Hamilton '09) Galactic Heavyweights: Constraints on Hadronic Modified Dispersion Relations from UHE Cosmic Rays Summer Research Project 2006 
In 1966, Kenneth Greisen, Vadim Kuzmin and Georgiy Zatsepin found that the upper threshold energy for charged particles from distant sources is 5 x 10^19 eV. However, current data indicates that some ultra high energy cosmic rays possess more energy than this GZK threshold. One way to solve this conundrum is through a modified, Lorentzviolating dispersion relation that involves a third order correction term. A new constraint on this correction term was derived from the kinematics of proton decay. 

Rob Silversmith (Clinton HS '07) A NoGo Theorem for the qLoop Algebra and Braided Preons as a Basis for the Standard Model Summer Research Project 
In the first part of the summer the consistency of the combinatorics of the TemperleyLieb algebra and a deformation of the qloop loop algebra was studied. It was found that no nontrivial, consistent deformation exists. In the second part, the prion model of BilsonThompson was investigated including problems of the model such as the modeling of mass and higher generations were explored. 

Alice Francis (Hamilton '06) Quantum Cosmological Effects on the Primordial Stochastic Gravitational Wave Background Senior Project 
A stochastic background of gravitational waves  spacetime oscillations is expected to originate just after the big bang. Little is understood about the physics of universe at this very early period, which makes the gravitational background a source of great interest within the physics community. If detectable, these primordial gravity waves would reveal information about the geometry of the universe at the time of emission. Different cosmological models produce different gravitational wave spectra, and therefore it might be possible to determine which model generated these gravitational waves. Two particular models are studied, Loop Quantum Cosmology and the HusainWinkler Cosmology. Modified wave equations, including corrections, are presented and the gravity wave energy densities for each are calculated. Although the order of the corrections differ, the result indicates that primordial gravitational waves in both cosmological models are observationally indistinguishable from those generated by the standard de Sitter model.


Ben Auerbach (Hamilton '05) Anisotropic Mass Effects on a Foucault Pendulum Senior Project and Summer 2004 
More than 40 years ago Cocconi and Salpeter proposed a formulation of Mach's Principle and considered how a nearby massive object, the Galactic Center, could produce a anisotropy of inertial mass. They provided a matrix for the inertial mass of an object for this case. This matrix represents the inertial mass relative to the direction the object moves. We investigated the Foucault pendulum in a mass anisotropic scenario. We found that a Foucault pendulum that has anisotropic effects would process faster than the classical Foucault pendulum. At the end of one day the anisotropic Foucault pendulum is processing at a rate equal to one half of a percent faster than the classical Foucault pendulum.


Sean McGovern (Hamilton '07) Numerical Experiments in Spin Network Dynamics Senior Project and Summer 2004 
In one approach to quantum gravity, space is represented as labeled graphs or spin networks. I wrote a program for a model 2D quantum gravity in which space is a trianglular lattice. The program simulates quantum gravity dynamics, as done by Borissov and Gupta. It is set up such that each vertex is trivalent and the numbers on each of the three edges must obey consistency conditions. The conditions on the edge values represent the restrictions imposed by angular momentum conservation or gauge invariance. In the dynamic process when a random edge gets its value changed, potentially a neighboring edge must be changed in order to restore gauge invariance. This process can propagate through many vertices. Some of these vertices may be distant. We hope to observe long range cascades of this type which may be an indication of gravity, a long range force.


Nancy Shaw (Hamilton '07)
Energy and Momentum in Modified Special Relativity Summer 2004 
Special Relativity and Quantum Theory are not entirely compatible. Certain modifications of Special Relativity, were looked at in an attempt to reconcile these theories. Specifically, modifications were made to Special Relativity to include a minimum length. Beginning with modified energy momentum transformations an attempt was made to derive spacetime transformations which incorporate the set minimum length. 

Dan Heyman (Hamilton '03)
Is Doubly Special Relativity Consistent with the Relativity Principle? Phys.Rev. D69 (2004) 105016 
We investigate the implications of adding a second observer independent scale to Einstein's theory of Special Relativity. To observe whether the new theories, known as Doubly Special Relativity, are consistent with the relativity principle we calculate particle process kinematics for two of these theories. We demand d that all inertial observers must agree on the occurrence of physical processes. The results of our analysis suggest that one DSR is consistent with the relativity principle while another is not.


Tomasz Konopka (Hamilton '02)
Quantum Gravity Effects on Ultra High Energy Particles New J. Phys. 4 (2002) 57 
The discrete structure of space predicted by loop quantum gravity implies modified particle dispersion relations and deviation from Lorentz symmetry. Observations of ultra high energy particles allow a phenomenological study of the quantum gravity effects and pose constraints on parameters characterizing particles' interactions with discrete space. Constraints in a three dimensional space (corresponding to parameters for hadrons, leptons and massless particles) show that current observational data is consistent with the quantum gravity model. The allowed parameter space is strongly constrained in the positive region where the particles would be able to move propagate at speeds faster than light, and the sign ambiguity for the hadronic parameter is removed altogether.


Kevin Setter (Swarthmore '02)
On the Statistical Mechanics of Quantum Geometry Class. Quant. Grav. 18 (2001) 51255141 
This thesis presents an original derivation of the BekensteinHawking formula for the entropy of a black hole within the framework of loop quantum gravity. This derivation differs from preceding ones in that it models the black hole as a grand canonical ensemble and makes use of a recently introduced quasilocal energy operator. It is shown that the statistical mechanics of the model reduces to that of a simple noninteracting gas of distinguishable particles with spin. For temperatures low in comparison with the Planck temperature and boundaries large in comparison with the Planck area, the entropy of the system is shown to be proportional to area (with a logarithmic correction), providing a simple derivation of the BekensteinHawking result (for a certain choice of the Immirzi parameter). Also in this limit, the quantum geometry on the boundary forms a "condensate" in the lowest energy level (j = 1/2). Finally, we relate our description, in terms of the grand canonical ensemble, to previous geometric entropy calculations, which made use of area ensembles.


Michael Seifert (Swarthmore '01)
Angle and Volume Studies in Quantized Space Class. Quant. Grav. 19 (2002) 22112227 
"Spatial observables" such as area, volume, and angle are given by the eigenvalues of Hermitian operators on spin network states in loop quantum gravity. We present results obtained in our investigations of the angle and volume operators, two operators which act on the vertices of spin networks. We find that the minimum observable angle is inversely proportional to the square root of the total spin of the vertex, a fairly slow decrease to zero. We also present numerical results indicating that the angle operator can reproduce the classical angle distribution. The volume operator is significantly harder to investigate analytically; however, we present analytical and numerical results indicating that the volume of a region scales as the 3/2 power of its bounding surface, which corresponds to the classical model of space. 
Last modified 27June 2014