Research
Art Credit: BLEACH Volume 70
Art Credit: BLEACH Volume 70
My research interests lie broadly in theoretical physics. While my past research experiences predominantly fall under the labels of high energy theory and quantum gravity, I am interested in condensed matter theory, quantum information science, and other facets of theoretical physics.
Below is a description of some topics I am interested in and what I have worked on in these areas. More information on most of my projects mentioned below can be found here.
It is well known that the unification of gravity and quantum mechanics has been difficult to accomplish. There are theories that manage to do this, but there is the additional challenge of determining the one that applies to our universe - especially given its cosmological behavior. Two candidate theories of quantum gravity are holographic spacetime (HST) and loop quantum cosmology (LQC).
HST attempts to address issues with string theory: it is difficult to apply to universes with a positive cosmological constant or without supersymmetry. HST takes a more general approach than string theory using the strong holographic principle to allow for these properties, since our universe possesses them. This theory has been successful in constructing a model of cosmology that recovers the flatness, homogeneity, and isotropy observed today. In addition, there exist primordial black holes in its post-inflation stages that are candidates for dark matter. One of my projects involved determining if primordial black holes in the early matter-dominated universe - a prediction of HST - is consistent with data observed today. That is, if such objects coalesced, their radiation would be visible in the Cosmic Microwave Background today. Professor Tom Banks and I found using an N-Body simulation that they do not merge.
LQC uses the techniques of loop quantum gravity and is a non-perturbative canonical quantization of homogenous spacetime. In this theory, the area of holonomies used to construct loops have a finite minimum, meaning loops cannot be arbitrarily small. Such a property lends itself to LQC in the form of a quantum difference equation that captures the dynamics of spacetime. Numerical simulations have shown that this equation leads to a generic resolution of singularities. In addition, a Friedmann Equation in the isotropic model of LQC, dubbed the Modified Friedmann Equation, has explicitly shown that the Hubble factor is bounded, meaning the Big Bang singularity is avoided. There has also been considerable work put into studying anisotropic models of LQC. With Professor Singh, I derived the Modified Friedmann Equations for the Bianchi-I LRS (Locally Rotation Symmetric) model of LQC, where only two of the three spatial dimensions are identical. As part of my undergraduate thesis, I also explained how to find such equations for a setting with three, independent spatial dimensions.
Quantum Field Theory (QFT) has been one of the most successful frameworks in physics. It has been used to model the behavior of particles in the standard model and many phenomena in other contexts like condensed matter systems and statistical physics. Given that QFT utilizes quantum mechanics and special relativity, it is prevalent in nearly all aspects of theoretical physics.
The Hubbard model is a system in QFT that is particularly exciting. This model is a lattice model that is particularly general, but it has not been analytically solved due to mathematical complexity. Thus, it has been probed with numerical methods extensively. Many actively studied problems, such as the homogenous electron gas cloud, are connected to this model, so the Hubbard model has wide-ranging applications. A project I am currently working on with Professor Banks is regarding his proposed approximation in this model when the number of fermions is large. His idea turns a set of infinite, implicit equations into a set of finite, explicit equations. This allows one to solve for the Legendre Transform (and its 2nd order derivative terms) of the generating functional.
Matrix models are quantum gauge theories in 0 + 1 dimension that are useful in many areas of physics. In these models, fundamental parameters like position and momentum are N x N Hermitian matrices rather than scalars. Observables and other interesting quantities can be found through integrating over these matrices. A famous example of a matrix model is the BFSS model, which sheds insight into M-theory when probed in the right limit. My project with Professor Mukund Rangamani is with the closely related D0-Brane matrix model. We are looking to apply the numerical technique known as bootstrap to this model to find information on the quasinormal mode frequencies of its holographic dual.
Mathematics has always played a pivotal role in physics. Even subfields like number theory and statistics, which I used to study the Ducci's Four-Number Game with Professors Domonic Klyve and Sooie-Hoe Loke, are necessary to reach today's level of understanding of the universe. There has also been a revitalization in how areas such as discrete mathematics are utilized in physics. This is exciting for the future of physics.
With advancements in computing, people have begun constructing physics through discrete structures. An example of this is the Wolfram Physics Project, in which the universe is structured as an algorithmic model that evolves via rules. In fact, a project I did with Professor Xerxes Arsiwalla as part of the Wolfram Physics School involved reproducing features of a theory of quantum gravity known as causal set theory using rewriting rules of causal graphs in Mathematica. Work done through the Wolfram Physics Project may shed light on how different theories of quantum gravity are connected from a computational perspective. This approach to physics is less conventional, but it showcases the many ways mathematics can influence physics.
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