# Cynthia Yan

Harvey Mudd College Mathematics 2017

Thesis: | Mathematics of Emergent Gravity Based on Quantum Entanglement |
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Thesis Advisor: | Prof. Weiqing Gu |

Second Reader: | Prof. Vatche Sahakian |

E-Mail: | xyan@g.hmc.edu |

## Abstract

Quantum Mechanics suggests that gravity is some long distance approximation of a quantum observable of some collective degrees of freedom. There are many ways proposed to resolve gravity at the microscopic scale. In particular, there is a notation that gravity comes from Von Neumann entanglement entropy. In this thesis, we focus on understanding the mathematics behind this proposal, i.e., the mathematical connection between gravity and quantum entanglement entropy in Matrix theory. As a physical example, we focus on a relatively simple model made up of a noncommutative spherical membrane in light-cone M theory, stabilized by an external force with two massless probes far away from the source. We compute the Lagrangian of this system under the BFSS theory which is a dimensional reduction of 10d super Yang-Mills. However, there is no tidal force extracted between the two probes, which suggests a mismatch between the trace of gravity and the entanglement entropy. From our analysis of this example, we gain a deeper understanding of the problem of emergent gravity from a mathematical perspective.