## Physics & Astronomy ETDs

#### Publication Date

Fall 12-2020

#### Abstract

Time derivatives of scalar fields occur quadratically in textbook actions. A simple Legendre transformation turns the Lagrangian into a Hamiltonian that is quadratic in the momentum. The partition function over the momentum is Gaussian. Mean values of operators are basically euclidian path integrals of their classical counterparts with positive weight functions. Monte Carlo simulations can estimate such mean values. This familiar framework falls apart when the time derivatives do not occur quadratically. The Legendre transformation becomes difficult or so intractable that the Hamiltonian can’t be determined. Even if the Hamiltonian is found, it usually is so complicated that the partition function can’t be integrated over the momentum to get a euclidian action with a positive weight function. Monte Carlo simulations don’t work when the weight functions have negative or complex values. This work solves both problems. It shows how to make the partition function without knowing the Hamiltonian. It also shows how to estimate complex weight functions by combining the Monte Carlo method with numerical integration and a lookup table. The “Atlantic City” Method or AC Method estimates the energy densities of theories that, unlike those with quadratic time derivatives, might have finite energy densities. The approximation of multiple integrals over weight functions that are negative or complex is the long-standing sign problem. The AC Method solves it for problems in which numerical integration leads to a positive weight function.

#### Degree Name

Physics

#### Level of Degree

Doctoral

#### Department Name

Physics & Astronomy

#### First Committee Member (Chair)

Dr. Kevin Cahill

#### Second Committee Member

Dr. Rouzbeh Allahverdi

#### Third Committee Member

Dr. Huaiyu Duan

#### Fourth Committee Member

Dr. James Degnan

#### Language

English

#### Keywords

Monte Carlo, Scalar Fields, Born-Infeld, Quantum Field Theory

#### Document Type

Dissertation

#### Recommended Citation

Amdahl, David John. "Monte Carlo Simulations of Awkward Actions." (2020). https://digitalrepository.unm.edu/phyc_etds/258