Quantum Mechanics by Numerical Simulation of Path Integral

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2017, Tutor: Federico MesciaThe Quantum Mechanics formulation of Feynman is based on the concept of path integrals, allowing to express the quantum transition between two space-time points without using the bra and ket formalism in the Hilbert space. A particular advantage of this approach is the ability to provide an intuitive representation of the classical limit of Quantum Mechanics. The practical importance of path integral formalism is being a powerful tool to solve quantum problems where the analytic solution of the Schrödinger equation is unknown. For this last type of physical systems, the path integrals can be calculated with the help of numerical integration methods suitable for implementation on a computer. Thus, they provide the development of arbitrarily accurate solutions. This is particularly important for the numerical simulation of strong interactions (QCD) which cannot be solved by a perturbative treatment. This thesis will focus on numerical techniques to calculate path integral on some physical systems of interest

New Aspects of Scattering Amplitudes, Higher-k Amplitudes, and Holographic Quark Gluon Plasmas

We present new results on different aspects of quantum field theory, which are divided into three main parts. In part I, we find and prove a new behavior of massless tree-level scattering amplitudes, including the biadjoint scalar theory, the U(N) non-linear sigma model, and the special Galileon, within specific subspaces of the kinematic space. We also derive new formulas for the double-ordered biadjoint scalar and $\phi^p$ amplitudes, which can be obtained as integrals over the positive tropical Grassmannian and under limiting procedures on the kinematic invariants. This reveals surprising connections with cubic amplitudes. We also present alternative versions of the formulas for $\phi^p$ amplitudes from combinatorial considerations in terms of non-crossing chord diagrams. In part II, we investigate the generalization of quantum field theory introduced by Cachazo, Early, Guevara and Mizera (CEGM) in 2019. We use soft limits to determine the number of singular solutions of the generalized scattering equations in certain cases and propose a general classification of all configurations that can support singular solutions. We also describe the generalized Feynman diagrams that compute CEGM amplitudes. These are planar arrays of Feynman diagrams satisfying certain compatibility conditions, and we propose combinatorial bootstrap methods to obtain them. Finally, in part III, we analyze different types of quark gluon plasmas in the presence of a background magnetic field using top-down holographic models. We explore conformal and nonconformal theories as consistent truncations of ${\cal N}=8$ gauged supergravity and identify a universal behavior in the ${\cal N}=2^*$ gauge theory