Supersymmetry & Grand Unification: Lecture 1

The paragraph introduces the concept of renormalization in quantum field theory. It explains that renormalization involves eliminating irrelevant small distances and applying dimensional analysis to solve complex problems in quantum field theory. The speaker uses the analogy of a nucleus being made of quarks but for many purposes, it's more useful to consider it as a collection of protons and neutrons. The paragraph also touches on the idea of solving quantum chromodynamics to compute the properties of protons and neutrons.

This paragraph delves into the application of renormalization in atomic physics. It discusses how atomic properties can be calculated using nuclear physics and that the mass and charge of the nucleus are key to atomic physics. The speaker emphasizes that once these properties are known, the details of protons and neutrons can be disregarded. The paragraph also illustrates the process of renormalization by comparing it to moving from a description of atoms to a more coarse-grained description that is more useful for certain purposes.

The speaker outlines the quantum mechanical approach to the problem at hand, starting with the Hamiltonian, which represents the energy of the system. The paragraph discusses the non-relativistic quantum mechanics and the kinetic energy of the nuclei. It also covers the Coulomb force between the nuclei and the electrons' kinetic energy. The speaker then moves on to discuss the forces between the electrons and the force between the protons and electrons, emphasizing the slow movement of the nuclei compared to the electrons.

The focus of this paragraph is on solving the Schrodinger equation to find the ground state energy of the electrons in a fixed background of nuclei. The speaker describes the process of fixing the positions of the nuclei and solving for the lowest energy state, which allows for the elimination of the electrons from the problem. The resulting Hamiltonian is then a function of the positions of the nuclei only, leading to an effective potential energy that includes the influence of the electrons.

The paragraph provides a break from renormalization theory to focus on dimensional analysis in physics. It emphasizes the importance of specifying one dimension in particle physics, which can be mass, energy, momentum, length, or time. The speaker also discusses how mass and energy are equivalent in units due to the famous equation E=mc^2, and how setting h-bar and c equal to one simplifies the problem, leaving only one dimensional quantity to specify.

This paragraph introduces a scalar quantum field theory with a single scalar field Phi, and discusses the Lagrangian from which Fineman diagrams are derived. The speaker explains the components of the Lagrangian, including the kinetic term and potential energy. The paragraph also covers the concept of action in quantum field theory and the importance of the Lagrangian density, which must have inverse length to the fourth power.

The focus is on the significance of dimensionless coupling constants in renormalization theory. The speaker discusses the units of various terms in the Lagrangian and how they relate to mass and energy. The paragraph also explains the structure of Fineman diagrams, which are built from vertices and propagators, and how these contribute to the amplitude of a process in quantum field theory.

The paragraph explores the renormalization of mass within scalar field theory. It explains that the mass term in the Lagrangian can be affected by quantum fluctuations, leading to a need for renormalization. The speaker uses the concept of a cutoff scale to ignore small-scale physics and shows how the original mass term can be combined with quantum corrections to form an effective mass term, which is measurable in experiments.

This paragraph discusses the appearance of an infinite series of terms in the process of renormalization, each contributing to the mass correction with powers of the coupling constant over the cutoff scale. The speaker explains that while higher-order diagrams may contribute less, they still must be considered in the renormalization process. The paragraph also touches on the concept of dimensional analysis in evaluating these contributions.

The speaker addresses the renormalization of the coupling constant lambda and the concept of fine-tuning. It is explained that every term in the Lagrangian can be renormalized, and the effective parameters measured in experiments account for all the short-distance physics. The paragraph also highlights the fine-tuning problem associated with the Higgs boson mass and the hierarchy problem in particle physics.

The paragraph contrasts the behavior of fermions and gauge bosons with scalar particles in terms of renormalization. It is shown that fermions and gauge bosons, if initially massless, remain massless even after renormalization. The speaker explains that the fine-tuning problem is primarily associated with the Higgs boson and that supersymmetry might offer a potential solution to this problem.

The final paragraph discusses the vacuum energy and its role in the fine-tuning problem. It is explained that while vacuum energy is not significant in most physics, it becomes important when considering gravity, as it contributes to the gravitational field. The speaker highlights the need for fine-tuning the vacuum energy to an extremely high precision and mentions the cosmological constant problem, which is related to dark energy.