Lecture 2 | String Theory and M-Theory

The paragraph introduces basic calculus formulas essential for understanding string theory. It discusses the function X of Sigma, an angular variable representing half a cycle around a circle. The concept of approximating a continuous function by a discrete one is explained, using calculus to refine the approximation by adding more points. The importance of smooth differentiable functions is highlighted, and integral calculus is used to sum up function values across an interval. The preliminary ends with a focus on mathematical equations that will be repeatedly used in the discussion of string theory.

This section delves into Fourier analysis, explaining how any function defined on the interval from 0 to Pi can be expressed as a sum of sines and cosines. It emphasizes the importance of boundary conditions, specifically Dirichlet and Neumann conditions, which describe the behavior of functions at the endpoints of the interval. The Dirichlet conditions are related to fixed endpoints, such as those of a violin string, while Neumann conditions apply to functions where the derivative is zero at the endpoints. The paragraph also touches on the physical implications of these conditions in the context of string theory.

The paragraph explores the behavior of functions in gravitational fields, addressing the continuity and differentiability of these functions. It discusses the concept of discontinuities, which are generally avoided in physics due to their implication of infinite energy. The text also touches on the idea of piecewise continuous functions and the conditions under which they are acceptable. The discussion then shifts to the properties of an idealized string system, highlighting the difference between Dirichlet and Neumann boundary conditions in the context of a string's physical behavior.

This part of the script introduces the concept of harmonic oscillators and their properties, including kinetic and potential energy. It defines the frequency of an oscillator and connects it to the spring constant and mass. The text also explores the philosophical question of what constitutes a particle, addressing common misconceptions about particles being point-like. Instead, it explains that particles are defined by their unique energy and mass spectra, which are discrete and require a significant amount of energy to excite to a higher state.

The paragraph discusses the nature of quantum strings and their distinction from particles. It emphasizes that strings are not point-like and are instead made up of many smaller components. The focus is on the energy spectrum of these components, which determines whether an object behaves like a particle or a composite entity. The text also touches on the concept of scale, suggesting that if observed at a closer scale, the structure of particles made of many pieces would become evident, but the energy required to excite these components remains significant.

The text explains the concept of the infinite momentum frame, a technique used to describe systems moving at speeds close to the speed of light using non-relativistic physics. It clarifies that the infinite momentum frame is not truly infinite, but rather a limit where the momentum along a chosen axis is made much larger than any other momentum components. This allows for an exact non-relativistic description of the system's motion in the plane perpendicular to the chosen axis. The paragraph also touches on the implications of this frame for understanding the properties of strings.

This section delves into the mathematics of string theory, focusing on the properties of strings moving in two dimensions. It discusses how the motion of strings is constrained in a way that all their degrees of freedom are perpendicular to a chosen axis, leading to the idea that the motion in that direction is entirely determined by the motion in other directions. This peculiar property of strings is linked to the holographic principle, which suggests that all the information contained in a volume of space can be represented on the boundary to the same space.

The paragraph discusses the energy of a string in terms of its kinetic and potential components. It describes how the potential energy of a string can be thought of as similar to that of a wave field, satisfying a wave equation. The text also addresses the boundary conditions at the ends of a string, explaining how Newton's laws apply and how they lead to either Dirichlet or Neumann boundary conditions, which affect how waves reflect off the ends of the string.

This section focuses on the quantum mechanical aspects of strings. It outlines the process of quantizing the energy of a string by expressing it as an infinite collection of harmonic oscillators. Each oscillator corresponds to a harmonic of the string and has a distinct frequency. The paragraph explains that the energy levels of these oscillators are quantized, leading to a particle-like behavior with discrete energy gaps. The goal is to compute these energy levels to understand the masses of the objects represented by the strings.

The text discusses the process of identifying particles from the vibrational modes of strings. It explains that certain vibrational states of the string correspond to particles that are recognized in physics, while others may be considered unwanted. The paragraph also introduces the concept of open and closed strings, highlighting that open strings often behave like photons, while closed strings behave like gravitons. It concludes by emphasizing the inevitability of closed strings in string theory and their significance in the theory's predictive power.