Lecture 8 | String Theory and M-Theory

The first paragraph introduces the concept of conformal mappings and their significance in mathematical physics, particularly in two-dimensional electrostatic problems. It explains that in a hypothetical two-dimensional world, the Coulomb force would be inversely proportional to R (one over R) instead of R squared, leading to different electrostatic equations. The paragraph also discusses the electrostatic potential, its relation to the electric field, and the equation for the electric field in electrostatics. It concludes by touching on the invariance of these equations under conformal transformations.

The second paragraph delves into the applications of conformal mappings, especially in the context of cartography and fluid flow. It highlights that conformal mappings preserve angles but not necessarily sizes, which is why they are useful for creating maps. The paragraph also mentions that Laplace's equation, which describes fluid flow, is related to electrostatic equations and that the curl in two dimensions is a scalar quantity. It concludes by emphasizing the utility of conformal mappings in deriving multiple solutions to electrostatic problems from a single solution through coordinate transformations.

The third paragraph explores the concept of complex functions and their derivatives. It discusses the conditions for a function to have a unique derivative, which is dependent on the Cauchy-Riemann equations. The paragraph explains that for a function to be analytic (i.e., have a unique derivative from every direction), the real and imaginary parts must satisfy these coupled equations. The paragraph concludes by noting that the existence of a unique derivative is a necessary and sufficient condition for a function to be analytic.

The fourth paragraph establishes the connection between analytic functions and Laplace's equation. It states that an analytic function implies two real solutions to Laplace's equation. The paragraph also discusses the properties of analytic functions, such as the fact that the sum and ratio of two analytic functions are also analytic, with the caveat that the denominator must not vanish. It concludes by emphasizing that most simple functions one might encounter are analytic.

The fifth paragraph focuses on proving that conformal mappings are indeed conformal, meaning they preserve angles. It uses the properties of complex numbers and their representation in polar coordinates to demonstrate that the ratio of two small displacements in the complex plane is equivalent to the ratio of their images under a conformal mapping. The paragraph concludes by showing that the angles between curves are preserved, which is the defining characteristic of a conformal mapping.

The sixth paragraph provides examples of conformal mappings, including the mapping of a half-plane to a strip and the transformation of the entire plane into a cylinder. It discusses how these mappings can be used to describe string theory concepts, such as the worldsheet of a string. The paragraph also explores the implications of these mappings for the Laplace equation and the importance of conformal mappings in string theory due to their invariance properties.

The seventh paragraph examines a specific conformal mapping from the complex plane to a unit disk. It discusses how different elements of the complex plane, such as radial lines and circles, are transformed under this mapping. The paragraph also touches on the utility of this mapping in solving Laplace's equation in various contexts and how it can be used to transform and solve a multitude of electrostatics problems.

The eighth paragraph connects the concepts of conformal mappings to string theory. It describes how the worldsheet of a string can be represented on different geometries, such as a half-plane or a disk, using conformal mappings. The paragraph also explains how the string worldsheet can be thought of as an infinite cylinder, which is useful for describing an open string with two ends. It concludes by noting the mathematical fact that allows for the description of closed strings on the whole plane without boundaries.

The ninth paragraph draws a connection between electrostatics problems and the calculation of scattering amplitudes in string theory. It explains that the scattering amplitude for particles in string theory can be thought of as solving an electrostatics problem with multiple types of charges. The paragraph details how the components of the momenta of the particles act as charges in this electrostatics problem and how the integration over the positions of these 'charges' gives the scattering amplitude. It concludes by noting that the final result is an exponentiation of the electrostatic energy.

The tenth paragraph discusses the mathematical underpinnings of string theory, emphasizing the simplicity of the theory at its core, which is likened to solving an electrostatics problem. It explains that the theory involves integrating over the positions of particles, which are injected at the boundaries of the string worldsheet. The paragraph also touches on the concept of compactification, which is a complex process that deals with the extra dimensions in string theory. It concludes by acknowledging the current state of string theory as a work in progress with many versions and complexities yet to be fully understood or experimentally verified.

The eleventh paragraph continues the discussion on the mathematical aspects of string theory, focusing on the integration process and the concept of compactification. It explains how, through conformal invariance, one can integrate over the positions of the particles and how this process is related to the number of dimensions in the theory. The paragraph also mentions the challenges in compactification and its implications for the number of possible models in string theory. It concludes by stating the vast number of potential theories that can arise from string theory and the ongoing efforts to find a model that aligns with experimental results.

The twelfth paragraph addresses the theoretical and experimental challenges faced by string theory. It acknowledges the discrepancy in the number of dimensions predicted by string theory compared to the three dimensions observed in the physical world. The paragraph discusses the importance of compactification in reconciling this difference and the mathematical complexity involved in doing so. It concludes by emphasizing the current theoretical nature of string theory and the significant work that remains to be done before it can be considered a complete and experimentally verified model of particle physics.