Lecture 7 | String Theory and M-Theory

The first paragraph introduces the concept of describing an ordinary particle's trajectory within classical and relativistic physics. It explains the use of three-dimensional position 'X' as a function of time 'T' in classical mechanics and the four-dimensional position with space and time components 'X mu' as a function of proper time 'tau' in relativistic physics. The principle of least action is used to define the motion of a particle, with the action 'S' being the integral over time of the kinetic energy for a non-relativistic particle. The paragraph also touches on the transition to quantum mechanics, where the focus shifts from finding a trajectory to calculating the amplitude for a particle to move from one point to another, using the action within an exponential function.

The second paragraph delves into quantum mechanics, contrasting it with classical mechanics by focusing on the amplitude of a particle's movement rather than its trajectory. It discusses the concept of path integrals, where the amplitude is calculated by integrating over all possible trajectories, which is a complex, infinite-dimensional integral. The paragraph also introduces the idea of a propagator in relativistic physics, which calculates the amplitude for a particle to move from one point to another and can be used to consider more complex processes involving particle interactions. A trick used by physicists to deal with these complex integrals is also mentioned, which involves focusing on the classical trajectories that correspond to the straight-line motion, as these are deemed most important.

The third paragraph explains a trick used in physics to deal with the complex integrals encountered in path integrals. This trick involves changing the integration variable 'tau' to a new variable 's', which is a multiple of 'tau'. The choice of the multiplier 'alpha' is left arbitrary and is intended to be chosen later for convenience. The paragraph discusses the mathematical manipulation involved in this change of variables and the implications it has for the action within the integral. The trick is said to always work and provide the correct answer once justified, despite being as challenging to understand as the original integral.

The fourth paragraph continues the discussion on path integrals, focusing on the Wiener integral, a form of path integral used in statistical mechanics. It explains how by choosing a specific value for 'alpha', the integral can be made more manageable by suppressing contributions from trajectories that deviate significantly from the classical path. The paragraph also touches on the process of analytic continuation, a method used by physicists to deal with the complexities of path integrals by evaluating them at complex points. This technique allows for the calculation of path integrals by turning the time variable on its side and then extrapolating the results to the desired complex values.

The fifth paragraph introduces string theory, where particles are not points but one-dimensional objects—strings. It discusses the action of a string, which is described by its position 'X' as a function of two variables, 'tau' and 'sigma', parameters of the string's worldsheet. The action is defined as an integral over the worldsheet, involving derivatives of 'X' with respect to 'tau' and 'sigma'. The paragraph explains that the string's path integral involves integrating over all possible worldsheet configurations, which is a complex task due to the infinite variability of the surfaces. The paragraph concludes by noting that the action for a string can be generalized to include interactions, such as two strings joining or splitting.

The sixth paragraph explores the generalization of string interactions beyond a single string to multiple strings. It discusses how the path integral approach can be used to calculate the amplitude for strings to interact, such as two strings joining to form a third or splitting into multiple strings. The paragraph emphasizes the complexity of these calculations due to the infinite variety of possible worldsheet surfaces. It also mentions the possibility of including surfaces with holes in the path integral, which corresponds to summing over all Feynman diagrams with various topologies and loop numbers in quantum field theory.

The seventh paragraph discusses the concept of conformal invariance in the context of string theory. It explains that the action for a string, which is invariant under certain transformations of the worldsheet coordinates, allows for a variety of coordinate systems that will all yield the same physical results. The paragraph also introduces the idea of the Laplace equation in two dimensions, which is derived from the string's action and is invariant under conformal transformations. These transformations preserve angles and can map the worldsheet onto different shapes while maintaining the form of the Laplace equation.

The eighth paragraph delves deeper into conformal mappings, explaining how they can be used to transform the worldsheet into different shapes without altering the underlying physics described by the Laplace equation. It discusses the properties of conformal mappings, such as preserving angles and shapes for small enough regions. The paragraph also touches on the application of conformal mappings in areas like cartography, where they are used to preserve the shape of small features on a map. It concludes by mentioning the use of conformal mappings in string theory to describe the splitting and joining of strings, highlighting the one-parameter family of such transformations.

The ninth paragraph explains the concept of channel duality in the context of scattering amplitudes in string theory. It discusses how the integral over the time interval between the joining and splitting of strings can be translated into a one-parameter integral over the positions of the points on the worldsheet. The paragraph highlights the symmetry between the S-channel and T-channel descriptions of particle interactions, which is a consequence of conformal symmetry. It concludes by noting that this duality was initially considered a remarkable feature of string theory, as it showed that one integral could account for both the formation and decay of particles as well as the exchange of particles between them.

The final paragraph reflects on the initial perception of channel duality in string theory as a miraculous symmetry between the S-channel and T-channel. It discusses how this symmetry was later understood to arise from conformal symmetry, which allows for a more symmetric representation of the interactions. The paragraph also touches on the role of the coupling constant in these interactions and how it enters as a multiplicative factor in the integrals. It concludes with a prompt for further information, directing interested individuals to Stanford's website.