\[
f = \frac1{2L}\sqrt{\frac T\mu}
\]
\[
\newcommand{\normord}[1]{:\mathrel{#1}:}
\def\ff#1#2{{\textstyle\frac{#1}{#2}}}
\begin{split}
Q &= \sum_n\left(L^{(\alpha)}_{n} + L^{(a)}_{n}\right)c_n
+ \sum_r G^{(\alpha a)}_{r}\gamma_r
 \ff12\sum_{m,n}(mn)\normord{c_{m}c_{n}b_{m+n}}\\
&\hspace{10mm}+
\sum_{m,r}\left(\ff32m+r\right)\normord{c_{m}\beta_{r}\gamma_{m+r}}
 \sum_{r,s}\gamma_{r}\gamma_{s}b_{r+s}  ac_0,
\end{split}
\]
Quantum Mechanics and String Theory (Math 5902)
Fall 2022
Instructor:
Paul Aspinwall
Credits: 1.00, Hours: 03.0
Time: MW 1:45PM  3:00PM
Location: Physics 119
Requirements
Office Hours
 2:00 to 3:00pm each Tuesday
 10:00 to 11:00am each Friday
Prerequisits
Math 212/219/222 and Math 221, or consent from me.
Homework
is weekly. See gradescope in Sakai.
Exams
There will be a final exam.
Notes
 See "Resources" in Sakai.
Synopsis
A rough outline is as follows
 Classical Mechanics (a quick review)
 Newtonian
 Lagrangian
 Hamiltonian
 A String made of particles
 Classical analysis of vibrations of a violin string
 Open and Closed Strings
 Quantum Mechanics
 Hilbert spaces and operators
 Canonical quantization
 Quantization of the vibrating string
 Symmetries in Quantum mechanics
 Groups and Lie groups
 Projective representations and central extensions
 Wigners Theorem
 Gauge symmetries
 Cohomology and BRST quantization
 Quantization of a fundamental string
 The Virasoro algebra
 BRST quantization and the criticial dimension
 The string spectrum
 Supersymmetry
 The superstring
 Critical dimension and the GSO projection
 Modular Invariance
 Bosons
 Fermions
 A circle
 Tori
 The Heterotic String
 The 2torus and Mirror Symmetry
Textbooks
This course will not be based on a textbook as there is none which is suitable.
Two main references (at a more advanced level) are
 M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory,
(Volumes 1 and 2), Cambridge 1987
 J. Polchinski, String Theory, (Volumes 1 and 2), Cambridge 1998.
There is also a lower level book, but this takes a quite different
route than this course:
 B. Zweibach, A First Course in String Theory, Cambridge 2009.
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