\[
f = \frac1{2L}\sqrt{\frac T\mu}
\]
\[
\newcommand{\normord}[1]{:\mathrel{#1}:}
\def\ff#1#2{{\textstyle\frac{#1}{#2}}}
\begin{split}
Q &= \sum_n\left(L^{(\alpha)}_{-n} + L^{(a)}_{-n}\right)c_n
+ \sum_r G^{(\alpha a)}_{-r}\gamma_r
- \ff12\sum_{m,n}(m-n)\normord{c_{-m}c_{-n}b_{m+n}}\\
&\hspace{10mm}+
\sum_{m,r}\left(\ff32m+r\right)\normord{c_{-m}\beta_{-r}\gamma_{m+r}}
- \sum_{r,s}\gamma_{-r}\gamma_{-s}b_{r+s} - ac_0,
\end{split}
\]
Quantum Mechanics and String Theory
(Math 590-2)
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 2-torus 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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