18.238 | Fall 2002 | Undergraduate

Geometry and Quantum Field Theory

Lecture Notes

The lecture notes are part of a book in progress by Professor Etingof. Please refer to the calendar section for reading assignments for this course. 

Chapter 1: Generalities on Quantum Field Theory (PDF)

1.1 Classical Mechanics   
1.2 Classical Field Theory   
1.3 Brownian Motion   
1.4 Quantum Mechanics   
1.5 Quantum Field Theory

Chapter 2: The Steepest Descent and Stationary Phase Formulas (PDF)

2.1 The Steepest Descent Formula   
2.2 Stationary Phase Formula   
2.3 Non-analyticity of I(h) and Borel Summation   
2.4 Application of Steepest Descent   
2.5 Multidimensional Versions of Steepest Descent and Stationary Phase

Chapter 3: Feynman Calculus (PDF)

3.1 Wick’s Theorem   
3.2 Feynman’s Diagrams and Feynman’s Theorem   
3.3 Another Version of Feynman’s Theorem   
3.4 Proof of Feynman’s Theorem   
3.5 Sum Over Connected Diagrams   
3.6 Loop Expansion   
3.7 Nonlinear Equations and Trees   
3.8 Counting Trees and Cayley’s Theorem   
3.9 Counting Trees with Conditions   
3.10 Counting Oriented Trees   
3.11 1-particle Irreducible Diagrams and the Effective Action   
3.12 1-particle Irreducible Graphs and the Legendre Transform

Chapter 4: Matrix Integrals (PDF)

4.1 Fat Graphs   
4.2 Matrix Integrals in Large N limit and Planar Graphs   
4.3 Integration Over Real Symmetric Matrices   
4.4 Application to a Counting Problem   
4.5 Hermite Polynomials   
4.6 Proof of Theorem 4.7

Chapter 5: The Euler Characteristic of the Moduli Space of Curves (PDF)

5.1 Euler Characteristics of Groups   
5.2 The Mapping Class Group   
5.3 Construction of the Complex Y

5.4 Enumeration of Cells in Y1g

5.5 Computation of ∑n(−1)n−1g(n)/2n)

Chapter 6: Matrix Integrals and Counting Planar Diagrams (PDF)

6.1 The Number of Planar Gluings   
6.2 Proof of Theorem 6.1

Chapter 7: Quantum Mechanics (PDF)

7.1 The Path Integral in Quantum Mechanics   
7.2 Wick Rotation   
7.3 Definition of Euclidean Correlation Functions   
7.4 Connected Green’s Functions   
7.5 The Clustering Property   
7.6 The Partition Function   
7.7 1-particle Irreducible Green’s Functions   
7.8 Momentum Space Integration   
7.9 The Wick Rotation in Momentum Space   
7.10 Quantum Mechanics on the Circle   
7.11 The Massless Case   
7.12 Circle Valued Quantum Mechanics   
7.13 Massless Quantum Mechanics on the Circle

Chapter 8: Operator Approach to Quantum Mechanics (PDF)

8.1 Hamilton’s Equations in Classical Mechanics   
8.2 Hamiltonians in Quantum Mechanics   
8.3 Feynman-Kac Formula   
8.4 Proof of the Feynman-Kac Formula in the Free Case   
8.5 Proof of the Feynman-Kac Formula (General Case)   
8.6 The Massless Case

Chapter 9: Fermionic Integrals (PDF)

9.1 Bosons and Fermions   
9.2 Supervector Spaces   
9.3 Supermanifolds   
9.4 Supermanifolds and Vector Bundles   
9.5 Integration on Superdomains   
9.6 The Berezinian of a Matrix   
9.7 Berezin’s Change of Variable Formula   
9.8 Integration on Supermanifolds   
9.9 Gaussian Integrals in an Odd Space   
9.10 The Wick Formula in the Odd Case

Chapter 10: Quantum Mechanics for Fermions (PDF)

10.1 Feynman Calculus in the Supercase   
10.2 Fermionic Quantum Mechanics   
10.3 Super Hilbert Spaces   
10.4 The Hamiltonian Setting for Fermionic Quantum Mechanics

Chapter 11: Free Field Theories in Higher Dimensions (PDF)

11.1 Minkowski and Euclidean Space   
11.2 Free Scalar Bosons   
11.3 Spinors   
11.4 Fermionic Lagrangians   
11.5 Free Fermions

Course Info

As Taught In
Fall 2002
Learning Resource Types
Lecture Notes
Presentation Assignments