《QUANTUMFIELDTHEORY》青少年教育丛书.pdf

QUANTUM FIELD THEORY
Notes taken from a course of R. E. Borcherds, Fall 2001, Berkeley
Richard E. Borcherds,
Mathematics department, Evans Hall, UC Berkeley, CA 94720, .
e–mail: ******@
home page: /~reb
Alex Barnard,
Mathematics department, Evans Hall, UC Berkeley, CA 94720, .
e–mail: ******@
home page: /~barnard
Contents
1 Introduction 2
Life Cycle of a Theoretical Physicist . . . . . . . . . . . . . . . . . . . . . . . 2
Historical Survey of the Standard Model . . . . . . . . . . . . . . . . . . . . . 3
Some Problems with Neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Elementary Particles in the Standard Model . . . . . . . . . . . . . . . . . . . 8
2 Lagrangians 9
What is a Lagrangian? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
The General Euler–Lagrange Equation . . . . . . . . . . . . . . . . . . . . . . 11
3 Symmetries and Currents 12
Obvious Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Not–So–Obvious Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
The ic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1
Converting Classical Field Theory to Homological Algebra . . . . . . . . . . . 17
4 Feynman Path Integrals 19
Finite Dimensional Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
The Free Field Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Free Field Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
The Non-Free Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 0-Dimensional QFT 23
Borel Summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Other Graph Sums . . . . . . . . . . .