An Invitation to Quantum Field Theory
This book provides an introduction to Quantum Field Theory (QFT) at an elementary level—with only special relativity, electromagnetism and quantum mechanics as prerequisites. For this fresh approach to teaching QFT, based on numerous lectures and courses
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For the benefit of the reader we summarize in this Appendix the main conventions used throughout the book.
A.1 Covariant Notation We have used the ‘‘mostly minus’’ metric 0 1 0 0 B 0 1 0 glm ¼ B @ 0 0 1 0 0 0
1 0 0 C C: 0 A 1
ðA:1Þ
Derivatives with respect to the four-vector xl ¼ ðct; xÞ are denoted by the shorthand o 1o ;r : ol l ¼ ox c ot Sporadically we have used the notation $
f ðxÞ o l gðxÞ ¼ f ðxÞol gðxÞ ol f ðxÞgðxÞ:
ðA:2Þ
As usual space-time indices will be labelled by Greek letters (l; m; . . . ¼ 0; 1; 2; 3) while Latin indices will be used for spatial directions (i; j; . . . ¼ 1; 2; 3). We reserved a, b for Dirac and a; b; c; . . . for Weyl spinor indices. The electromagnetic four-vector potential Al is defined in terms of the scalar u and vector potential A by Al ¼ ðu; AÞ:
L. Ávarez-Gaumé and M.Á. Vázquez-Mozo, An Invitation to Quantum Field Theory, Lecture Notes in Physics 839, DOI: 10.1007/978-3-642-23728-7, Springer-Verlag Berlin Heidelberg 2012
ðA:3Þ
275
276
Appendix A: Notation, Conventions and Units
e lm ¼ The components of the field strength tensor Flm ¼ ol Am om Al and its dual F 1 rk are given respectively by 2 elmrk F 0 1 0 1 0 Ex E y Ez 0 Bx By Bz B E B Bx 0 0 Bz By C e Ez Ey C Flm ¼ @ Ex B 0 Bx A; F lm ¼ @ By Ez 0 Ex A; ðA:4Þ y z Ez By Bx 0 Bz Ey Ex 0 with E ¼ ðEx ; Ey ; Ez Þ and B ¼ ðBx ; By ; Bz Þ the electric and magnetic fields. Similar expressions are valid in the nonabelian case.
A.2 Pauli and Dirac Matrices We have used the notation rl ¼ ð1; ri Þ where ri are the Pauli matrices 0 1 0 i 1 0 ; r2 ¼ ; r3 ¼ : ðA:5Þ r1 ¼ 1 0 i 0 0 1 They satisfy the identity ri rj ¼ dij 1 þ eijk rk ;
ðA:6Þ
from where their commutator and anticommutator can be easily obtained. Dirac matrices have always been used in the chiral representation 0 rl l c ¼ : ðA:7Þ rlþ 0 The chirality matrix is normalized as c25 ¼ 1 and defined by c5 ¼ ic0 c1 c2 c3 . In many places we have used the Feynman’s slash notation a= ¼ cl al .
A.3 Units Unless stated otherwise, we work in natural units h ¼ c ¼ 1. Electromagnetic Heaviside-Lorentz units have been used, where the Coulomb and Ampère laws take the form F¼
1 qq0 r; 4p r 3
dF 1 II 0 ¼ : d‘ 2pc2 d
ðA:8Þ
In these units the fine structure constant is a¼
e2 : 4p hc
ðA:9Þ
The electron charge in natural units is dimensionless and equal to e 0:303.
Appendix B A Crash Course in Group Theory
Group theory is one of the most useful mathematical tools in Physics in general and in quantum field theory in particular. To make the presentation self-contained we summarize in this Appendix some basic facts about group theory. Here we limit ourselves to the statement of basic results. Proofs and more detailed discussions can be found in the many books on the subject, such as the ones listed in Ref. [1, 2, 3, 4].
B.1 Generalities Physical transformations have a number of interesting properties. To have an intuitive example in mind let us think of rotations in three-dimensional space. These transformations have inter
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