I have written up some notes which may be helpful as a background
in mathematical techniques, and some that give more technical details
of things presented in the lecture notes without all the details.
These are notes on mathmatical techniques which should be part of your
background. Some are not directly relevant to this course.
- Notes on using indices correctly: “On Indices and
(view) or (print).
Some cautionary notes on how indices are used and how to avoid
making nonsense when evaluating expressions with dummy indices.
This seems trivial, but I have seen many students have difficulties.
- Using ε's and determinants
- εijk and cross products
in 3-D Euclidean space:
Notes on totally antisymmetric tensors, or Levi-Civita symbols,
- εμνρ... in higher dimensional
Euclidean or Minkowski space:
including their use with matrices and determinants. The
Levi-Civita symbol is also essential in curved spaces, but that is
for another course.
Also, on ε and determinants, "Properties of Determinants":
- The gradient operator view
- On differential forms. If you have not seen differential forms
before, you might want to look at
my notes from 507, pages 153-4 on 1-forms and pp 167-175 on
higher k-forms, or
They are discussed in advanced calculus texts, e.g.
Buck, "Advanced Calculus".
- Vector Identities from cover of
Jackson, or all on one
- The beta function B(x,y) and Γ(ε) for ε ≅ 0:
- Note on the Surface "Area" of a D-dimensional ball, and on the Euler
Γ function: “Γ(N/2) and the Volume
of SD-1”: (view or
- Notes on Bessel functions (view or
print), giving in particular the
- Notes on Lie Algebras and Groups:
You really ought to take a course on group theory, but if
you haven't, here are
- “ Lightning review of groups”:
Their definition and representations, the connection of continuous
groups to Lie algebras, Killing forms and Casimir operators.
- “Note on Representations, the Adjoint rep, the Killing
form, and antisymmetry of cijk”:
Here are some notes that go a bit beyond what we discussed in class.
- Extra note on Noether's Theorem:
- Stress-Energy tensor for Maxwell Theory:
- Schwinger trick and Feynman Parameters:
- A “Little Note on Fierz”:
Here are some notes on more advanced topics in QFT
Here are some notes on subjects other than field theory,
but which might be helpful.
- Group Invariant Metric,
a metric gμν
on the group with the property that
lengths are preserved under multiplication by a common group element.
As a consequence g1/2 is the Haar measure.
This is only for those interested, and is not of any
direct relevance to this course.
- Comment on Integrals over Grassman
Last modified: Thu Jul 25 13:38:04 2013