Search Rutgers Search Physics Physics 615 Fall 2013 Supplementary Notes

 Different books use different conventions, and sometimes even my notes may differ. Here is a possibly useful table. Here are some notes expanding on explicit pages of the text book: Energy in Bremsstrahlung, Note on p. 179 (view) or (print). On Evaluating I(v,v') . Kinematics of p. 218 .
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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 Arguments”: (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, (view), (print)
• εμνρ... in higher dimensional Euclidean or Minkowski space: (view), (print), 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": (view), (print)
• Differentials
• The gradient operator view or print.
• 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 Zapolsky's notes. They are discussed in advanced calculus texts, e.g. Buck, "Advanced Calculus".
• Vector Identities from cover of Jackson, or all on one sheet version.
• The beta function B(x,y) and Γ(ε) for ε ≅ 0: view or print.
• 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 print).
• Notes on Bessel functions (view or print), giving in particular the orthonormality properties.
• 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”: (view or print). 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”: (view or print).

## Here are some notes that go a bit beyond what we discussed in class.

• Extra note on Noether's Theorem: view or print.
• Stress-Energy tensor for Maxwell Theory: view or print.
• Schwinger trick and Feynman Parameters: view or print.
• A “Little Note on Fierz”: view or print.

## Here are some notes on more advanced topics in QFT

Topic One Column Two Columns
Path Integral Formulation of Quantum Mechanics view print
Functional Integral for a Scalar Field view print
Gauge Theory on a Lattice view print
A short note on Fadeev-Popov Ghosts view or print
some comments on BRST cohomology view or print
Generating Functions view print

### Supplementary Notes

Here are some notes on subjects other than field theory, but which might be helpful.
• Group Invariant Metric, which discusses 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 Variables (view, print)

Joel Shapiro (shapiro@physics.rutgers.edu)