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Published in:
A Quick Introduction to Tensors Read chapter Read first chapter

2015 | OriginalPaper | Chapter

6. The Representation Operator and Its Applications

Author : Nadir Jeevanjee

Published in: An Introduction to Tensors and Group Theory for Physicists

Publisher: Springer International Publishing

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Abstract

This chapter applies the material of the previous chapters to some particular topics, specifically the Wigner–Eckart theorem, selection rules, and gamma matrices and Dirac bilinears. We begin by discussing the perennially confusing concepts of vector operators and spherical tensors, and use these to give a quick overview of the Wigner–Eckart theorem and selection rules. These latter subjects are then made precise using the notion of a representation operator. We conclude by showing that Dirac’s famous gamma matrices can be understood in terms of representation operators, which then immediately gives the transformation properties of the “Dirac bilinears” of QED.

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previous chapter Basic Representation Theory
Footnotes
1
Note that the ‘l’ in the superscript of T m (l) is in parentheses; this is because the l isn’t really an active index, but just serves to remind us which SO(3) representation we’re dealing with.
 
2
These are known as Wigner functions or Wigner D-matrices.
 
3
Also, note the analogy between (6.21) and (6.14); in fact, the Pauli matrices can be thought of as a lower-dimensional analog of the gamma matrices, and it’s no coincidence that the gamma matrices γ i in (6.19) are built out of the Pauli matrices! For more on this see Frankel [6].
 
Literature
6.
T. Frankel, The Geometry of Physics, 1st edn. (Cambridge University Press, Cambridge, 1997)
17.
J.J. Sakurai, Modern Quantum Mechanics, 2nd edn. (Addison Wesley Longman, Reading, MA, 1994)
Metadata
Title
The Representation Operator and Its Applications
Author
Nadir Jeevanjee
Copyright Year
2015
Book
An Introduction to Tensors and Group Theory for Physicists

Print ISBN: 978-3-319-14793-2

Electronic ISBN: 978-3-319-14794-9

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-14794-9_6

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