Basic Representation Theory

  • Nadir Jeevanjee


Chapter 5 discusses representation theory, which formalizes the notion of an object that transforms in a certain way under a given transformation (e.g. vectors under rotations, or antisymmetric tensors under boosts). We begin by defining a representation of a group as a vector space on which that group acts, and we give many examples, using the vector spaces we met in Chap.  2 and the groups we met in Chap.  4. We then discuss how to take tensor products of representations, and we see how this reproduces the additivity of quantum numbers in Quantum Mechanics. We then define irreducible representations, which are in a sense the ‘smallest’ ones we can work with, and we compute these representations for SU(2). These just end up being the familiar spin j representations, where j is a half-integer. We then use these results to compute the irreducible representations of the Lorentz group as well.


Tensor Product Irreducible Representation Invariant Subspace Double Cover Equivalent Representation 
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  1. 4.
    T. Frankel, The Geometry of Physics, 1st ed., Cambridge University Press, Cambridge, 1997 Google Scholar
  2. 6.
    H. Goldstein, Classical Mechanics, 2nd ed., Addison-Wesley, Reading, 1980 Google Scholar
  3. 7.
    M. Göckeler and T. Schücker, Differential Geometry, Gauge Theories, and Gravity, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1987 Google Scholar
  4. 8.
    B. Hall, Lie Groups, Lie Algebras and Representations: An Elementary Introduction, Springer, Berlin, 2003 Google Scholar
  5. 9.
    I. Herstein, Topics in Algebra, 2nd ed., Wiley, New York, 1975 Google Scholar
  6. 11.
    A.L. Onishchik, Lectures on Real Semisimple Lie Algebras and Their Representations, ESI Lectures in Mathematics and Physics, 2004 Google Scholar
  7. 14.
    J.J. Sakurai, Modern Quantum Mechanics, 2nd ed., Addison-Wesley, Reading, 1994 Google Scholar
  8. 16.
    S. Sternberg, Group Theory and Physics, Princeton University Press, Princeton, 1994 Google Scholar
  9. 17.
    V.S. Varadarajan, Lie Groups, Lie Algebras and Their Representations, Springer, Berlin, 1984 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of CaliforniaBerkeleyUSA

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