Unitary representations of the Poincaré group

This is the website containing bibliography on the representation theory of the Poincaré group. Its purpose is to collect useful references to aid in the study of this subject.

Notes

These are notes of mine on this and related topics. Some are quite old, some are newer.

• First lecture of Principles of Mathematical Physics (2004-2005 session) (pdf)
  The end of the lecture -- which was never actually given -- contains the group theoretical foundations of galilean relativity.
• The theory of induced representations in field theory (pdf)
  Some old notes of mine written when I lectured about this topic in 1987 while I was a student at the Institute for Theoretical Physics in Stony Brook.

Problem sheets

Problem sheets will appear here in due time.

1. Problem Sheet 1
2. Problem Sheet 2

References

• The Principle of Relativity (djvu)
  Hendrik A. Lorentz, Albert Einstein, Hermann Minkowski and Hermann Weyl
  This contains a wonderful compilation of translations of essays and papers on the special and general theories of relativity by some of the people who developed them. Minkowski's 1908 talk "Space and Time" is worth reading if only for the poetic language.
• On unitary representations of the inhomogeneous Lorentz group (pdf)
  Eugene P. Wigner
  This is the original paper on this subject.
• Group theoretical discussion of relativistic wave equations (pdf)
  Valentine Bargmann and Eugene P. Wigner
  Simplified treatment of Wigner's original paper.
• Irreducible unitary representations of the Lorentz group (pdf)
  Valentine Bargmann
  The unitary irreducible representations of SL(2,ℝ) and SL(2,ℂ).
• Theory of group representations and applications (djvu)
  Asim O. Barut and Ryszard Rączka
  Chapter 17 discusses the relevant topics, but they depend on quite a lot which has gone before. Still, this book is a gem and has a lot of results on the applications of group theory (especially infinite-dimensional representations of noncompact Lie groups) to Physics.
• Lie groups for physicsts (djvu)
  Robert Hermann
  Despite the condescending-sounding title, it's far from a watered-down version of Lie theory. It is one of many books by Robert Hermann on Lie theory and physics. Chapter 9 in this book contains the geometric theory of induced representations.
• Fourier analysis on groups and partial wave analysis
  Robert Hermann
  This is in the library, but I have not looked at it in more than 20 years!
• Induced representations of groups and quantum mechanics
  George W. Mackey
  Mackey developed the theory of induced representations in the infinite-dimensional setting required by quantum mechanics. I happen to have a copy of this book in my office.
• Harmonic analysis as the exploitation of symmetry -- a historical survey (pdf)
  George W. Mackey
  A leisurely, surprisingly accessible survey about many topics in representation theory and physics from one of the masters.
• Realizations of the unitary representations of the inhomogeneous space-time groups I: General structure (pdf) II: Covariant realizations of the Poincaré group (pdf)
  Ulrich H Niederer and Lochlainn O'Raifeartaigh
  These two papers have a reasonably modern discussion of the Wigner/Mackey method of induced representations applied to the Poincaré group.
• Quantum theory of massless particles
  Steven Weinberg
  I can't seem to find this reference in the library or anywhere online. I have not seen it since the late 1980s, so perhaps it's superseded by his book (see below) on Quantum Field Theory.
• Quantum Theory of Fields, I: Foundations (djvu)
  Steven Weinberg
  The second chapter has a good discussion (in the usual Weinberg style) of the group theoretical underpinnings of quantum field theory on Minkowski spacetime. (Recommended by Joan Simón.)
• Group Theory in Physics, Vol. II (djvu)
  JF Cornwell
  Chapter 17 is the relevant one here. (Recommended by Antonella D'Avanzo.)
• Group theory and Physics (djvu)
  Shlomo Sternberg
  Chapter 3.9 contains a good modern treatment emphasising the geometric language of homogeneous vector bundles.
• Un soupçon de théorie des groupes: groupe des rotations et groupe de Poincaré (pdf)
  Bertrand Delamotte
  Undergraduate notes from the French DEA in theoretical physics. (Recommended by Elena Méndez-Escobar.)
• The unitary representations of the Poincaré group in any spacetime dimension (pdf)
  Xavier Bekaert and Nicolas Boulanger
  Recent paper (in the hep-th style) on the representations of the Poincaré group in diverse dimensions. (Recommended by Paul de Medeiros.)