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Questions:

  1. What is the connection between representation theory of complex semisimple Lie groups and representations of (maybe "proper") Lorentz groups?

  2. Why should one read Bargmann's paper on irred. unitary representations of Lorentz group if one wants to know unitary representation?

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One possible answer to this question is that the Lorentz group (in dimension at least 3) is semisimple and not compact, and it is a somewhat paradigmatic example. The Lorentz group is dimension 4 (which is what is treated in Bargmann's paper) is locally isomorphic to $SL(2,\mathbb{C})$. Perhaps this group plays a similarly motivating rôle as $SU(2)$ plays in studying the representation theory of compact Lie groups. –  José Figueroa-O'Farrill Nov 26 '10 at 17:20
    
The word proper is overloaded in mathematics and very overloaded in special relatvity. As well as the usual "proper Lorentz group" there is Ungar's proper-time proper-velocity Lorentz group, which could also be called for short "proper Lorentz group". "The relativistic proper-velocity transformation group", A Ungar, Progress In Electromagnetics Research, 2006, pier.engg.hku.hk/pier/pier60/04.0512151.Ungar.pdf –  Roy Maclean Nov 27 '10 at 22:27

2 Answers 2

up vote 3 down vote accepted

Weyl's theorem states that any finite dimensional representation of a compact Lie group is completely reducible. The Lorentz group is not compact, but its maximal compact subgroup is $SU(2)$. This is why there is a 1-1 correspondence between the representations of the Lorentz group (algebra) and those of $SU(2)$ (respectively $su(2)$).

You can find more details about this relation in

  • R. O. Wells, Jr. Differential analysis on complex manifolds. Published 1980 by Springer-Verlag in New York. I quote from page 173:

Proposition 3.1: The mappings $r_1$, $r_2$ and $d$ in (3.7) are all bijective, i.e., there is a one-to-one correspondence between representations of $SL(2,\mathbb C)$, $sl(2,\mathbb > C)$, $SU(2)$ and $su(2)$.

The representations of $SU(2)$ and $su(2)$ are treated in most books on representation theory.

Indeed, Wigner's and Bargmann's articles are useful if you are interested in how the spin particles occur from representations of the Lorentz group:

  • E. P. Wigner. On unitary representations of the inhomogeneous Lorentz group. Annals of Mathematics, (40):149{204, 1939.
  • V Bargmann. On Unitary Ray Representations of Continuous Groups. Ann. of Math., 59:1{46,
  • E. P. Wigner. Group Theory and its Application to Quantum Mechanics of Atomic Spectra. Academic Press, New York, 1959.

The main idea is that the wavefunctions should transform in wavefunctions at a Poincare transformation, and the transformation should be unitary. So, we need unitary representations of the Poincare group.

In order to classify the irreducible representations of a group, one can use the Casimir invariants. The Lie algebra of the $ISL(2,\mathbb C)$ group, $isl(2,\mathbb C )$ (isomorphic to the Poincare Lie algebra $so(1,3)$) has two Casimir invariants, namely $m^2=p^a p_a$ and the squared angular momentum about the center of mass, $S^2=s(s+1)$, where the spin $s$ takes semi-integer values. Usually is considered that only the representations corresponding to $m^2\geq 0$ have physical meaning, the ones with $m^2<0$ being tachyonic. For the case $m^2>0$ $s$ is of the form $0,\frac 1 2, 1, \frac 3 2, \ldots \frac n 2 \ldots$. For the case $m^2=0$, $s$ can be $0,\pm\frac 1 2, \pm1, \pm\frac 3 2, \ldots\pm\frac n 2 \ldots$. In this last case there exists also representations with continuous spin, but no physical evidence support this kind of representations.

Added. The nice relation between the representations of $SL(2,\mathbb C)$ and $SU(2)$ refers, as I stated, to the finite dimensional case. But what's the connection between the finite-dimensional and the infinite-dimensional representations? The infinite-dimensional reps of $SL(2,\mathbb C)$ which are of interest in quantum mechanics are spinor fields. That is, they are superpositions of sections in finite-dimensional complex vector bundles which are associated to $SL(2,\mathbb C)$. To construct such an associated finite-dimensional bundle, you start with a finite-dimensional representation. Strictly speaking, the things are more complicated for infinite-dimensional representations, but for quantum mechanical systems (with a finite number of particles), there is this nice connection between infinite-dimensional and finite-dimensional representations.

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-1. I'm sorry to say that this answer is largely incorrect. Why has this been accepted? Or even upvoted? The OP is not talking about the Poincaré group (a.k.a. the "inhomogeneous" Lorentz group in the old literature). The unitary irreps of the Poincaré group are indeed found in the papers by Wigner mentioned here, but also in more modern work, e.g. papers of Niederer and O'Raifeartaigh in the mid 1970s. The Casimirs of the Poincaré Lie algebra are not the spin and momentum, but the mass and then either spin or helicity for massive or massless reps, respectively. (continued below) –  José Figueroa-O'Farrill Nov 27 '10 at 17:45
    
(continued from above) Wells's book seems like an odd reference for the question. In Wells's book, what you find is a description of the Hodge-Lefschetz theory for Kähler manifolds, on which you do have an action of $sl(2,\mathbb{C})$ in the cohomology, but the ensuing representaiton is certainly not unitary, which is what this question is about. It cannot be because Hodge theory tells you the cohomology of a compact Kähler manifold is finite-dimension, yet unitary representations of $SL(2,\mathbb{C})$ (or any noncompact Lie group) are necessarily infinite-dimensional. (continued below) –  José Figueroa-O'Farrill Nov 27 '10 at 17:49
    
(and finally) There is no one-to-one correspondence between unitary irreps of $\su(2)$ and those of $\sl(2,\mathbb{C})$. In fact, Bargmann's paper proves that the unitary irreps of $\sl(2,\mathbb{C})$ come in two families, one labelled by a positive real number and the other by pair $(r,k)$ consisting of a real number $r$ and where $0< 2k \in \mathbb{Z}$. Any of these representations contains an infinite number of irreps of the maximal compact $su(2)$ subalgebra. –  José Figueroa-O'Farrill Nov 27 '10 at 17:57
    
@José Figueroa-O'Farrill Actually, R.O. Wells's book contains not only the Lefshetz decomposition, but an introduction to the representations of $sl(2,\mathbb C)$. Thanks for your comments. I updated my answer with the correct Casimir invariants. –  Cristi Stoica Nov 27 '10 at 19:28
    
@Cristi: Wells only discusses finite-dimensional reps of $sl(2,\mathbb{C}$, since those are the only relevant reps in the Hodge-Lefschetz theory of compact Kähler manifolds. There is nothing in that book (at least in the second edition, which is the one I have) about the unitary representation theory of $sl(2,\mathbb{C})$. –  José Figueroa-O'Farrill Nov 27 '10 at 19:54
  1. The Lorenz group is essentially a semidirect product of $SL(2,\mathbb C)$ and a four dimensional abelian group. (I am only considering the connected component of identity, but that is not a big deal.) Now, there are general results of George Mackey which describe unitary representations of a semidirect product in terms of those of each factor. A good place to read about Mackey theory is Varadarajan's book Geometry of quantum theory. It also has a chapter on representations of the Lorenz group.

  2. To work with unitary representations you don't need to read Bargmann's paper. There are many other sources which explain the representation theory of $SL(2,\mathbb R)$ and $SL(2,\mathbb C)$ in more modern language. See R. Howe's book, Nonabelian harmonic analysis, S. Lang's book $SL(2,\mathbb R)$, or M. Taylor's book Noncommutative harmonic analysis.

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Yeah. Howe & Lang's books are preferable. –  Alex Nov 28 '10 at 1:22
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Sorry, but what you are calling the Lorentz group is actually the Poincaré group. The connected component of the identity of the Lorentz group is $SO(3,1)_0$ whose universal covering group is $SL(2,\mathbb{C})$. It was Wigner who solved the problem of classifying the unitary irreps of the Poincaré group and Mackey who later generalised this result to other groups of that type. –  José Figueroa-O'Farrill Nov 28 '10 at 2:03
    
Yes, you're right that what I described should be called the Poincare group. –  Hadi Nov 29 '10 at 21:24

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