Representations of Lorentz group

2010-11-26T14:39:08Z

Questions:

What is the connection between representation theory of complex semisimple Lie groups and representations of (maybe "proper") Lorentz groups?
Why should one read Bargmann's paper on irred. unitary representations of Lorentz group if one wants to know unitary representation?

Answer by Cristi Stoica for Representations of Lorentz group

2010-11-27T11:42:09Z

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.

Answer by Hadi for Representations of Lorentz group

2010-11-27T21:59:40Z

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.
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.

Representations of Lorentz group - MathOverflow

Representations of Lorentz group

Answer by Cristi Stoica for Representations of Lorentz group

Answer by Hadi for Representations of Lorentz group