Representations of Lorentz group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:50:31Z http://mathoverflow.net/feeds/question/47433 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/47433/representations-of-lorentz-group Representations of Lorentz group Alex 2010-11-26T14:39:08Z 2010-11-29T21:15:58Z <p>Questions:</p> <ol> <li><p>What is the connection between representation theory of complex semisimple Lie groups and representations of (maybe "proper") Lorentz groups?</p></li> <li><p>Why should one read Bargmann's paper on irred. unitary representations of Lorentz group if one wants to know unitary representation?</p></li> </ol> http://mathoverflow.net/questions/47433/representations-of-lorentz-group/47501#47501 Answer by Cristi Stoica for Representations of Lorentz group Cristi Stoica 2010-11-27T11:42:09Z 2010-11-27T21:20:20Z <p>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 $su(2)$).</p> <p>You can find more details about this relation in</p> <ul> <li>R. O. Wells, Jr. Differential analysis on complex manifolds. Published 1980 by Springer-Verlag in New York. I quote from page 173:</li> </ul> <blockquote> <p><strong>Proposition 3.1:</strong> 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 $su(2)$.</p> </blockquote> <p>The representations of $SU(2)$ and $su(2)$ are treated in most books on representation theory.</p> <p>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:</p> <ul> <li>E. P. Wigner. On unitary representations of the inhomogeneous Lorentz group. Annals of Mathematics, (40):149{204, 1939.</li> <li>V Bargmann. On Unitary Ray Representations of Continuous Groups. Ann. of Math., 59:1{46,</li> <li>E. P. Wigner. Group Theory and its Application to Quantum Mechanics of Atomic Spectra. Academic Press, New York, 1959.</li> </ul> <p>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. </p> <p>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 $m^2&lt;0$ 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.</p> <p><strong>Added.</strong> 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. </p> http://mathoverflow.net/questions/47433/representations-of-lorentz-group/47542#47542 Answer by Hadi for Representations of Lorentz group Hadi 2010-11-27T21:59:40Z 2010-11-29T21:15:58Z <ol> <li><p>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 <em>Geometry of quantum theory</em>. It also has a chapter on representations of the Lorenz group.</p></li> <li><p>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, <em>Nonabelian harmonic analysis</em>, S. Lang's book <em>$SL(2,\mathbb R)$</em>, or M. Taylor's book <em>Noncommutative harmonic analysis</em>.</p></li> </ol>