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  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 dwork 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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  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 dwork 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.