I'm aware that the finite dimensional irreducible representations of the universal cover of the Lorentz group, ${\rm SL}(2,\mathbb{C})$, can be found in terms of angular momentum representations. The usual trick is to complexify the Lorentz algebra and recognize it as a direct sum of two complexified ${\frak su}(2)$ algebras. We then construct representations of ${\rm SL}(2,\mathbb{C})$ by picking two irreducible representations of angular momentum indexed by spins $A$ and $B$. This results in the $(A,B)$ representations of the Lorentz group. In particular this classifies the various fields according to their behavior under Lorentz transformations. These representations can't be unitary because they are finite-dimensional by construction and ${\rm SL}(2,\mathbb{C})$ is non-compact.

Now, I'm aware that we can also study the infinite-dimensional unitary representations of ${\rm SL}(2,\mathbb{C})$. This is in particular mentioned in the Wikipedia page on representations of the Lorentz group.

I'm interested in understanding these infinite-dimensional unitary representations of ${\rm SL}(2,\mathbb{C})$. In particular what are the irreducible ones and how the reducible ones are expressed in terms of these. Another question that I have in mind is how these relate to the ones we find in Wigner's classification of relativistic particles.

In that case we study representations of the Poincaré group and we do so by first and foremost diagonalizing the translations. After that we find how the Lorentz group is represented, so all these spaces we find in Wigner's classification carry in particular unitary representations of ${\rm SL}(2,\mathbb{C})$. I wonder how these relate to the "principal series" mentioned in Wikipedia's page.

Given this description of what I'm looking for, what would be a good reference to study all of that?


1 Answer 1


There may be other accounts for the infinite-dimensional representations of $\text{SL}(2,\mathbb C)$, but I think that the one that builds everything thoroughly is still the Naimark classic:

M.A. Naimark - Linear Representations of the Lorentz group, Pergamon Press, 1964 (translation of the original Russian edition of 1958).

Probably all the other accounts have been inspired by this book (I am thinking of M. Carmeli's books, for example).


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