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My lecture notes state that we need to classify all finite-dimensional irreducible representations of the proper, orthochronous Lorentz group in order to formulate a QFT for particles with non-zero spin.

This is done by characterising the Lorentz algebra by the eigenvalues $a (a + 1)$ and $b (b + 1)$ of the square of the operators $$ \vec{A} = \frac{1}{2} (\vec{J} + i \vec{K}) \\ \vec{B} = \frac{1}{2} (\vec{J} - i \vec{K}) , $$ where $\vec{J}$ is the generator of rotation and $\vec{K}$ the generator of boosts.

The corresponding representation of the Lorentz group is then obtained by taking the exponential map of particular operators like $\frac{\vec{\sigma}}{2}, 0$ for $a = \frac{1}{2}, b = 0$.

Can $\vec{A}^2$ an $\vec{B}^2$ be understood as the Casimirs of the Lie algebra or do they have something in common with the concept (I am missing some understanding here)?

How can I guarantee that taking the exponential map of an irreducible representation of the Lie algebra gives me an irreducible representation in the corresponding Lie group?

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  • $\begingroup$ I've asked a related question on Mathematics -math.stackexchange.com/q/2316362 - perhaps it is of some help. $\endgroup$ – user1620696 Sep 14 '17 at 13:44
  • $\begingroup$ Comment to the question (v2): Are you talking about the restricted Lorentz group/algebra, or its complexification? $\endgroup$ – Qmechanic Sep 14 '17 at 13:48
  • $\begingroup$ To be honest, I can't tell. We have not dived that deep into group theory in our theoretical particle physics course. I came across this complexification thing, when searching here and on Wikipedia, but wasn't able to wrap my head around it. Please feel free to explain how those things relate! $\endgroup$ – rgba Sep 14 '17 at 13:55
  • $\begingroup$ WP not helpful? $\endgroup$ – Cosmas Zachos Sep 14 '17 at 14:21
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    $\begingroup$ Two good references for projective representations are Weinberg's The Quantum Theory of Fields, Vol I, chapter 2, and Brian C. Hall's Lie groups, Lie algebras, and Representations: An Elementary Entroduction. (The first one is not an easy read, but it covers infinite-dimensional representations.) There is also a detailed version of the Wikipedia article on the Lorentz group representations at Wikiversity. It follows Hall's construction. $\endgroup$ – YohanN7 Sep 15 '17 at 7:18
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1) Finite-dimensional irreducible

  • (i) representations of the double cover $Spin(1,3)\cong SL(2,\mathbb{C})$ of the restricted Lorentz group $SO^+(1,3;\mathbb{R})$,

  • (ii) representations of the corresponding Lie algebra $so(1,3;\mathbb{R})$,

  • (iii) projective representations of the restricted Lorentz group $SO^+(1,3;\mathbb{R})$,

are all labelled by two non-negative half-integers $$(a,b)~\in~\frac{1}{2}\mathbb{N}_0 \times\frac{1}{2}\mathbb{N}_0.$$

See also e.g. this Phys.SE post and links therein.

2) If $a+b~\in~\mathbb{N}_0$ is an integer, it is also a group representation of the restricted Lorentz group $SO^+(1,3;\mathbb{R})$ itself.

3) $A_i$ and $B_i$, $i\in\{1,2,3\},$ are the $3+3=6$ generators of the complexified Lie algebra $$so(1,3;\mathbb{C})~\cong~sl(2,\mathbb{C})_A\oplus sl(2,\mathbb{C})_B,$$ with quadratic Casimirs $\vec{A}^2$ and $\vec{B}^2$.

4) The exponential map $\exp: so(1,3;\mathbb{R})\to SO^+(1,3;\mathbb{R})$ for the restricted Lorentz group is surjective, cf. e.g. this Phys.SE post.

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