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?
No comments:
Post a Comment