+ 2 like - 0 dislike
86 views

The set of Dirac spinors is a representation of the Lorentz algebra, whose generators are represented by the Dirac matrices $\gamma^\mu$ via the commutators, ie an element of the the Lorentz group is $\exp\left(\frac{1}{2}[\gamma_\mu, \gamma_\nu]\omega^{\mu\nu}\right)$ where $\omega^{\mu\nu}$ are the parameters.

The Dirac adjoint $\bar{\psi}=\psi^\dagger\gamma_0$ is defined such that the bilinear form $\bar{\psi}\psi$ is an invariant under transformations of the Lorentz group.

In the theory of compact Lie groups, we know there is a unique measure which induces a unique bilinear form on the Lie algebra.

Is this the same case for the Lorentz group? If not what are the other invariants? Is this the case for all (spin-) representations of the Lorentz group?

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOv$\varnothing$rflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.