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Uniqueness of the Dirac adjoint

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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?

asked Jul 24 in Mathematics by Albert Zhou [ revision history ]
edited Aug 2

In fact, Dirac spinors form an irreducible representation of the full Poincare group, whereas the restriction of the representation to the Lorentz group is highly reducible. Please rethink your question in this light.

Could you explain how reducibility affects my question?

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