![]() ![]() Non-trivial unitary representations of $SL(2,\mathbb C)$ are necessarily infinite dimensional. An easy argument extends the negative result to its universal covering $SL(2, \mathbb C)$. The orthochronous proper Lorentz group has this property. The most famous theorem by Wigner states that, in a complex Hilbert space $H$, every bijective map sending rays into rays (a ray is a unit vector up to a phase) and preserving the transition probabilities is represented (up to a phase) by a unitary or antiunitary (depending on the initial map if $\dim H>1$) map in $H$.ĭealing with spinors $\Psi \in \mathbb C^4$, $H= \mathbb C^4$ and there is no Hilbert space product (positive sesquilinear form) such that the transition probabilities are preserved under the action of $S(\Lambda)$, so Wigner theorem does not enter the game.įurthermore $S$ deals with a finite dimensional Hilbert space $\mathbb C^4$ and it is possible to prove that in finite-dimensional Hilbert spaces no non-trivial unitary representation exists for a non-compact connected semisimple Lie group that does not include proper non-trivial closed normal subgroups. (6) To derive the identity, use the completeness. Proof the Fierz identity for generators of the fundamental representation of SU(N) ta ij t a kl 1 2 il jk 1 N ij kl. Derive the value of C F by taking the trace in (5) using (2). That is, the spinors can be arranged to appear in an order that makes them natural for a trace operation.įor example, with $U,V,W$ as unspecified operators and $\theta,\psi$ as spinors, one might have: fundamental and the adjoint representation, i.e. They seem to arise in QFT calculations and (perhaps as a result) they can always be put into "pure density matrix" form. Some notes to be deleted after we have a solution: The article implies that they arise from the blade structure of a Clifford algebra. Fierz identities are discussed in the wikipedia article:īut the article doesn't give any derivation. ![]()
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