# Tag Info

3

Each separable infinite-dimensional Hilbert space carries both irreducible and reducible representations of any noncompact Lie groups you can name. But this information in itself is of little use. The Hilbert spaces in quantum mechanics always come with distinguished representations that give certain operators an interpretation as distinguished ...

1

Prahar has already given a good answer. Here we will instead focus on the pertinent Lie group (as opposed to the Lie algebra and its generators). The Lie group $SL(2,\mathbb{C})$ is the double cover of the restricted Lorentz group $SO^+(3,1)$, cf. e.g. this Phys.SE post. The fundamental/defining representation $V\cong\mathbb{C}^2$ of the Lie group ...

0

It precisely one of the Wightman axioms that the infinite-dimensional unitary representations1 $U : \mathrm{SO}(1,3)\to\mathrm{U}(\mathcal{H})$ on the space of states $\mathcal{H}$ of the theory upon which the field act as operator is compatible with the field transformation law under the finite-dimensional representation \$\rho_\text{fin}: ...

Top 50 recent answers are included