Applied Math PhD comprehensive exam | Giuseppe Sellaroli, Tensor operators for non-compact groups in loop quantum gravity

Tensor operators for non-compact groups in loop quantum gravity

Tensor operators, in particular spinor ones, for SU(2) can been used in Euclidean loop quantum gravity as the building blocks for observables, in what is known as "spinor approach". In the (2+1) Lorentzian case, SU(2) must be replaced by the non-compact group SL(2,R), the double cover of the Lorentz group. I present the generalization of the usual results for tensor operators in this non-compact case; in particular I give an analogue of the well-known Wigner-Eckart theorem and the Jordan-Schwinger representation, and use them to construct the spinor formalism for LQG.