Exact Results in Quantum Theory & Gravity

#1: My talk will focus on the theory of classical representations of SL(2,R). First I will present the representations of Lie algebra sl(2,R). Then I will present a transition from the representation of Lie algebra to the representation of Lie group, discussing possible problems in this procedure. This requires a brief discussion on the representations of universal cover of SL(2,R). With the knowledge of the action of Lie algebra representations I will derive conditions for the unitarity of SL(2,R) representations. At the end of my talk I will summarize the series of unitary representations of SL(2,R).

#2: The purpose of the talk will be to show that the three operators: the Schroedinger operator with inverse square potential, the dilation operator and operator of multiplication squared generate unitary representations of the universal cover of SL(2,R).