A novel realization of the Virasoro algebra in number state space

Abstract

The group of diffeomorphisms is crucial in quantum computing. Representing it by vector fields over a d-manifold, d ⩾ 2 , accounting for both projective action and conformal symmetry at the quantum mechanical level, requires the direct-sum decomposition of tensor product for non-compact algebras, viable only for s u ( 1 , 1 ) . As a step towards the solution, a realization of the ( d = 1 ) Virasoro algebra Vir ∼ Diff + ( S ( 1 ) ) in the universal envelope of su ( 1 , 1 ) (and h ( 1 ) ) is presented, which is simple in the discrete positive series irreducible unitary representation D κ ( + ) of s u ( 1 , 1 ) .