Conformal field theory
The Hilbert space of physical states is a unitary module of the Virasoro algebra corresponding to a fixed value of "c". Stability requires that the energy spectrum of the Hamiltonian be nonnegative. The modules of interest are the highest weight modules of the Virasoro algebra.
Hilbert space
The adjoint of a densely defined unbounded operator is defined in essentially the same manner as for bounded operators. Self-adjoint unbounded operators play the role of the "observables" in the mathematical formulation of quantum mechanics. Examples of self-adjoint unbounded operators on the Hilbert space are:These correspond to the momentum and position observables, respectively. Note that neither nor is defined on all of , since in the case of the derivative need not exist, and in the case of the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of .