The Liouville theory is quantized with use of Fock-space methods, an infinite set of charges $L_n$, $n=0, \pm 1, \dots$, is constructed which represents the conformal algebra in two dimensions, and consequences of this algebra are discussed. It is then argued, with use of variational methods in Fock space, that the spectrum of the Liouville Hamiltonian is continuous, and that there exist energy eigenstates obeying the constraints $L_n|E〉=0\mathrm{}$, $n>\mathrm{}0\mathrm{}$.

Supersymmetric two-dimensional sigma models which include Wess-Zumino topological terms are constructed, analyzed, and interpreted by means of torsion on the field manifold. One-loop renormalization results are presented geometrically, revealing that an infrared fixed point exists when the torsion parallelizes the manifold. The O(4)/O(3) model is used to illustrate general results.

The Wigner phase-space distribution function provides the basis for Moyal’s deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. The general features of time-independent Wigner functions are explored here, including the functional (“star”) eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux (“supersymmetric”) isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the Pöschl-Teller potential, and the Liouville potential.