Abstract:
Quantization of a general nonlinear phase manifold X in the quasicIassical approximation leads to the two-dimensional analog of the Bohr–Sommerfeld conditions, in which the form pdq is replaced by dpΛdq and the vacuum energy h/2 by hν/2, where ν is the index of two-dimensional noncontractable cycles in X . A study is made of smooth manifolds X on which the index ν is integral and manifolds with conical singularities, on which ν can take half-integral values. Smooth functions f on X are associated with operators ˆf that act on the sections of a ertain sheaf and locally have the form
ˆf=f(q,−ih∂/∂q), h→0.
Citation:
M. V. Karasev, V. P. Maslov, “Quantization of symplectic manifolds with conical points”, TMF, 53:3 (1982), 374–387; Theoret. and Math. Phys., 53:3 (1982), 1186–1195