Abstract:
Quantization of a general nonlinear phase manifold $\mathfrak X$ in the quasicIassical approximation leads to the two-dimensional analog of the Bohr–Sommerfeld conditions, in which the form $pdq$ is replaced by $dp\Lambda dq$ and the vacuum energy $h/2$ by $h\nu/2$, where $\nu$ is the index of two-dimensional noncontractable cycles in $\mathfrak X$ . A study is made of smooth manifolds $\mathfrak X$ on which the index $\nu$ is integral and manifolds with conical singularities, on which $\nu$ can take half-integral values. Smooth functions $f$ on $\mathfrak X$ are associated with operators $\hat{f}$ that act on the sections of a ertain sheaf and locally have the form
$\hat{f}=f(q,-ih\partial/\partial q)$, $h\to0$.
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
This publication is cited in the following 6 articles:
D. B. Zot'ev, “Kostant Prequantization of Symplectic Manifolds with Contact Singularities”, Math. Notes, 105:6 (2019), 846–863
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D. A. Sadovskií, D. N. Kozlov, P. P. Radi, “Direct absorption transitions to highly excited polyads 8, 10, and 12 of methane”, Phys. Rev. A, 82:1 (2010)
M. V. Karasev, V. P. Maslov, “Asymptotic and geometric quantization”, Russian Math. Surveys, 39:6 (1984), 133–205
M. V. Karasev, “Asymptotic behavior of the spectrum of mixed states for self-consistent field equations”, Theoret. and Math. Phys., 61:1 (1984), 1034–1040
M. V. Karasev, V. P. Maslov, “Pseudodifferential operators and a canonical operator in general symplectic manifolds”, Math. USSR-Izv., 23:2 (1984), 277–305