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Universality of Fedosov's Construction for Star Products of Wick Type on Pseudo-K\"ahler Manilfolds
In this paper we construct star products on a pseudo-K\"ahler manifold
using a modification of the Fedosov method based on a different
fibrewise product similar to the Wick product on . In a first step
we show that this construction is rich enough to obtain star products of every
equivalence class by computing Deligne's characteristic class of these
products. Among these products we uniquely characterize the ones which have the
additional property to be of Wick type which means that the bidifferential
operators describing the star products only differentiate with respect to
holomorphic directions in the first argument and anti-holomorphic directions in
the second argument. These star products are in fact strongly related to star
products with separation of variables introduced and studied by Karabegov. This
characterization gives rise to special conditions on the data that enter the
Fedosov procedure. Moreover, we compare our results that are based on an
obviously coordinate independent construction to those of Karabegov that were
obtained by local considerations and give an independent proof of the fact that
star products of Wick type are in bijection to formal series of closed
two-forms of type on . Using this result we finally succeed in
showing that the given Fedosov construction is universal in the sense that it
yields all star products of Wick type on a pseudo-K\"ahler manifold.Comment: terminology corrected, typos removed, appendix adde
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Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states
Contrary to the finite dimensional case, Weyl and Wick quantizations are no
more asymptotically equivalent in the infinite dimensional bosonic second
quantization. Moreover neither the Weyl calculus defined for cylindrical
symbols nor the Wick calculus defined for polynomials are preserved by the
action of a nonlinear flow. Nevertheless taking advantage carefully of the
information brought by these two calculuses in the mean field asymptotics, the
propagation of Wigner measures for general states can be proved, extending to
the infinite dimensional case a standard result of semiclassical analysis.Comment: 39 page
Internationalization manager’s identity and authority identification:staffing of the export department in Greek internationalized manufacturing companies
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