Generalization of the Berezin Formula to the Noncommutative Case

Rukmini Dey

In this paper, we define Berezin-type and Odzijewicz-type quantizations on compact smooth manifolds. The method is as follows: we embed the smooth manifold of real dimension [Formula: see text] into [Formula: see text] and induce the quantizations from there. The standard way by which reproducing kernel Hilbert spaces are defined on submanifolds gives a way to define the pullback coherent states. In Berezin-type quantization, the Hilbert space of quantization is the pullback (by the embedding) of the Hilbert space of geometric quantization of [Formula: see text]. In the Odzijewicz-type quantization, one has to consider a tensor product of the geometric quantization line bundle with holomorphic [Formula: see text]-forms. In the Berezin case, the operators that are quantized are those induced from the ambient space [Formula: see text]. The Berezin-type quantization exhibited here is a generalization of an earlier work of the author and Ghosh. In both Berezin- and Odzijewicz-type quantizations, we first exhibit this quantization explicitly on [Formula: see text] and we induce the quantization on the smooth compact embedded manifold from [Formula: see text].

Dror Bar-Natan

V. P. Belavkin

A universal functional Ito formula, recently discovered within the *-Ito noncommutative algebraic approach, is described herein. The formula unifies the classical Ito formulas for Wiener, Poisson, and quantum cases, extends the chain rule df(x1,...,xl) = ∑f′i(x)dxi to the case of noncommuting stochastically differentiable operators Xi(t), and also admits a nonadapted generalization.

C. N. Ktorides, Loucas C. Papaloucas

V. P. Belavkin, M F Brown

We study the continuity property of multiple Q-adapted quantum stochastic integrals with respect to noncommuting integrands given by the non-adapted multiple integral kernels in Fock scale.The noncommutative algebra of relatively (exponentially) bounded nonadapted quantum stochastic processes is studied in the kernel form as introduced in [10].The differential Q-adapted formula generalizing Itô product formula for adapted integrals is presented in both strong and weak sense as a particular case of the quantum stochastic nonadapted Itô formula.

Mark K. Transtrum, Jean-François S. Van Huele