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Summary

The general concept of quantization is discussed, which provides the starting point for the further developments in this book. Starting with the concepts familiar from quantum mechanics, a number of quantum spaces defined via explicit operators on Hilbert space are discussed in detail, including compact and non-compact examples.

In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form

$$Au(x)=\int_{{\open R}^n}\int_{{\open R}^n}e^{{\rm i}(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)\,{\rm d}y\,{\rm d}\xi,$$

where

$\tau :{\open R}^n\to {\open R}^n$ is a general function. In particular, for the linear choices

$\tau (x)=0$,

$\tau (x)=x$ and

$\tau (x)={x}/{2}$ this covers the well-known Kohn–Nirenberg, anti-Kohn–Nirenberg and Weyl quantizations, respectively. Quantizations of such type appear naturally in the analysis on nilpotent Lie groups for polynomial functions τ and here we investigate the corresponding calculus in the model case of

${\open R}^n$. We also give examples of nonlinear τ appearing on the polarized and non-polarized Heisenberg groups.

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3 - Quantization of symplectic manifolds

Summary

Pseudo-differential operators with nonlinear quantizing functions

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2 results

3 - Quantization of symplectic manifolds

Summary

Pseudo-differential operators with nonlinear quantizing functions