Born–Jordan Quantization

Abstract

This chapter is in a sense the keystone of this book. Using the theory of the Cohen class previously studied we give a first working definition of Born–Jordan quantization by selecting a particular Cohen kernel. We state and prove some important properties of the associated Born–Jordan operators, and discuss some unexpected properties of these operators; for instance we will show that Born–Jordan quantization is not one-to-one: the zero operator is the quantization of infinitely many classical phase space functions. Another approach, based on Shubin’s theory of pseudo-differential operators, will be developed in the forthcoming chapters.