Chapter 2: A quantum mechanical prototype

The development of physics in the second half of the 20th century gave rise to quantum physical interpretations of several mathematical theories. The theory of theta functions is such an example. To help understand how theta functions arise from quantum mechanics, we first present a fundamental example, that of the quantization of a finite number of free one-dimensional particles. We do this also because this example is prototypical for more general cases of Witten's Chern-Simons theory (cf. [Gelca and Uribe (2010)])

The Stone-von Neumann theorem states that up to a unitary equivalence the quantization model is unique, so our task is apparently simple. But the model can be realized in many ways, each of which offering a different perspective on the subject. The purpose of this chapter is to guide the reader through a variety of models for the quantization of finitely many one-dimensional particles, in order to build the necessary background for subsequent chapters.