7 - Discrete Quantum Mechanics

Abstract

This chapter discusses the way to implement discrete quantum mechanics. The usual continuum quantum mechanics can be approximated arbitrarily closely by a discrete quantum mechanics. There are reasons for believing that natural phenomena are intrinsically discrete and should not be based upon a continuum spacetime model. Such continuum models are frequently plagued with infinities and singularities and they break down at small distances and high energies. An example of this can be seen in the path integral formalism. There is a speculation that there exists in nature an elementary length and an elementary time and that all length and time measurements should be integer multiples of these. As the existence of an elementary length does not automatically lead to a discrete spacetime, it is not clear whether discrete quantum mechanics will play the role of a fundamental physical theory.