Quantization: History and Problems

TL;DR

This paper reviews the historical development of quantization, discusses key theoretical challenges like the Groenewold-Van Hove theorem, and compares various quantization methods including Weyl and Geometric Quantization.

Contribution

It provides a comprehensive overview of the history, problems, and different approaches to quantization, highlighting the limitations of Dirac's original scheme.

Findings

Groenewold-Van Hove theorem shows Dirac's quantization map is inconsistent.

Weyl quantization and Geometric Quantization offer alternative frameworks.

Operator ordering and coordinate issues are central challenges in quantization.

Abstract

In this work, I explore the concept of quantization as a mapping from classical phase space functions to quantum operators. I discuss the early history of this notion of quantization with emphasis on the works of Schr\"odinger and Dirac, and how quantization fit into their overall understanding of quantum theory in the 1920's. Dirac, in particular, proposed a quantization map which should satisfy certain properties, including the property that quantum commutators should be related to classical Poisson brackets in a particular way. However, in 1946, Groenewold proved that Dirac's mapping was inconsistent, making the problem of defining a rigorous quantization map more elusive than originally expected. This result, known as the Groenewold-Van Hove theorem, is not often discussed in physics texts, but here I will give an account of the theorem and what it means for potential "corrections" to Dirac's scheme. Other proposals for quantization have arisen over the years, the first major one being that of Weyl in 1927, which was later developed by many, including Groenewold, and which has since become known as Weyl Quantization in the mathematical literature. Another, known as Geometric Quantization, formulates quantization in differential-geometric terms by appealing to the character of classical phase spaces as symplectic manifolds; this approach began with the work of Souriau, Kostant, and Kirillov in the 1960's. I will describe these proposals for quantization and comment on their relation to Dirac's original program. Along the way, the problem of operator ordering and of quantizing in curvilinear coordinates will be described, since these are natural questions that immediately present themselves when thinking about quantization.