Abstract
Canonical quantization has served wonderfully for the quantization of a vast number of classical systems. That includes single classical variables, such as p and q, and numerous classical Hamiltonians H(p,q), as well as field theories, such as π(x) and φ(x), and many classical Hamiltonians H(π,φ. However, in all such systems, there are situations for which canonical quantization fails. This includes certain particle and field theory problems. Affine quantization involves a simple recombination of classical variables that lead to a new chapter in the process of quantization, and which is able to solve a vast variety of normally insoluble systems, such as quartic interactions in scalar field theory in spacetime dimensions 4 and higher, as well as the quantization of Einstein’s gravity in 4 spacetime dimensions.
Highlights
Canonical quantization is the leading quantization formulation, while there are other procedures that claim to be a substitute for canonical quantization
Affine quantization involves a simple recombination of classical variables that lead to a new chapter in the process of quantization, and which is able to solve a vast variety of normally insoluble systems, such as quartic interactions in scalar field theory in spacetime dimensions 4 and higher, as well as the quantization of Einstein’s gravity in 4 spacetime dimensions
Affine quantization does not pretend to take the place of canonical quantization, but instead, it expands similar procedures of canonical quantization to solve problems that canonical quantization can not solve
Summary
Canonical quantization is the leading quantization formulation, while there are other procedures that claim to be a substitute for canonical quantization. This article is focussed on how classical phase-space procedures can point to an alternative set of basic quantum operators instead of the traditional basic quantum operators, namely, the momentum P and the position Q, which satisfy [Q, P] = i 1l. In a sense, these operators are the foundation of canonical quantization. While the references above offer fairly full stories, we will—in keeping with the purpose of the present paper which is designed to offer a beginner-level tutorial on affine quantization—present a basic outline of all three of the examples listed above in this paper
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