Abstract
Classical phase-space variables are normally chosen to promote to quantum operators in order to quantize a given classical system. While classical variables can exploit coordinate transformations to address the same problem, only one set of quantum operators to address the same problem can give the correct analysis. Such a choice leads to the need to find the favored classical variables in order to achieve a valid quantization. This article addresses the task of how such favored variables are found that can be used to properly solve a given quantum system. Examples, such as non-renormalizable scalar fields and gravity, have profited by initially changing which classical variables to promote to quantum operators.
Highlights
Classical phase-space variables are normally chosen to promote to quantum operators in order to quantize a given classical system
Conventional phase-space variables, such as p and q, where −∞ < p, q < ∞, with Poisson brackets {q, p} = 1, are natural candidates to promote to basic quantum operators in the procedures that canonical quantization employs
Traditional coordinate transformations, such as p → p and q → q, with −∞ < p, q < ∞ and Poisson brackets {q, p} = 1 are qualified, in principle, as natural candidates to promote to basic quantum operators
Summary
A flat surface and the choice of Cartesian coordinates, with or without a and b shifts, lead to acceptable classical variables to promote to become the basic quantum operators. Q P p, q =ω ( P + p1l ) ω =p and p, q Q p, q =ω (Q + q1l ) ω =q , which is a clear connection between the quantum and classical basic variables We finalize this connection with a Fubini-Study metric [2] that involves a tiny ray distance (minimized over non-dynamical phases) between two infinitely close canonical coherent states given by dσ Observe that this process has given us Cartesian coordinates. These favored coordinates are Cartesian coordinates and they are promoted to valid basic quantum operators, as Dirac had predicted
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