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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.

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. However, traditional coordinate transformations, such as p → p ¯ and q → q ¯ , with − ∞ < p ¯ , q ¯ < ∞ and Poisson brackets { q ¯ , p ¯ } = 1 are also qualified, in principle, as natural candidates to promote to basic quantum operators. This article resolves the problem of deciding which pair of classical variables to promote to basic quantum operators in order to achieve a valid quantization.

Dirac gave us the clue about which classical variables to choose for canonical quantization [

The only problem deals with how that flat surface, which is linked to both the classical and quantum realms, comes about. As a potential link to other forms of quantization (the spin and affine versions), we call our flat, two-dimensional surface a “constant zero curvature”.

The Role of Canonical Coherent StatesWe choose to name the favored classical variables that are promoted to quantum operators as p and q, and the valid quantization of these variables named as P and Q. We confirm this choice by first introducing canonical coherent states given by

| p , q 〉 ≡ e − i q P / ℏ e i q Q / ℏ | ω 〉 , (1)

where ( Q + i P / ω ) | ω 〉 = 0 . It follows that 〈 p , q | P | p , q 〉 = 〈 ω | ( P + p 1 l ) | ω 〉 = p and 〈 p , q | Q | p , q 〉 = 〈 ω | ( Q + q 1 l ) | ω 〉 = q , which is a clear connection between the quantum and classical basic variables. We finalize this connection with a Fubini-Study metric [

d σ ( p , q ) 2 ≡ 2 ℏ [ ‖ d | p , q 〉 ‖ 2 − | 〈 p , q | d | p , q 〉 | 2 ] = ω − 1 d p 2 + ω d q 2 . (2)

Observe that this process has given us Cartesian coordinates. These variables clearly are invariant even if we choose p + a and q + b . These favored coordinates are Cartesian coordinates and they are promoted to valid basic quantum operators, as Dirac had predicted.

The surface of an ideal three-dimensional ball is two-dimensional and spherical with a constant radius; we can say that it has a “constant positive curvature”. Again, like a flat space, the properties at any point on the spherical surface are exactly like those at any other point on the surface. This is the space on which the spin variables appear. There are three spin operators, S 1 , S 2 , and S 3 , which belong to the groups S O ( 3 ) or S U ( 2 ) . These operators obey certain rules, such as [ S 1 , S 2 ] = i ℏ S 3 , and natural permutations, as well as

S 1 2 + S 2 2 + S 3 2 = ℏ 2 s ( s + 1 ) 1 l 2 s + 1

where 2 s + 1 is the dimension of the vectors, and the spin s values are ( 1,2,3, ⋯ ) / 2 . Some basic vectors are | s , m 〉 , where S 3 | s , m 〉 = m | s , m 〉 and − s ≤ m ≤ s , ( S 1 + i S 2 ) | s , m 〉 = | s , m + 1 〉 , as well as ( S 1 + i S 2 ) | s , s 〉 = 0 .

The Role of Spin Coherent StatesThe spin coherent states are given by

| θ , φ 〉 ≡ e − i φ S 3 / ℏ e − i θ S 2 / ℏ | s , s 〉 , (3)

where − π < φ ≤ π , and 0 ≤ θ ≤ π . We also introduce q = ( s ℏ ) 1 / 2 φ and p = ( s ℏ ) 1 / 2 c o s ( θ ) . It follows that

d σ ( θ , φ ) 2 ≡ 2 ℏ [ ‖ d | θ , φ 〉 ‖ 2 − | 〈 θ , φ | d | θ , φ 〉 | 2 ] = ( s ℏ ) [ d θ 2 + sin ( θ ) 2 d φ 2 ] , (4)

or we can say that

d σ ( p , q ) 2 ≡ 2 ℏ [ ‖ d | p , q 〉 ‖ 2 − | 〈 p , q | d | p , q 〉 | 2 ] = ( 1 − p 2 / s ℏ ) − 1 d p 2 + ( 1 − p 2 / s ℏ ) d q 2 . (5)

Equation (4) makes it clear that we are dealing with a spherical surface with a radius of ( s ℏ ) 1 / 2 . Equation (5) makes it clear that if s → ∞ , in which case both p and q span the real line, we will have recovered properties of canonical quantization.

So far we have obtained surfaces with a constant zero curvature and a constant positive curvature. Could there be more? Could there be surfaces with a “constant negative curvature”?

One of the simplest problems to quantize is an harmonic oscillator for which − ∞ < p , q < ∞ , but it is not so simple if 0 < q < ∞ . To solve the second version requires a new method of quantization called affine quantization, which, as we will discover, involves a constant negative curvature. To introduce this procedure let us focus on the classical term p d q which is part of a classical action functional. Instead of these variable’s range being − ∞ < p , q < ∞ , let us assume q is limited to 0 < q < ∞ , and we want to change variables. As a first step, let us consider p d q = p q d q / q = p q d ln ( q ) . While q must be positive, l n ( q ) covers the whole real line. Although that p → p q and q → l n ( q ) both cover the whole real line, we instead just choose pq and q as our new variables. A potential quantization of this pair of variables could involve q → Q , with 0 < Q < ∞ , and p q → ( P Q + Q P ) / 2 ≡ D . Note, if 0 < Q < ∞ , then P cannot be self adjoint; however, thanks to Q, D can be self adjoint, which is a very important advantage. The two basic operators for affine quantization then are D and Q, for which [ Q , D ] = i ℏ Q ^{1}.

The affine coherent states, where both q and Q have been chosen dimensionless for simplicity, are given by

| p ; q 〉 ≡ e i p Q / ℏ e − i ln ( q ) D / ℏ | b 〉 , (6)

where [ ( Q − 1 ) + i D / b ℏ ] | b 〉 = 0 . For these variables we find that 〈 p ; q | Q | p ; q 〉 = 〈 b | q Q | b 〉 = q and 〈 p ; q | D | p ; q 〉 = 〈 b | D + p q Q | b 〉 = p q . It follows that

d σ ( p , q ) 2 ≡ 2 ℏ [ ‖ d | p , q 〉 ‖ 2 − | 〈 p , q | d | p , q 〉 | 2 ] = ( b ℏ ) − 1 q 2 d p 2 + ( b ℏ ) q − 2 d q 2 . (7)

This expression for the Fubini-Study metric is that of a constant negative curvature, an amount of − 2 / ( b ℏ ) , which is also geodetically complete [

Our quantization of classical variables has now been completed. It was shown that classical variables that represent the coordinates of constant positive, zero, and negative curvatures, complete the natural forms of surface and these three divisions include a different variety of classical systems that can be quantized. Affine quantization, as a special procedure to quantize systems, has not yet become universally well-known and exploited; it deserves more attention. Besides the harmonic oscillator with 0 < q < ∞ (see [

Others are encouraged to see what affine quantization can do for their own quantization problems.

The author declares no conflicts of interest regarding the publication of this paper.

Klauder, J.R. (2020) The Favored Classical Variables to Promote to Quantum Operators. Journal of High Energy Physics, Gravitation and Cosmology, 6, 828-832. https://doi.org/10.4236/jhepgc.2020.64055