# The Angular Momentum Dilemma and Born–Jordan Quantization

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## Abstract

The rigorous equivalence of the Schrödinger and Heisenberg pictures requires that one uses Born–Jordan quantization in place of Weyl quantization. We confirm this by showing that the much discussed “ angular momentum dilemma” disappears if one uses Born–Jordan quantization. We argue that the latter is the only physically correct quantization procedure. We also briefly discuss a possible redefinition of phase space quantum mechanics, where the usual Wigner distribution has to be replaced with a new quasi-distribution associated with Born–Jordan quantization, and which has proven to be successful in time-frequency analysis.

## Keywords

Weyl quantization Born–Jordan quantization Angular momentum## 1 Introduction

To address quantization problems in these “ Times of Entanglement” is not very fashionable: everything seems to have been said about this old topic, and there is more or less a consensus about the best way to quantize a physical system: it should be done using the Weyl transformation. The latter, in addition to being relatively simple, enjoys several nice properties, one of the most important being its “ symplectic covariance”, reflecting, at the quantum level, the canonical covariance of Hamiltonian dynamics. Things are, however, not that simple. If one insists in using Weyl quantization, one gets inconsistency, because the Schrödinger and Heisenberg pictures are then not equivalent. Dirac already notes in the Abstract to his paper [9] that “...*the Heisenberg picture is a good picture, the Schrödinger picture is a bad picture, and the two pictures are not equivalent*...”. This observation has also been confirmed by Kauffmann’s [17] interesting discussion of the non-physicality of Weyl quantization. This non-physicality is made strikingly explicit on an annoying contradiction known as the “ angular momentum dilemma”: the Weyl quantization of the squared classical angular-momentum is not the squared quantum angular momentum operator, but it contains an additional term \(\frac{ 3}{2}\hbar ^{2}\). This extra term is actually physically significant, since it accounts for the nonvanishing angular momentum of the ground-state Bohr orbit in the hydrogen atom. This contradiction has been noted by several authors^{1}, to begin with Linus Pauling in his *General Chemistry *[20]; it is also taken up by Shewell [21], and discussed in detail by Dahl and Springborg [7] and by Dahl and Schleich [8].

*V*are irregular (we are using generalized coordinates \(x=(x_{1},\ldots ,x_{n})\), \(p=(p_{1},\ldots ,p_{n})\)). One can therefore wonder whether it is really necessary to write yet another paper on quantization rules, just to deal with a quasi-philosophical problem (the equivalence of two pictures of quantum mechanics) and one little anomaly (the angular momentum dilemma, we will discuss below). The quantum world is however more subtle than that. The problem is that if we stick to Weyl quantization for general systems, another inconsistency appears, which could have far-reaching consequences. It is due to the fact the commonly used phase space picture of quantum mechanics, where the Wigner distribution plays a central role, is intimately related to Weyl quantization. In short, if we change the quantization rules, we also have to change the phase space picture, thus leading not only to a redefinition of the Wigner distribution, but also to substantial changes in related phase space objects, such as, for instance the Moyal product of two observables, which is at the heart of deformation quantization.

## 2 BJ Versus Weyl: The Case of Monomials

*equally weighted*average of all the possible operator orderings. Weyl [22] proposed, independently, some time later a very general rule: elaborating on the Fourier inversion formula, he proposed that the quantization \(\widehat{A}\) of a classical observable

*a*(

*x*,

*p*) should be given by

*Fa*(

*x*,

*p*) is the Fourier transform of

*a*(

*x*,

*p*) and \( d^{n}x=dx_{1}\cdot \cdot \cdot dx_{n}\), \(d^{n}p=dp_{1}\cdot \cdot \cdot dp_{n}\); applying this rule to monomials yields (McCoy [19])

*xp*corresponds \(\frac{1}{2}(\widehat{x}\widehat{p}+\widehat{p}\widehat{x}\) ) in both cases, but

## 3 Generalized BJ Quantization

*a*(see [10, 11, 18]). Here \(\widehat{T}(z)=e^{-i\sigma (\widehat{z} ,z)/\hbar }\) is the usual Heisenberg–Weyl operator; \(\sigma \) is the standard symplectic form defined by \(\sigma (z,z^{\prime })=px^{\prime }-p^{\prime }x\) if \(z=(x,p)\), \(z^{\prime }=(x^{\prime },p^{\prime })\). The natural generalization of the \(\tau \)-rule (5) is obtained [12, 15] by replacing \(\widehat{T}(z)\) with \(=e^{-i\sigma _{\tau }( \widehat{z},z)/\hbar }\) where

*a*:

*a*(

*x*,

*p*), but rather of the observable

*sinus cardinalis*function familiar from Fraunhofer diffraction [5].

^{2}

*x*,

*p*) into two sets of independent coordinates \(z^{\prime }=(x^{\prime },p^{\prime })\) and \(z^{\prime \prime }=(x^{\prime \prime },p^{\prime \prime })\). Let \(b(z^{\prime })\) be an observable in the first set, and \( c(z^{\prime \prime })\) an observable in the second set, and define \(a=b\otimes c\); it is an observable depending on the total set variables \( (x,p)=(x^{\prime },x^{\prime \prime },p^{\prime },p^{\prime \prime })\). We obviously have \(a_{\sigma }=b_{\sigma }\otimes c_{\sigma }\) and \(\widehat{T} (z)=\widehat{T}(z^{\prime })\otimes \widehat{T}(z^{\prime \prime })\) (because the symplectic form \(\sigma \) splits in the sum \(\sigma ^{\prime }\oplus \) \( \sigma ^{\prime \prime }\) of the two standard symplectic forms \(\sigma ^{\prime }\) and \(\sigma ^{\prime \prime }\) defined on, respectively, \(z^{\prime }\) and \( z^{\prime \prime }\) phase spaces). It follows from formula (6) that we have \(\widehat{A}=\widehat{B}\otimes \widehat{C}\);

*i.e.*Weyl quantization preserves the separation of two observables. This property is generically not true for Born–Jordan quantization: because of the presence in formula (8) of the function \(\Theta (z)\) we cannot write the integrand as a tensor product, and hence we have in general

## 4 The Angular Momentum Dilemma

*s*state. Let

^{3}. Now, “ dequantizing” \(\widehat{\ell }^{2}\) using the Weyl transformation leads to the function \(\ell ^{2}+\tfrac{3}{2}\hbar ^{2}\) (as was already remarked by Shewell [21], formula (4.10)), which gives the “ wrong” value \(\tfrac{3}{2}\hbar ^{2}\) for the Bohr angular momentum. However, if we view \(\widehat{\ell }^{2}\) as the Born–Jordan quantization of \(\ell ^{2}\), then we recover the Bohr value \( \hbar ^{2}\). Let us show this in some detail. It suffices of course to study one of the three terms appearing in the square of the vector (15 ), say

*a*, it follows that the difference

## 5 The BJ-Wigner Transform

*a*with its Born–Jordan quantization \(\widehat{A}_{ \mathrm {BJ}}=\mathrm{Op}_{\mathrm {BJ}}(a)\), then formula (22) becomes

## 6 Discussion

*A*which is sufficiently smooth can be viewed as the Weyl transform of some classical observable

*a*, and conversely. However, this is not true of the BJ quantization scheme: it is not true that to every quantum observable (or “ operator”) one can associate a classical observable. In fact, rewriting formula (11) as

*a*(

*x*,

*p*)) if we know the Weyl transform \(a_{\mathrm {W}}\) of

*A*, and this because the function \(\Theta (x,p)=\sin (px/2\hbar )/(px/2\hbar )\) has infinitely many zeroes: \(\Theta (x,p)=0\) for all phase space points (

*x*,

*p*) such that \(p_{1}x_{1}+\cdot \cdot \cdot +p_{n}x_{n}=0\). We are thus confronted with a difficult division problem; see [6]. It is also important to note that we loose uniqueness of quantization when we use the Born–Jordan “ correspondence”: if \((a_{ \mathrm {W}})_{\sigma }(x,p)=0\) there are infinitely many Weyl operators who verify (28). These issues, which might lead to interesting developments in quantum mechanics, are discussed in detail in our book [14]. The BJ-Wigner transform and its relation with what we call “ Born–Jordan quantization” has been discovered independently by Boggiatto and his coworkers [1, 2] who were working on certain questions in signal theory and time-frequency analysis; they show—among other things—that the spectrograms obtained by replacing the standard Wigner distribution by its modified version \(W_{\mathrm {BJ}}\psi \) are much more accurate. The properties of \(W_{\mathrm {BJ}}\psi \) are very similar to those of \(W\psi \); it is always a real function, and it has the “ right” marginals and can thus be treated as a quasi-distribution, exactly as the traditional Wigner distribution does.

## Footnotes

- 1.
It is also mentioned in Wikipedia’s article [16] on geometric quantization.

- 2.
I thank Basil Hiley for having drawn my attention to this fact.

- 3.
Of course, their argument is heuristic, because there is

*in general*no relation between the eigenvalues of a quantum operator and the values of the corresponding classical observable.

## Notes

### Acknowledgments

Open access funding provided by University of Vienna. This work has been supported by the Grant P-27773 of the Austrian Research Foundation FWF.

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