# Born–Jordan Quantization and the Equivalence of the Schrödinger and Heisenberg Pictures

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## Abstract

The aim of the famous Born and Jordan 1925 paper was to put Heisenberg’s matrix mechanics on a firm mathematical basis. Born and Jordan showed that if one wants to ensure energy conservation in Heisenberg’s theory it is necessary and sufficient to quantize observables following a certain ordering rule. One apparently unnoticed consequence of this fact is that Schrödinger’s wave mechanics cannot be equivalent to Heisenberg’s more physically motivated matrix mechanics unless its observables are quantized using *this* rule, and *not* the more symmetric prescription proposed by Weyl in 1926, which has become the standard procedure in quantum mechanics. This observation confirms the superiority of Born–Jordan quantization, as already suggested by Kauffmann. We also show how to explicitly determine the Born–Jordan quantization of arbitrary classical variables, and discuss the conceptual advantages in using this quantization scheme. We finally suggest that it might be possible to determine the correct quantization scheme by using the results of weak measurement experiments.

## Keywords

Heisenberg picture Schrödinger picture Quantization Born and Jordan Dequantization## 1 Introduction

*same*rules. A consequence of this property is that if we believe that Heisenberg’s “matrix mechanics” is correct and is equivalent to Schrödinger’s theory, then the Hamiltonian operator appearing in the Schrödinger equation (2)

*must*be quantized using the Born–Jordan rule, and not, as is usual in quantum mechanics, the Weyl quantization rule.

### **Notation 1**

Real position and momentum variables are denoted \(q,p\); more generally, for systems with \(n\) degrees of freedom we write \(q=(q_{1},..,q_{n})\), \(p=(p_{1},\ldots ,p_{n})\). The boldface letters \(\mathbf {q},\mathbf {p}\) are used to denote the corresponding quantum observables. Similarly, the quantum operator associated with a classical observable \(A\) is denoted by \(\mathbf {A}\) and we write \(A\longleftrightarrow \mathbf {A}\). It is assumed throughout that this correspondence (“quantization”) is *linear*.

## 2 The Born and Jordan Argument

We begin by shortly exposing the main arguments in Born and Jordan’s paper [2].

*i.e.*scalars; we will call these infinite matrices (for which we always use boldface letters)

*observables*. In particular Born and Jordan introduce momentum and position observables \(\mathbf {p}\) and \(\mathbf {q}\) and matrix functions \(\mathbf {H}(\mathbf {p,q)}\) of these observables, which they call “Hamiltonians”. Following Heisenberg, they assume that the equations of motion for \(\mathbf {p}\) and \(\mathbf {q}\) are formally the same as in classical theory, namely

*only*possible choice is

## 3 Born–Jordan Quantization

*mutatis mutandis*to systems with an arbitrary number of degrees of freedom. We will call this rule (and its extension to higher dimensions) the

*Born–Jordan*(BJ)

*quantization rule*. Weyl [25] proposed, independently, some time later (1926) another rule leading to the replacement of (18) with

^{1}Both quantizations are thus not equivalent; as Kauffmann [16] observes, Weyl’s rule is the single most symmetrical operator ordering, whereas the BJ quantization is the equally weighted average of all the operator orderings.

*equivalent*, then we

*must*quantize the Hamiltonian in Schrödinger’s equation using BJ quantization. In fact, recall from formula (7) that the Heisenberg and Schrödinger Hamiltonians are related by

An obvious consequence of these considerations is that if one uses in the Schrödinger picture the Weyl quantization rule (or any other quantization rule), we obtain two different renderings of quantum mechanics. This observation seems to be confirmed by Kauffmann’s [16] interesting discussion of the non-physicality of Weyl quantization.

## 4 Generalization to Arbitrary Observables

We have been considering the quantization of polynomials for simplicity; in de Gosson and Luef [12] and de Gosson [6] we have shown in detail how to Born–Jordan quantize arbitrary functions of the position and momentum variables.

^{2}). We have

*i.e.*that one can derive Schrödinger’s equation from Hamilton’s equations of motion, and vice versa. Canonical covariance means the following: let \(\mathrm{* }{Sp}(n)\) be the symplectic group of the \(n\)-dimensional configuration space; it consists of all linear canonical transformations of the corresponding \(2n\)-dimensional phase space (we have given an elementary construction of \(\mathrm{* }{Sp}(n)\) in de Gosson [8]). The elements of \(\mathrm{* }{Sp}(n)\) are identified with \(2n\times 2n\) matrices \(S\) (“symplectic matrices”) satisfying the condition \(S^{T}JS=J\) where the superscript \(T\) indicates transposition and \(J= \begin{pmatrix} 0 &{}\quad I\\ -I &{}\quad 0 \end{pmatrix} \) where \(0\) and \(I\) are the zero and identity \(n\times n\) matrices. Now, to every symplectic matrix \(S\) one can associate two unitary operators \(\pm \widehat{S}\) acting on \(L^{2}(\mathbb {R}^{n})\) (the square integrable functions); the set of all these operators form a group, the

*metaplectic group*\(\mathrm{* }{Mp}(n)\) (see de Gosson [4] for a detailed study of that group). The property of canonical covariance for a quantization rule \(A\longleftrightarrow \mathbf {A}\) means that for every symplectic matrix \(S\) we must have \(A\circ S\longleftrightarrow \widehat{S} \mathbf {A}\widehat{S}^{-1}\) (\(A\circ S\) is the new observable \(A\circ S(q,p)=A(S(q,p))\)). (Thus, a symplectic transformation of the coordinates in a classical observable corresponds at the operator level to conjugation by the corresponding metaplectic operator.) Now it is a mathematical theorem that there is only one quantization rule which enjoys this property: namely Weyl quantization. Therefore, if we use BJ quantization in place of Weyl quantization, we will lose canonical covariance for all observables which are not quantized the “Weyl way”. But this observation has no drastic consequences because, as we just mentioned, the Weyl and BJ quantizations of all physical Hamiltonians (28) are the same, and will thus have the property of canonical covariance. And there is another case where this remains true: formula (20) implies that monomials \(q_{j}^{2}\), \(p_{j}^{2}\), \(p_{j}q_{j}\) (and, of course \(p_{j}q_{k})\) have the same quantization in both schemes; it easily follows that the same is true for the generalized harmonic oscillator

## 5 Discussion

*dequantization*(or “classicization”). Besides being canonically covariant, the Weyl rule has a very important, but rather unwelcome, property: it is one-to-one invertible because

*every*continuous operator can be written uniquely as a Weyl operator (for a mathematical proof see e.g. de Gosson [4, 5]). This invertibility means that every quantum observable has a (unique) classical counterpart, and this is physically not tenable. The situation is very different when one uses BJ quantization. Let us explain this in some detail. We begin with the following observation, which is simple and subtle at the same time. Consider the BJ quantization \(\mathbf {A}_{\mathrm {BJ} }\overset{\mathrm {BJ}}{\longleftrightarrow }A\) of some classical observable \(A\). Born–Jordan operators are continuous operators, hence we can also view \(\mathbf {A}_{\mathrm {BJ}}\) as a Weyl operator: \(\mathbf {A}_{\mathrm {BJ} }=\mathbf {B}_{\mathrm {W}}\overset{\mathrm {W}}{\longleftrightarrow }B\) where \(B\) is generally different from \(A\). In de Gosson [6] and de Gosson and Luef [12] we have proven that the phase space Fourier transforms \(\mathcal {F}A\) and \(\mathcal {F}B\) of the classical observables \(A\) and \(B\) are related by the formula

It would certainly be interesting and useful to have explicit examples; the calculations are rather technical, and part of work in progress [13].

## Footnotes

## Notes

### Acknowledgments

This work has been supported by a grant from the Austrian Research Agency FWF (Projektnummer P20442-N13). I take the opportunity to extend my warmest thanks to Glen Dennis, who was kind enough to point out errors and misprints.

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