Born--Jordan Quantization and the Equivalence of Matrix and Wave Mechanics

TL;DR

This paper revisits Born-Jordan quantization, demonstrating its necessity for the equivalence of matrix and wave mechanics, and discusses its advantages over Weyl quantization, with potential experimental verification via weak measurements.

Contribution

It clarifies the importance of Born-Jordan quantization for the consistency of quantum mechanics and provides methods to explicitly compute it for arbitrary variables.

Findings

Born-Jordan quantization ensures energy conservation in matrix mechanics.

Wave mechanics aligns with matrix mechanics only when using Born-Jordan quantization.

Weak measurement experiments could identify the correct quantization scheme.

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 Schr\"odinger'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.