Conservation Laws For Every Quantum Measurement Outcome

TL;DR

This paper argues that conservation laws in quantum mechanics can be applied to individual measurement outcomes, not just statistically over many measurements, and demonstrates this for angular momentum, with implications for local conservation.

Contribution

It introduces a framework showing conservation laws hold in each individual quantum measurement, extending beyond traditional statistical interpretations, with general applicability.

Findings

Conservation applies to each individual measurement outcome.

Conservation can be localized to the system and its reference frame.

The approach is demonstrated for angular momentum on a circle.

Abstract

In the paradigmatic example of quantum measurements, whenever one measures a system which starts in a superposition of two states of a conserved quantity, it jumps to one of the two states, implying different final values for the quantity that should have been conserved. The standard law of conservation for quantum mechanics handles this jump by stating only that the total distribution of the conserved quantity over repeated measurements is unchanged, but states nothing about individual cases. Here however we show that one can go beyond this and have conservation in each individual instance. We made our arguments in the case of angular momentum of a particle on a circle, where many technicalities simplify, and bring arguments to show that this holds in full generality. Hence we argue that the conservation law in quantum mechanics should be rewritten, to go beyond its hitherto statistical formulation, to state that the total of a conserved quantity is unchanged in every individual measurement outcome. As a further crucial element, we show that conservation can be localised at the level of the system of interest and its relevant frame of reference, and is independent on any assumptions on the distribution of the conserved quantity over the entire universe.