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arXiv:2401.14261 · Proceedings of the National Academy of Sciences

Conservation laws and the foundations of quantum mechanics

Yakir Aharonov, Sandu Popescu, Daniel Rohrlich

Published Oct 2, 2023• 13 citations
arXivHTMLPDF

Abstract

e1921529118 (2021)], it was argued that while the standard definition of conservation laws in quantum mechanics, which is of a statistical character, is perfectly valid, it misses essential features of nature and it can and must be revisited to address the issue of conservation/nonconservation in individual cases. Specifically, in the above paper, an experiment was presented in which it can be proven that in some individual cases, energy is not conserved, despite being conserved statistically. It was felt however that this is worrisome and that something must be wrong if there are individual instances in which conservation does not hold, even though this is not required by the standard conservation law. Here, we revisit that experiment and show that although its results are correct, there is a way to circumvent them and ensure individual case conservation in that situation. The solution is however quite unusual, challenging one of the basic assumptions of quantum mechanics, namely that any quantum state can be prepared, and it involves a time-holistic, double nonconservation effect. Our results bring light on the role of the preparation stage of the initial state of a particle and on the interplay of conservation laws and frames of reference. We also conjecture that when such a full analysis of any conservation experiment is performed, conservation is obeyed in every individual case.

DOI
10.1073/pnas.2220810120

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e1921529118 (2021)], it was argued that while the standard definition of conservation laws in quantum mechanics, which is of a statistical character, is perfectly valid, it misses essential features of nature and it can and must be revisited to address the issue of conservation/nonconservation in individual cases. Specifically, in the above paper, an experiment was presented in which it can be proven that in some individual cases, energy is not conserved, despite being conserved statistically. It was felt however that this is worrisome and that something must be wrong if there are individual instances in which conservation does not hold, even though this is not required by the standard conservation law. Here, we revisit that experiment and show that although its results are correct, there is a way to circumvent them and ensure individual case conservation in that situation. The solution is however quite unusual, challenging one of the basic assumptions of quantum mechanics, namely that any quantum state can be prepared, and it involves a time-holistic, double nonconservation effect. Our results bring light on the role of the preparation stage of the initial state of a particle and on the interplay of conservation laws and frames of reference. We also conjecture that when such a full analysis of any conservation experiment is performed, conservation is obeyed in every individual case.
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.
In quantum mechanics, conservation laws are usually understood only statistically and only in the context of unitary evolution: during such evolution, if an observable commutes with the Hamiltonian, its expectation value and Born-rule probability distribution remain unchanged. Individual measurement outcomes, however, disrupt even this weak form of statistical conservation. Recently, though, it has been argued that a stronger, non-statistical form of conservation, valid at the level of individual measurement outcomes, can be recovered. The proposal relies on the claim that any superposition of eigenstates of a conserved quantity must ultimately be prepared relative to another physical system, which serves as the relevant reference frame and remains correlated with the system being prepared. On this enlarged description, the change observed in the measured system is claimed to be exactly compensated by an opposite change in the preparer, thereby restoring conservation at the level of individual measurement outcomes. We analyze this proposal and argue that it faces significant conceptual difficulties. We show that the proposed mechanism is not adequately justified and, moreover, does not establish conservation in individual measurement outcomes as claimed.
Significance Conservation laws are one of the most important aspects of nature. As such, they have been intensively studied and extensively applied, and are considered to be perfectly well established. We, however, raise fundamental question about the very meaning of conservation laws in quantum mechanics. We argue that, although the standard way in which conservation laws are defined in quantum mechanics is perfectly valid as far as it goes, it misses essential features of nature and has to be revisited and extended.
It is often claimed that Einstein is wrong about quantum mechanics. However, when comparisons are made with respect to theoretical foundations rather than experimental results Einstein’s theories are found to be superior. Although statistical formulations of quantum mechanics correctly describe energy and momentum exchange with remarkable accuracy there are individual cases, paradoxically, that violate the conservation laws. A recent study finds that in order to account for observed violations frames of reference involved in preparation of the initial state must be explicitly taken into account [1]. We show here that the same conclusion was arrived at in a much simpler way by Einstein over 100 years ago. Deficiencies in the foundations of non-relativistic theory are corrected by introducing Hamilton’s principle and deriving relativistic equations of motion. This allows quantum mechanical formalism in abstract space to be replaced by the superposition of particle field geometries in real space. Applying the calculus of variations to the Schrödinger wave equation reveals that the wave function is incomplete because it requires twice the allowable action minimum.
As in classical physics, also in quantum mechanics: physical processes are bound to obey conservation laws. However, there is an important difference. While in classical physics we can suppose that all variables have definite values (irrespective of whether we know them or not), quantum mechanics allows them to be intrinsically uncertain. In those cases conservation laws apply only to ensemble averages (expectation values), and hence are less informative. They show their full strength in measurement processes. Then the measured values satisfy the respective conservation laws in any individual experiment. Let me remind you of spin measurements on a system of two spin-½ particles, being in a state with zero total spin, which we used in Section 5.5 to illustrate the EPR paradox. The spin components, with respect to an arbitrarily chosen direction, of the two particles are indefinite; however, measuring them yields definite values (+½ or −½) that add up to zero, thus confirming the angular conservation law. In the following, we will consider some realistic experiments for which conservation laws provide a simple explanation. Energy and momentum conservation For an understanding of the interaction between matter and radiation, Einstein's photon concept (actually, he spoke of ‘light quanta’) proved to be very helpful. He assumed photons to be particle-like constituents of electromagnetic radiation, in particular light. They have the energy h ν, where ν is the frequency of the radiation field. (Since the electromagnetic spectrum extends over an extremely wide range, there is a huge number of different sorts of photons.) The crucial point is that photons are generated or annihilated only as a whole. This property gives rise to typical quantum effects.
Conservation principles are essential to describe and quantify dynamical processes in all areas of physics.Classically, a conservation law holds because the description of reality can be considered independent of an observation (measurement).In quantum mechanics, however, invasive observations change quantities drastically, irrespective of any classical conservation law.One may hope to overcome this nonconservation by performing a weak, almost noninvasive measurement.Interestingly, we find that the nonconservation is manifest even in weakly measured correlations if some of the other observables do not commute with the conserved quantity.Our observations show that conservation laws in quantum mechanics should be considered in their specific measurement context.We provide experimentally feasible examples to observe the apparent nonconservation of energy and angular momentum.