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arXiv:1609.05041 · Proceedings of the National Academy of Sciences

On conservation laws in quantum mechanics

Yakir Aharonov, Sandu Popescu, Daniel Rohrlich

Published Dec 28, 2020• 50 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1073/pnas.1921529118

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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.
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.
Although quantum mechanics correctly predicts what it is possible to observe (the emissions) it ignores the other half of natural phenomena, what cannot be observed (the absorptions), thereby violating the conservation laws. By describing only one-half of quantum mechanics conceptual difficulties such as wave function collapse, infinite paths, non-locality, and abstract mathematical models appear out of nowhere. The deficiencies are corrected in non-relativistic theory by showing that the unobserved absorption energies of off-diagonal elements in matrix mechanics are essential and that matrix and wave mechanics are equally important for formulating a complete quantum mechanics. Due to the conservation of energy absorption must be carried out to completion before energy emission can begin. When relativistically correct equations of motion are derived they show that the wave function is incomplete because it describes the excitation and decay of an electron with twice the allowable action minimum. Einstein’s unfinished quantum theory forms the mathematical foundation leading up to and underlying the proposed mathematical model. Keywords Conservation laws; Quantum mechanics; Hamilton’s principle; Energy; Momentum; Path integral formulation; Matrix mechanics
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.
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 quantum mechanics correctly describes energy exchange due to emissions with remarkable accuracy it ignores momentum exchange (the absorptions), thereby failing to recognize their causal relationship due to the conservation laws. The deficiencies in non-relativistic theory are corrected by introducing Hamilton’s principle and deriving relativistic equations of motion. This allows wave function behavior to be described as the combined action of a particle and an associated localized field. The calculus of variations is applied to the Schrödinger wave equation to show that it is incomplete because it yields twice the allowable action minimum.
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.