Review of Conservation laws and the foundations of quantum mechanics, Yakir Aharonov, Sandu Popescu and Daniel Rohrlich, Proceedings of the National Academy of Sciences 2023 Vol. 120 No. 41 e2220810120;

Aharonov, Popescu and Rohrlich (APR) address the important issue of conservation laws for quantum processes and whether they hold only on average over an ensemble of trials or whether instead they hold on every trial. To discuss this, they consider a thought experiment involving a particle moving in a circle with angular momentum but with an unusual initial wavefunction which is a superposition of angular momentum eigenstates but also containing a superoscillating region, “superoscillating” being a phenomenon in which frequency variations can occur faster than the fastest Fourier component of a band-limited function. The experiment consists in measuring via a projector if the particle is in the superoscillatory region. Notably, when the particle is found there, it can have angular momentum which is much higher than all the angular momentum components that it initially had due to the effects of the superoscillating region, suggesting that the angular momentum may not be conserved on every trial. However, it was found that the state preparation process does provide a compensating angular momentum which can then be transferred to the particle at the time of measurement. The state preparation is the source of the high angular momentum gained by the particle by the superoscillations, while the measuring device subsequently allows transfer of this angular momentum to the particle. Thus, in a detailed analysis of both the measurement process and the initial state preparation process, APR find that the angular momentum of the system is indeed conserved on every trial even in the presence of superoscillations.

Superoscillations were previously introduced to the physics community by Aharonov and coworkers but they have been known in signal processing, for example in the context of oversampling in which the sampling frequency is significantly higher than the Nyquist rate [1]. Ordinarily, a signal bandlimited to μ/2 Hz is not expected to oscillate at frequencies higher than μ Hz. However, for signals that have amplitudes of widely different scales, it is actually possible to have finite energy signals, called superoscillations, that oscillate arbitrarily fast over long time intervals [2]. The reason that superoscillations do not contradict information theory is that the superoscillations’ amplitude must decrease exponentially with the length of the superoscillating interval. This allows superoscillatory information compression to an arbitrary extent but demands that the signal power grows exponentially with the length of the part of the signal that is superoscillatory.

The present paper is a follow-up to a previous paper by the same authors [3] in which they came to the opposite conclusion, that conservation held only on average and not on every trial, though they stated that the issue needs to be revisited.

In the earlier paper, they examined energy conservation of a photon in a box, again with an initial wavefunction which is a superposition of energy eigenstates with a maximum energy Emax but also containing a superoscillating region where the wavefunction oscillates faster than any of the actual Fourier components, as indicated in the sketch. The photon is given a chance to escape by the opening of a shutter in which case the escaping photon can apparently have energy higher than it had inside, E>Emax , which would violate energy conservation on every trial. The authors further show that the process of opening the shutter does not provide any compensating energy. This previous APR paper suggested that superoscillations may provide a loophole allowing conservation on every trial to be violated but yet still allowing conservation to hold on-average.

The APR result is a useful demonstration of how conservation holds on every trial for quantum systems in situations in which the wavefunction contains superoscillating regions. Although the particle can gain high angular momentum from the superoscillating region, they identify a mechanism involving state preparation by which the high angular momentum is compensated during the measurement process. This mechanism applies both to the angular momentum conservation thought experiment of the present paper and to the energy conservation thought experiment of the previous APR paper which was thought to violate conservation on every trial. The result is therefore useful in demonstrating that superoscillations do not provide a loophole for conservation on every trial.

APR analyze only a particular model of angular momentum conservation with a wavefunction exhibiting superoscillations regions and demonstrate conservation on every trial only in this case. They do not investigate conservation laws within the context of a general theory of the measurement problem and thus cannot conclude how general this result is or identify general principles within a measurement theory that would require conservation on every trial. Nevertheless, they do conjecture that the result is general, but without further justification:

“In particular, we conjecture that when such a full analysis of any conservation experiment is performed conservation is obeyed in any individual case, not only statistically.”

On-average conservation is not the same as strict energy conservation on individual trials in the historic development of quantum mechanics. The conservation laws had the status of a fundamental law since the mid-19th century but would not be tested experimentally at the level of individual microscopic processes such as atomic transitions or collisions of electrons until 1925 and this played an important role in the Bohr-Kramers-Slater debate [4, p.276]. In 1924 a theory by Bohr, Kramers, and Slater (BKS) [5] was put forward in which the electromagnetic field was treated as a wave and not as a particle. BKS were not able to develop a theory that would conserve energy on all individual scattering events and at the same time retain the property that the electromagnetic field acted as a wave. BKS allowed for individual quantum transitions to violate conservation of energy, but energy would still be conserved on average. BKS tried to take the duality between particle and wave-models as the starting point for the interpretation of quantum theory. The waves would play the role of a probability field, even though this forces energy conservation for individual processes to be abandoned. The theory was opposed by Einstein, and experiments were conducted by Bothe and Geiger and independently by Compton and Simon in 1924-1925 that examined individual scattering events. It was found that energy and momentum are conserved on individual scattering events. BKS theory was thus found to be incorrect and this added insight into the issue of wave-particle duality. For the most part, from 1925 through to the present there appears to be agreement on this.

More generally, based on our work on measurement theory we also have made the prediction that conservation and momentum are conserved not only on-average but also on all individual trials [4, p.413]:

The authors desire to follow-up their work by looking at implications of conservation on all trials as they state, “We take this as a good indicator of the validity of this line of enquiry and believe these results are only the tip of an iceberg, part of a more general structure concerning conservation laws.” We strongly agree here with APR, and this has been our own approach to resolving the measurement problem.

Although not part of the review, we would like to add an advertisement that the basic and intermediate theory of our investigation is largely now completed and currently being documented for publication. The approach can be seen at It is interesting to note that when we originally began our investigation, we also initially developed models for which energy and momentum were only conserved on-average and not on all individual trials. However, we changed our view to conservation on all individual trials as we refined our initial models and as internon theory progressed.


[1] Berry MV, “Faster than fourier,” in Quantum Coherence and Reality: In Celebration of the 60th Birthday of Yakir Aharonov , Anandan J.S. and Safko J.L., Eds., Singapore: World Scientific, 1994, pp. 55–65.
[2] P. J. S. G. Ferreira and A. Kempf, “Superoscillations: Faster than the Nyquist rate,” IEEE Transactions on Signal Processing, vol. 54, no. 10, pp. 3732–3740, Oct. 2006, doi: 10.1109/TSP.2006.877642.
[3] Y. Aharonov, S. Popescu, and D. Rohrlich, “On conservation laws in quantum mechanics,” Proceedings of the National Academy of Sciences, vol. 118, no. 1, pp. 1–7, 2020, doi: 10.1073/pnas.1921529118/-/DCSupplemental.y.
[4] M. Steiner and R. Rendell, The Quantum Measurement Problem. Inspire Institute, 2018.
[5] N. Bohr, H. A. Kramers, and J. C. Slater, “The quantum theory of radiation,” Philosophical Magazine and journal of science, vol. 47, pp. 785–802, 1924.

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