The physicists have been troubled by the same scenario for three decades.

They wrote about a wave phenomenon called superoscillation in 1990.

In December 2020, the trio published a paper in the Proceedings of the National Academy of Sciences explaining the problem of superoscillation in quantum systems. The law states that the energy of an isolated system is more than a physical principle. It is now understood that it is an expression of the fundamental symmetries of the universe.

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Physicists are divided on whether the new paradoxes exposes a genuine violation of the conserved energy. Their attitudes toward the problem depend on whether individual experimental outcomes in quantum mechanics should be considered seriously. The hope is that by resolving the puzzle, researchers will be able to clarify some of the most subtle and strange aspects of quantum theory.

Mirror Trick

The scenario in question is similar to opening a box full of red light, low-energy waves and a high-energy radiation shoot out. How can this happen?

The key ingredient is a phenomenon called superoscillation, which seems to contradict what a physics student learns about waves.

Any wave can be represented as a sum of waves of different frequencies. The students learned that a wave can only be oscillated as fast as its component. It should stay red if you combine a bunch of red light.

The collective wave that wiggles faster than any of the others was discovered by Aharonov and Popescu around 1990. Michael Berry showed that it is possible to combine sound waves below 1 hertz in order to play Beethoven's Ninth symphony. The rediscovery of superoscillation inspired physicists to invent new applications, from high-resolution images to new radio designs.

A portrait of Sandu Popescu, a mostly bald man with glasses wearing a suit and tie.

It doesn't contradict any laws of physics. The situation that they encountered when they applied the concept to quantum mechanics was downright strange.

In quantum mechanics, a particle is described by a wave function, which is a kind of wave that varies in amplitude. Wave functions can be expressed as sums of waves.

The wave's energy is proportional to its Frequency. When a wave function is a combination of multiple waves, the particle is in a position of energies. The wave function seems to collapse to one of the energies in the superposition when it is measured.

The paradoxes were exposed using a thought experiment. If you have a photon trapped inside a box, the wave function has a superoscillatory region. Put a mirror in the photon's path where the wave function superoscillates, and keep it there for a short time. The mirror will bounce the photon out of the box if the photon is close enough to the mirror.

We are dealing with the photon's wave function here. The wave function doesn't collapse since the bounce doesn't constitute a measurement. Most of the wave function remains in the box, but a small piece near where the mirror was inserted leaves the box and heads toward the detector.

The piece has been plucked from the rest of the wave function and is now identical to a photon of higher energy. The entire wave function collapses when this piece hits the detector. There is a small chance that the detector will register a high-energy photon when it does. It is like a box of red light coming out of the ground.

The measurement scheme gives more energy to the photon than the wave function would allow. Where did the energy come from?

Legal Ambiguities

In 1915, the mathematician Emmy Noether proved that the quantities of energy and momentum come from nature. The rule that the equations governing particles stay the same from moment to moment conserves energy. The stable quantity is energy. In situations where gravity warps the fabric of space-time, energy isn't conserved since this warping changes the physics in different places and times. Physicists agree that time-translation symmetry should hold for light in a box.

The equations of quantum mechanics are complicated to apply.

In classical mechanics, you can always check the initial energy of a system, let it evolve, then check the final energy, and you will find that the energy stays constant. The wave function of a quantum system is collapsing, preventing it from evolving as it otherwise would have. If you want to check the energy of a quantum system, you have to run an experiment many times and check the initial and final energy at the same time. The evolution of the system should match the distribution of energies.

Popescu says the thought experiment is compatible with this version of energy saving. Only on rare occasions will the photon be found in the superoscillatory region. The energy budget will stay balanced over the course of many runs.

Physicists object to the thought experiment because it suggests that energy saving can be done in individual instances. Even if it is hard to check, David Griffiths, a professor at Reed College in Oregon and author of standard textbooks on quantum mechanics, maintains that energy must be conserved in each individual experiment.

Marletto agrees. If it looks like your experiment is violating the law, you're not looking hard enough. There are a number of ways in which the excess energy could come from, one of which is not fully taking into account the environment.

Popescu and his colleagues thought that the photon would get more energy from the mirror, but they thought the mirror's energy wouldn't change.

The search continues for a solution to the apparent paradoxes and a better understanding of quantum theory. Physicists have used these puzzles in the past. John Wheeler once said, "No progress without a paradox!"

Popescu said, "You're never really going to understand what quantum mechanics is if you ignore such questions."