The Born Identity

In Quantum Mechanics by Brian Koberlein

Quantum theory is probabilistic by nature. Because of fuzzy effects of quantum indeterminacy, the equations of quantum mechanics can’t tell us exactly what an object is doing, but only what the likely outcome will be when we interact with it. This probability is determined by the Born rule (named after physicist Max Born). The rule has various forms, but in the most common approach it means that squaring the wavefunction of an object yields the probability of a particular outcome. The Born rule works extraordinarily well, making quantum theory the most accurate scientific theory we have, but it is also an assumption. It’s a postulate of quantum theory rather than being derived formally from the model. So what if it’s wrong. 

Even if it is wrong on some level, the great success of quantum physics demonstrates that it certainly works in most cases. But we scientists love to test our assumptions even when they work, so there have been attempts to disprove the Born rule. One approach looked at a triple-slit experiment, which is a variation of the famous double-slit experiment.

Interference patterns from a double slit experiment. Credit: Pieter Kuiper

Interference patterns from a double slit experiment. Credit: Pieter Kuiper

In the double slit experiment, quantum objects such as photons or electrons are beamed through two closely spaced slits. Since we don’t know which slit each object passes through, the possibilities overlap to produce an interference pattern rather than two sharp lines. According to the Born rule, even when we run the experiment one object at a time, the probability distribution of each object follows this pattern. This is exactly what we see experimentally, making it an excellent demonstration of quantum theory.

The triple slit experiment uses three small openings instead of two. While it seems like a trivial change, if done correctly it allows for secondary interactions that could in principle violate the Born rule. Basically, if you just square the total wavefunction of the three slits, you get one probability distribution. If you calculate the secondary interactions you get a different distribution. The difference is extremely small, but in 2010 the experiment was performed, and found the Born rule held within experimental limits.

While this would seem to confirm the Born rule, the precision of the experiment was only to 1 part in 100, which isn’t very high. Unfortunately, even getting that level of precision is difficult with the triple-slit experiment. That’s because the test only works if the wavelength of the experiment is less than the width of the slits. But now a new paper proposes a different approach that might yield even greater precision.

This new approach is a double slit experiment with a twist. Rather than simply letting an object pass through the two slits, an extra level is introduced to shift objects from one slit to the other. This can be done for either slit or both, even without knowing which slit the object passes through. According to the Born rule, any such shift should have no effect on the outcome. If the shift affects the outcome, then the Born rule is violated. Doing this kind of shift in a real experiment will be tricky, but it’s not limited by the wavelengths of the objects, so potentially it would be much more precise than previous experiments.

Given the power of the Born rule thus far, I wouldn’t bet on seeing a violation. But this kind of experiment is a win-win. Either the Born rule continues to reign, or we discover a subtle violation that could lead to a better understanding of things like quantum gravity.

Paper: Sinha, U., et al. Ruling Out Multi-Order Interference in Quantum Mechanics. Science 329, 418-421 (2010).

Paper: James Q. Quach. Which-way double slit experiments and Born rule violationarXiv:1610.06401 (2016).