The exact story is contested, but in every version, it has a grim ending. A man discovers a new mathematical truth and loses his life for it. In every version of the narrative, the ancient Greek philosopher Hippasus of Metapontum drowns at sea, punished either by the gods or by his peers.
In most tellings, it is discovering that there are numbers that can’t be expressed as an exact fraction, their decimal places running into oblivion without repeating. These irrational numbers – like the square root of 2 or pi – greatly upset Hippasus’s peers, the cultish Pythagoreans, who may have tossed him overboard for bringing them to light. More than 2000 years later, irrational numbers were embedded in the mathematical foundations of quantum mechanics.
The physicists who wrote the theory’s early formulas didn’t have to face stormy seas, but their peers called their work shocking, bordering on heretical. We have made our peace with irrational numbers for a century, even though the quantum realm that includes them is bizarre, hosting particles that may not exist if we aren’t looking at them, or entanglement that can tie two particles together across the cosmos – what Albert Einstein called “spooky action at a distance”. Why mathematicians want to destroy infinity – and may succeed But one physicist says it doesn’t have to be this way.
In a bold new proposal, he has drafted a way out of that unhappiness: banishing the irrational numbers that cost Hippasus his life. Palmer argues that this new theory wouldn’t suffer any oddness – gone is entanglement and its spookiness, away with the strangeness of superpositions that allow two contradictory possibilities at once. Crucially, his rational quantum mechanics (RaQM) would put a fundamental limit on the power of quantum computers, which means it could not only be tested in the near term, but may also imperil a billion-dollar industry.
In fact, if Palmer’s proposal is correct, quantum computers will eventually fail to run the famous Shor’s algorithm – a quantum recipe for factoring very large numbers that could break widely used encryption – dispelling decades of dreams that have motivated and justified their development since the 1990s. To be clear, Palmer is no Pythagorean. He has no problem with irrational numbers, per se.
For him, the true mathematical villain of the story of quantum mechanics is John von Neumann, who set out its first rigorous framework. Quantum mechanics classes still teach von Neumann’s maths, starting with the Hilbert space, the mathematical arena where all quantum action happens. To analyse the behaviour of something quantum like an electron, or to predict its future behaviour, physicists assign it a mathematical object that lives in Hilbert space.
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