SCHRÖDINGER'S UNPUBLISHED HIDDEN-VARIABLE MODEL, FOUND IN VIENNA

When you toss a fair coin, there is a fifty percent chance it lands heads. In classical physics, that chance only reflects ignorance: the force of the throw, the air resistance and so on decide the outcome, and with all of them in hand the toss would be predictable. Quantum mechanics is different. Its formalism usually predicts only probabilities. From the start, some physicists wondered whether it was incomplete — whether extra, hidden variables could restore a deterministic picture underneath.

In 1932, John von Neumann published what was widely taken as a proof that this was impossible. Grete Hermann spotted problems with its generality as early as 1935, but her criticism went almost unnoticed. The limits of the proof became widely recognised only after David Bohm’s deterministic hidden-variable theory of 1952 and John Bell’s critique of 1966. In between, according to Flavio Del Santo of the University of Geneva and Constructor University in Bremen, the search for hidden-variable models had essentially vanished for over a decade.

A photon that cannot know

The question came back in the middle of the war. On 18 May 1943, the mathematician and physicist Edmund Taylor Whittaker gave a lecture in London claiming that quantum mechanics had broken “the bonds of necessity”. On 3 January 1944, Satyendra Nath Bose said much the same in Delhi: “Fifty years ago the belief in causality and determination was absolute. To-day physicists have gained knowledge but lost their faith.”

Whittaker added a simple argument. A polarized photon passes a polarizer with probability cos²φ, where φ is the angle between its polarization and the polarizer: this is Malus’ law. Suppose each photon carries one hidden bit, 0 for “pass” and 1 for “blocked”. For N photons, N cos²φ of them must carry a 0. But you can turn the polarizer while the photons are already flying. Their bits cannot change, so the fraction that passes cannot change either — yet Malus’ law says it must. Hidden variables, Whittaker concluded, are impossible.

Eleven replies, and a twelfth

The argument set off a debate in Nature: eleven contributions in about a year, ten in that journal and one in the Proceedings of the Physical Society. The experimentalist Gilbert West objected that “there are many who, knowing how easily assumptions can lurk in analytical work, refuse to be mathematically bludgeoned.” He then made the key point: Whittaker had only shown that causality fails if all the hidden parameters belong to the photon; the crystal of the polarizer has its own. The mathematician Henry Piaggio closed the exchange in March 1945 with a truce, quoting Bertrand Russell: no decisive reason for complete determinism, but no reason against it either.

What nobody knew until now is that there was a twelfth reply. It survives as an undated handwritten draft titled “Collapse of Determinism” in the Erwin Schrödinger Archive of the Austrian Central Library for Physics, at the University of Vienna. It mentions the debate up to a letter of 25 November 1944, so it was written after that. Del Santo publishes its first full transcription and analysis.

A few lines that rescue determinism

“I too have been puzzled by this conclusion,” Schrödinger wrote. “For the derived absurdity is visibly produced by the simplifying assumption of two discrete parameter values only.”

His fix: give each photon a hidden real number λ between 0 and 1, spread evenly, and let the photon pass when λ ≥ sin²φ. The share of photons that pass is then exactly cos²φ — Malus’ law, with every outcome fixed in advance by the photon’s number and the polarizer’s angle. To handle a second polarizer, Schrödinger added a deterministic update: a transmitted photon’s number becomes λ′ = (λ − sin²φ)/cos²φ, which spreads the survivors evenly over 0 to 1 again. Del Santo shows that a second polarizer at angle φ′ then passes a fraction cos²(φ′ − φ) of them, as quantum mechanics predicts. Schrödinger also gave the rule for reflected photons: “I can see no inconsistency in assuming these changes.”

Del Santo adds a point that seems to have escaped everyone in the debate, Whittaker and Schrödinger included: there is nothing specifically quantum about one bit being too little. A coin also needs more than one bit if you can move the floor while it is in the air.

A determinist who wasn’t

The rest of the manuscript turns to philosophy. “Experimental evidence + logical deduction can never prove that either the deterministic or the indeterministic view is impossible,” Schrödinger wrote, and “the strongest plea in favour of indeterminism is older than quantum mechanics.”

For Del Santo, the model is therefore a proof of principle, not a conversion. Since the early 1920s, under Franz Exner’s influence, Schrödinger had held that “Nature’s laws are of thoroughly statistical character” and called determinism a “philosophic prejudice”.

Why did he never send it? Heinrich Pelzer had published hidden-variable answers to Whittaker in May 1944, cited in the Nature debate in January 1945. His quantitative version, in a short note added in proof, is equivalent to Schrödinger’s for a single polarizer, though without an explicit update rule for each photon. Del Santo suggests that Schrödinger, learning of it, may have judged the point already made.

The history that followed is in the paper too. Bohm revived hidden variables in 1952; in 1964 Bell showed that local hidden-variable models make testable predictions, and experiments have since ruled that class out. Einstein, Schrödinger and Bohm — the three names most often cast as champions of determinism — all built such models at some point. None of them, Del Santo argues, was simply a determinist.