Insider Brief
- A new study analyzes an unpublished Schrödinger manuscript that challenged claims that hidden information could not explain apparently random quantum outcomes.
- Schrödinger’s model reproduced the expected results of polarization measurements, serving as a mathematical counterexample rather than a complete theory of quantum physics.
- The manuscript illustrates the complexity of Schrödinger’s views, showing that he could construct a deterministic model while favoring fundamental randomness in nature.
- Image: Photo by Robert Stump on Unsplash
PRESS RELEASE — An unpublished manuscript by Erwin Schrödinger shows how apparently random quantum results could follow definite rules, challenging an argument that such explanations were impossible.
The handwritten draft, analyzed in a new historical study posted on arXiv, adds a missing contribution to a wartime dispute over whether quantum mechanics had ended the scientific case for a predictable universe. It also complicates a familiar picture of Schrödinger as a defender of determinism, the idea that events are fixed by preceding conditions.
According to physicist Flavio Del Santo, Schrödinger constructed his model to expose a weakness in an argument, rather than to establish how nature actually works. His draft combined a mathematical counterexample with a defense of the possibility that randomness is fundamental.
The study presents what Del Santo describes as the first transcription and analysis of the manuscript with a central finding concerning the limits of scientific claims. Schrödinger showed that ruling out one proposed explanation does not necessarily rule out every explanation of the same kind.
A Missing Contribution
The document survives in the Erwin Schrödinger Archive at the University of Vienna’s Austrian Central Library for Physics. Titled “Collapse of Determinism,” it consists of three handwritten pages and is undated.
Del Santo, affiliated with the University of Geneva and Constructor University in Germany, places it within a debate that unfolded in 1944 and 1945. References in the draft establish that Schrödinger wrote it after a letter published in Nature on Nov. 25, 1944.
The dispute grew out of one of quantum physics’ stickiest problems. The theory generally predicts the probabilities of measurement results, rather than specifying what will happen in each individual case.
Some physicists asked whether those probabilities reflected incomplete knowledge. It’s often explained through a simple coin toss. When you toss a coin and shout, “heads or tails,” the outcome appears random. But in a classical description, an array of subtle influences — from how hard the coin is tossed to the speed of the wind around it, technically influences the coin’s motion. If coin-tossers knew enough about these forces, the reasoning goes, could they also know the exact outcome of their throw?
In the quantum world, uncertainty seems to come with the territory, for better or worse. But, the question becomes could particles carry hidden information that determines what they do, even if scientists cannot observe it? Such proposed information became known as hidden variables.
In 1932, mathematician John von Neumann presented a result widely understood to rule out adding hidden variables to quantum mechanics. Del Santo recounts how that conclusion discouraged work on deterministic alternatives, although its generality had been challenged by mathematician Grete Hermann in 1935.
The issue resurfaced during World War II through contributions by British mathematician and physicist Edmund Taylor Whittaker and Indian physicist Satyendra Nath Bose. Both questioned classical determinism and invoked von Neumann’s work.
Whittaker also offered a simpler argument involving light — and it was that argument that became the target of Schrödinger’s unpublished response.
A Rule for Individual Photons
Whittaker’s example concerned polarized light passing through a polarizer, a device that selects light according to the orientation of its electric field.
The angle between the light’s polarization and the polarizer determines how much light passes through. At the level of individual photons, or particles of light, quantum mechanics expresses that relationship as a probability.
Whittaker considered a hidden property with only two possible values, determining whether a photon would pass through or be reflected. He argued that this arrangement could not reproduce the changing transmission rates when the polarizer was turned.
Schrödinger identified a restriction built into that reasoning. Why should the hidden property have only two possible values?
His alternative assigned each photon a number between zero and one, with the numbers evenly distributed across the incoming beam. Whether a photon passed through then depended on both its number and the polarizer’s angle.
Changing the angle changed the cutoff for transmission. The photon did not need advance knowledge of the setting.
Across many photons, the rule reproduced the expected transmission rate, a relationship known as Malus’ law. Yet each individual outcome was determined by the proposed number and the apparatus setting.
Schrödinger also addressed what would happen after the first measurement. Simply selecting the photons that passed through would leave a different distribution of hidden numbers, creating a problem for subsequent polarizers.
He supplied a definite mathematical rule that changed each photon’s number after transmission or reflection. That restored the distribution needed for further measurements.
Del Santo reconstructs this procedure and explicitly demonstrates how it produces the quantum prediction for a second polarizer. He also distinguishes that extended explanation from the construction that Schrödinger wrote down.
The result was a counterexample to Whittaker’s impossibility claim. It showed that the observed statistics in this particular arrangement could be reproduced by a deterministic model. It did not establish that photons actually carry such numbers.
Why the Draft Stayed Unpublished
The question then becomes why didn’t Schrödinger publish the paper. The most likely reason might be that someone already beat him to it.
In 1944, Austrian-born physicist Heinrich Pelzer published models challenging Whittaker’s argument. His paper included mechanical analogies and a brief mathematical construction that reproduced the transmission probability for a single polarizer.
According to Del Santo, that construction is mathematically equivalent to Schrödinger’s for the single-polarizer case, despite using a different description.
Pelzer also recognized that the hidden information would need to be redistributed after transmission. Schrödinger went further by specifying a deterministic rule for changing each photon’s hidden value, according to Del Santo.
Del Santo suggests that Schrödinger may have learned of Pelzer’s work through the continuing Nature correspondence and concluded that his main objection had been made.
That explanation remains a possibility, rather than a documented reason for abandoning publication. The manuscript shows that Schrödinger followed the debate through late November 1944, but the study does not establish what happened afterward.
Del Santo writes: “If he subsequently learned of Pelzer’s result, he may have regarded the essential point as already established and therefore decided not to pursue publication of his own reply. Schrödinger’s manuscript was nevertheless considerably broader in scope and philosophically more ambitious, combining a concrete hidden-variable model with a reflection on the limits of impossibility proofs and on the respective status of deterministic and indeterministic descriptions.”
The draft’s philosophical discussion also helps explain why it should not be read as a declaration that nature must be deterministic.
Del Santo connects it to Schrödinger’s earlier views that physical laws could be fundamentally statistical. On this reading, Schrödinger could reject an attempted proof against determinism while still favoring indeterminism himself.
Constructing a model was a way to test an argument, rather than an endorsement of the model as reality.
New Ground, or Old Sod?
The manuscript adds evidence that challenges to sweeping claims against hidden variables extended beyond the best-known published exchanges, but it also requires careful limits.
Schrödinger’s construction addressed a specific sequence of polarization measurements. The study does not present it as a theory accounting for all quantum phenomena, and it does not overturn later experimental findings.
In 1964, physicist John Bell showed that a class of hidden-variable theories makes predictions that can be tested against quantum mechanics. Subsequent experiments have all but ruled out that class, known as local hidden-variable models, Del Santo writes.
While the this abandoned paper may not have much of a scientific impact, the study does have historic and philosophical impact by exploring the ongoing — and, at times, vicious — debate between determinists and indeterminists.
“The debate between determinism and indeterminism has divided physicists for over a century and continues to do so today,” writes Del Santo. “From the outset, it has carried a strong ideological charge, often taking the form of a partisan confrontation between two opposing camps, each electing its own champions and, at times, instrumentalizing them in support of positions that ran against their own views.”
However, he also points out that some of the figures in that debate — scientists like Albert Einstein, Schrödinger and David Bohm — often cast as champions of determinism, held more complicated views. Einstein was uneasy about deterministic explanations of quantum mechanics, Bohm criticized a mechanically predetermined universe, and Schrödinger favored indeterminism.
“What they have in common is that they devoted much of their intellectual lives to questions of causality and indeterminism, which led them, at times, to explore deterministic models and their limits,” concludes Del Santo. “Such models could serve as proofs of principle against widespread beliefs or alleged impossibility claims, as strikingly illustrated by the case of Schrödinger reconstructed here, but never as expressions of a dogmatic commitment to determinism.”