In classical mechanics, whether a tossed coin lands heads or tails is considered to be determined by its initial conditions. The outcome only looks random because we cannot precisely know the force, direction, air resistance, and so on. It is not because nature itself is indeterminate. Quantum mechanics, established in the early 20th century, posed a major challenge to this deterministic view of nature. The results of measurements on photons, electrons, and other particles are predicted probabilistically, and the standard description of quantum mechanics alone cannot uniquely determine what happens in any individual measurement. This led to a debate over whether quantum mechanics might be missing physical quantities that have not yet been described: so-called "hidden variables."
In the 1940s, the British mathematician Edmund Taylor Whittaker used the polarization measurement of photons as an example to argue that it is impossible to reproduce quantum mechanical predictions with a deterministic hidden-variable model. At the time, the claim that quantum mechanics had overturned classical determinism was being debated in the journal Nature, and one of the physicists who questioned Whittaker's claim was Erwin Schrödinger (Erwin Schrödinger), the founder of wave mechanics. Schrödinger had shown mathematically that giving photons a hidden variable with continuous values could reproduce the statistical results of polarization experiments deterministically. However, the manuscript was kept in storage without being published, and more than 80 years later it was analyzed in detail by Flavio Del Santo of the University of Geneva.
Schrödinger's Unpublished Manuscript in the University of Vienna Archive
These findings are presented in the paper "Schrödinger's unpublished hidden-variable model in the 1944-45 debate over the 'Collapse of Determinism'" (arXiv:2610.20752, DOI: 10.48550/arXiv.2610.20752), which Flavio Del Santo posted to the preprint server arXiv on October 14, 2026. Del Santo transcribed an unpublished manuscript held in the Erwin Schrödinger Archive at the Central Library for Physics of the University of Vienna, and analyzed its relationship to the 1940s debate and the content of its mathematical model. This is research based on historical archival investigation and theoretical analysis; it did not involve new physical experiments or simulations. It is also currently a preprint, not a paper published in a peer-reviewed journal.
The manuscript under study is a handwritten document titled "Collapse of Determinism," held under the call number "W33-781/9." It carries no date of writing, but the text refers to a letter by Wilfred W. Barkas published in Nature on November 25, 1944. The manuscript was therefore written no earlier than that date. However, there is no evidence to establish that it was written in November 1944, and the exact time of writing is unknown.
The importance of this manuscript does not lie in Schrödinger having completed a new theory that overturned quantum mechanics. At the time, the physics community had widely accepted an argument that, because of the probabilistic nature of quantum mechanics, it was logically impossible to explain the behavior of nature deterministically. Schrödinger examined the reasoning behind that claim and presented a counterexample using a simple mathematical model. Del Santo places this manuscript not as an expression of support for determinism, but as a test of principle showing that a particular impossibility proof does not hold.
What Did Whittaker's Polarizer Argument Overlook?
The target of Schrödinger's rebuttal was the argument Whittaker presented in a 1943 lecture. Focusing on the probability that a photon passes through a polarizer, Whittaker claimed that the outcome could not be explained deterministically by hidden variables. At the heart of the argument was "Malus' law," under which the transmittance varies with the angle between the light's polarization and the polarizer.
When a linearly polarized photon is incident on a polarizer tilted by an angle relative to its polarization direction, the probability that the photon is transmitted is given by:
According to this formula, when the polarization direction and the polarizer's transmission axis coincide, the photon passes with 100% probability. At 90 degrees, where they are perpendicular, the transmission probability is 0%, and at 45 degrees it is 50%. The theoretical transmission probabilities for various angles are as follows.
| Polarizer angle | Transmission probability |
|---|---|
| 0° | 100% |
| 15° | about 93.3% |
| 30° | 75% |
| 45° | 50% |
| 60° | 25% |
| 75° | about 6.7% |
| 90° | 0% |
Whittaker considered whether this continuously varying transmission probability could be explained by a hidden variable a photon carries in advance. What he assumed was one bit of information, made up of the two values 0 and 1, that determines whether the photon "passes" or is "reflected or blocked" by the polarizer. If the photon has this information at the moment it leaves the source, the measurement result is predetermined. But what happens if the experimenter changes the polarizer's angle while the photon is in flight toward it?
Because the one bit of information the photon carries does not change, under this assumption the fraction of transmitted photons would not change when the polarizer is rotated. Under Malus' law, however, changing the polarizer's angle changes the transmission probability continuously. From this contradiction, Whittaker concluded that it is impossible to explain photon behavior deterministically with hidden variables.
However, as Del Santo points out, what this argument ruled out was only a specific model in which the hidden variable is restricted to the two values 0 and 1. Since the polarizer's angle varies continuously, it is not surprising that one bit of information is insufficient to determine the measurement result. What matters is that the result depends not only on the photon's own state but also on the settings of the measuring apparatus. Even in classical mechanics, for instance, if you change the height of the floor while a coin is in flight, the outcome cannot be determined from the coin's state alone. Whittaker's argument therefore did not show that determinism in general is impossible.
Reproducing Malus' Law with a Continuous Hidden Variable
In his unpublished manuscript, Schrödinger removed the one-bit constraint that Whittaker had imposed and constructed a model in which each photon is given a hidden variable with a continuous value. The variable introduced here is , a real number between 0 and 1. It is assumed that this value is uniformly distributed over the interval $[0,1]$ for all photons.
The probability density function of the hidden variable is expressed as follows.
Then, with the polarizer's angle set to , whether the photon is transmitted is determined by the following rule.
Here, $T=1$ represents transmission and $T=0$ represents blocking. Under this rule, once the photon's hidden variable and the polarizer's angle are fixed, the measurement result is uniquely determined. No probabilistic choice is made at the time of measurement for individual photons. What differs from Whittaker's model is that the outcome is determined by taking into account both the continuous value the photon carries and the setting of the measuring apparatus.
So what is the overall result when $N$ photons following this rule are sent into a polarizer? The fraction of photons that satisfy the transmission condition is obtained by integrating the distribution of the hidden variable from to 1. Letting be the number of transmitted photons, we get:
By the trigonometric identity , the fraction of transmitted photons matches Malus' law exactly. In other words, even in a model where the result for each individual photon is determined deterministically, the ensemble as a whole can reproduce the same transmission probability as quantum mechanics. This shows that the mere fact that probabilistic measurement results are obtained cannot rule out the possibility of an underlying deterministic mechanism.
However, to apply this model not just to a single polarizer but to cases where photons pass through multiple polarizers in succession, one more problem must be solved. Photons that have passed the first polarizer are limited to those satisfying . As a result, the hidden variable after transmission is distributed not over the whole original $[0,1]$ but over the narrower interval . Although the distribution remains uniform within this interval, its range has changed, so the same rule cannot be applied directly to the next polarizer.
Schrödinger noticed this problem as well and presented a rule that updates the hidden variable of a photon that has passed through a polarizer, as follows.
This formula is a linear transformation that maps the interval , over which the post-transmission variable is distributed, back onto $[0,1]$. The transformed is uniformly distributed from 0 to 1, so the same form of rule can be used in the next measurement. Moreover, this update is not a probabilistic operation: the result is determined by the pre-transformation value and the polarizer's angle. In other words, not only whether a photon passes through a polarizer but also what hidden state it has after transmission can be described deterministically.
Schrödinger's own manuscript ends its calculation at the point of presenting this update rule. Del Santo examined the continuation mathematically and confirmed that the quantum mechanical predictions are also reproduced for a second polarizer. If the angle of the first polarizer is and that of the second is , their relative angle is . The number of photons passing through the second polarizer is therefore given by:
This result matches the transmittance that quantum mechanics predicts for photons passing through two polarizers in succession. Applying the deterministic state update that Schrödinger introduced yields the correct statistics not only for the first polarizer but also for subsequent polarization measurements. However, this agreement is a theoretical one derived from the equations, not something verified by comparison with experimental data. Moreover, what this model explains is the polarization measurement of a single photon; it has not been shown to reproduce every phenomenon of quantum mechanics.
The relationship between the arguments discussed in this study and Bell's theorem, which appeared later, can be summarized as follows.
| Theory / argument | Main assumptions | What was shown | Scope and limitations |
|---|---|---|---|
| Whittaker's argument (1943) | The photon has a one-bit hidden variable of 0 or 1 | A fixed one bit of information cannot explain the transmittance as the polarizer's angle changes | Did not rule out deterministic models that depend on continuous values or on measuring-apparatus settings |
| Schrödinger's counterexample (1944 onward) | The photon has a hidden variable uniformly distributed over $[0,1]$ and updated deterministically after measurement | Reproduces Malus' law and the statistical predictions of successive polarization measurements | Does not show that nature is deterministic, nor does it explain entanglement of many particles |
| Bell's theorem (1964) | Assumes hidden-variable models satisfying specific conditions, such as locality between spatially separated particles | Correlations in local hidden-variable models are constrained by Bell's inequality | Does not rule out all deterministic theories, including nonlocal hidden-variable theories |
The 1944–1945 Debate and Pelzer's Earlier Work
Key to understanding why Schrödinger did not publish this model is the debate on determinism carried on in British science from 1943 to 1945. One of its triggers was Whittaker's Guthrie Lecture, "Chance, Freewill and Necessity in the Scientific Conception of the Universe," delivered on May 18, 1943. Another was the presidential address "The Classical Determinism and the Quantum Theory," delivered by the Indian physicist Satyendra Nath Bose at the Indian Science Congress on January 3, 1944. Both argued that the development of quantum mechanics had made it difficult to maintain classical determinism.
Following these discussions, in July 1944 the British mathematician H. T. H. Piaggio published an editorial in Nature titled "Collapse of Determinism." This prompted a continuing exchange among Gilbert D. West, Wilfred W. Barkas, Whittaker, and others, bringing renewed attention to the question of whether determinism could be logically refuted. The central points of contention were John von Neumann's 1932 argument that adding hidden variables to quantum mechanics is impossible, and Whittaker's polarizer example.
In this debate, important objections to the arguments against hidden-variable models had already been raised. In a January 1945 letter, West pointed out that Whittaker's argument considered only variables belonging to the photon itself and ignored physical quantities on the measuring-apparatus side, such as the state of the crystal making up the polarizer. Once the interaction between the photon and the measuring apparatus is considered, he argued, there is no need to restrict the factors determining the outcome to the photon alone. This observation shares its way of thinking with the model Schrödinger built in his unpublished manuscript.
Del Santo's investigation also revealed the existence of an even more interesting earlier work. Heinrich Pelzer of Electron Ltd. in Britain had submitted a paper on January 21, 1944, and published "Is an Interpretation of Quantum Phenomena by Hidden Parameters Impossible?" on May 1 of that year (Proceedings of the Physical Society, DOI: 10.1088/0959-5309/56/3/305). This paper presented a mechanical model and a mathematical hidden-variable model as counterexamples to Whittaker's claim.
In Pelzer's model, the photon is given a continuous parameter, and whether it is transmitted is determined by that value and the polarizer's angle. If the distribution of the parameter is set appropriately, Malus' law is reproduced as the transmission probability. According to Del Santo, Pelzer's model becomes mathematically equivalent to Schrödinger's for a single polarizer under the change of variable . In other words, the fact that Whittaker's argument does not rule out determinism in general had already been shown in published research, independently of Schrödinger's manuscript.
However, there is an important difference between the two models. Pelzer stated that when photons pass through multiple polarizers in succession, the distribution of the hidden variable must be reset. But he did not give a concrete deterministic transformation formula for how to update each photon's hidden variable. Schrödinger, on the other hand, specified the linear transformation to described above, building a model in which even the state update after transmission could be treated deterministically.
So why did Schrödinger not publish this result? Del Santo suggests that, in following the Nature debate, Schrödinger may have learned of Pelzer's earlier work and judged that the central point of the rebuttal had already been made. Indeed, West's January 1945 letter cited Pelzer's paper, so Schrödinger likely had the opportunity to learn of it. However, no direct evidence such as letters or diaries has been found in which he states his reason for abandoning publication. The explanation that learning of the earlier work led to the manuscript going unpublished is therefore only Del Santo's conjecture.
Was Schrödinger a Determinist?
The most important caution about this unpublished manuscript is not to take it as evidence that Schrödinger denied the probabilistic nature of quantum mechanics and supported determinism. Schrödinger is sometimes portrayed, alongside Albert Einstein, as skeptical of the indeterministic interpretation of quantum mechanics. But the manuscript Del Santo analyzed records a way of thinking that differs from such a simple understanding. What Schrödinger was questioning was not which of determinism or indeterminism is correct, but the validity of arguments that declare one of them logically impossible.
In the manuscript, Schrödinger states that combining observed facts with logical inference cannot prove that either determinism or indeterminism is impossible. Even a rigorous mathematical argument can apply its conclusion only within the scope of the assumptions made in the proof. Maintaining one position may require adding unnatural hypotheses, but that alone does not prove logical impossibility.
Schrödinger further points out that strong grounds for supporting indeterminism existed even before quantum mechanics was established. This point is also discussed in the Nature Reviews Physics commentary "Was Schrödinger ever a determinist?" (DOI: 10.1038/s42254-026-00968-z), published in 2026 by Del Santo and the physicist Nicolas Gisin. From the 1920s, influenced by the physicist Franz Serafin Exner, Schrödinger had been attentive to the possibility that the laws of nature are inherently statistical in character. From this standpoint, his construction of a hidden-variable model does not necessarily signify a conversion to determinism.
Rather, presenting a counterexample to an argument that logically excludes determinism and believing that nature is in fact deterministic are entirely separate matters. What Schrödinger showed was that a deterministic model reproducing the probabilistic predictions of quantum mechanics can be constructed, at least within the scope of single-photon polarization measurements. Such a model is important as a "proof of principle" that a theory can in principle hold, but it is not evidence that actual nature follows that mechanism.
Nor does this model fundamentally overturn the current debate over hidden-variable theories in quantum mechanics. In 1964, the physicist John Stewart Bell derived constraints that local hidden-variable models must satisfy regarding the correlations of multiple particles in entangled states. This is "Bell's theorem" (DOI: 10.1103/PhysicsPhysiqueFizika.1.195), and subsequent experiments have confirmed quantum correlations that local hidden-variable models cannot explain.
However, what Bell's theorem rules out are hidden-variable theories satisfying specific conditions such as locality, not all deterministic quantum theories. For example, it does not exclude deterministic interpretations with nonlocal properties, such as de Broglie–Bohm theory. Schrödinger's unpublished model, meanwhile, deals with polarization measurements of a single photon and does not address the entanglement correlations arising between multiple spatially separated particles. This model therefore does not refute Bell's theorem, nor does it prove that the probabilistic interpretation of quantum mechanics is mistaken.
The significance of this discovery lies not in a new experimental fact about quantum mechanics but in enabling a more accurate understanding of the history of debates over its foundations. Schrödinger was not trying to prove that determinism is the truth of nature; he was mathematically testing whether the claim that determinism is impossible actually holds. This unpublished manuscript, more than 80 years old, reaffirms the need to distinguish, when considering important questions in physics, between a theory's predictions holding true and the interpretation behind them being proven the only correct one.