Flavio Del Santo of the University of Geneva and Constructor University has unearthed a previously unknown manuscript revealing that Erwin Schrödinger drafted an explicit hidden-variable model in response to a 1944-45 debate in Nature over the foundations of quantum mechanics. This work demonstrates that hidden-variable models were not necessarily ruled out by the physics of the time.

Schrödinger’s Drafted Hidden-Variable Model Challenges Determinism

The model directly addressed an argument presented by J.W.T. Whittaker regarding Malus’ law, a principle governing the intensity of light after passing through a polarizer, and Schrödinger believed a simplifying assumption in Whittaker’s work led to a flawed conclusion. He detailed his counter-argument in a manuscript now fully transcribed and analyzed, revealing a subtle approach to the question of hidden variables and their compatibility with quantum theory.

I reconstruct the historical context of this forgotten manuscript and publish here the first transcription and analysis. Schrödinger’s proposal should be understood as a proof of principle rather than as evidence of his commitment to determinism. Schrödinger’s manuscript, however, went beyond simply offering an alternative model; it also engaged with the broader philosophical implications of determinism versus indeterminism, reflecting a period of intense scrutiny regarding the foundations of quantum mechanics.

The physicist’s analysis of Whittaker’s argument centered on the assumption of only two discrete parameter values, and Schrödinger proposed a more complex system to account for the observed phenomena. He believed this approach could resolve the apparent absurdity Whittaker identified, though he did not necessarily view this as conclusive proof of determinism.

Indeed, Schrödinger’s manuscript reveals a more complex and subtle position than often attributed to him, and he expressed a degree of skepticism about the ability to definitively prove or disprove either deterministic or indeterministic interpretations of quantum phenomena. The debate over hidden variables ultimately faded for a time, only to resurface in 1952 with David Bohm’s causal interpretation of quantum mechanics, but Schrödinger’s contribution offers a valuable insight into the early stages of this discussion.

Early Debate: Hidden Variables & Quantum Mechanics

The mid-1940s saw a renewed examination of deterministic hidden variables as a potential supplement to quantum mechanics, and Erwin Schrödinger actively contributed to this debate with a previously unknown model. While John von Neumann’s 1932 work was widely considered a disproof of such models, the ensuing discussion in Nature revealed subtle perspectives, including one from Schrödinger himself. He drafted a response that explicitly outlined a hidden-variable model, demonstrating that these models were not necessarily precluded by the existing physics.

This manuscript, recently unearthed, provides a more complex understanding of Schrödinger’s position than previously understood. Schrödinger’s analysis centered on the inadequacy of a one-bit hidden variable, arguing that the apparent absurdity Whittaker identified stemmed from this restriction. “For the derived absurdity is visibly produced by the simplifying assumption of two discrete parameter values only,” he wrote, suggesting that a more subtle approach might be viable. He even referenced Bertrand Russell’s 1936 opinion that there was no conclusive reason to favor complete determinism in physics, further demonstrating his open-mindedness.

Von Neumann’s 1932 Argument Against Hidden Variables

Von Neumann’s 1932 argument asserting the impossibility of deterministic hidden variables faced immediate, though largely overlooked, scrutiny. Grete Hermann, in 1935, identified limitations within the mathematical foundations of von Neumann’s proof, a critique that received minimal attention at the time, potentially due to Hermann’s position as a young woman and an outsider to the core physics discussions. This initial challenge set the stage for renewed debate in the mid-1940s, resurfacing the question of whether quantum mechanics required a strictly probabilistic interpretation or could accommodate underlying deterministic mechanisms.

The debate intensified with contributions from physicists like Whittaker, who strongly advocated for irreducible indeterminism, citing von Neumann’s work as definitive. However, this assertion would soon be challenged by a detailed, unpublished response from Erwin Schrödinger. Schrödinger’s model incorporated a deterministic update rule for a hidden variable after transmission through the first polarizer, ensuring a uniform distribution for subsequent measurements.

This update, expressed as a mathematical transformation, allowed the model to reproduce the statistical predictions of quantum mechanics, even with multiple polarizers. Schrödinger’s approach differed from that of Pelzer, who acknowledged von Neumann’s insight regarding successive polarizers but did not specify an explicit deterministic transformation law for the hidden variable. Schrödinger’s explicit construction, however, provided a concrete mechanism for maintaining the statistical behavior predicted by quantum theory.

He recognized the need to “reshuffle” the hidden variables after transmission to restore the initial distribution, a step important for accurately modeling the behavior of multiple polarizers. Schrödinger’s work, while not a complete theory in itself, demonstrated the viability of deterministic hidden-variable models. He argued that even the most careful logical demonstrations of impossibility were, at best, limited to the specific problem as formulated, leaving open the possibility of alternative formulations or approaches.

Schrödinger’s concern stemmed from a broader skepticism towards overly assertive claims of impossibility, believing that a more subtle and cautious approach was essential for scientific progress. “I wish to put forward a general argument against all supposed demonstrations of impossibility, such as von Neumann’s,” he wrote, emphasizing the need for continued investigation and open-mindedness in the face of complex scientific challenges.

The manuscript reveals Schrödinger’s intention was not necessarily to champion determinism, but to demonstrate that the claim of impossibility was not as secure as some physicists believed. He cautioned that noting the implications of abandoning deterministic principles and the resulting division among physicists.

Whittaker and Bose Revive Indeterminism Debate

The mid-1940s saw a resurgence of debate regarding deterministic hidden variables in quantum mechanics, prompted by contributions to Nature following lectures by Edmund Taylor Whittaker and Satyendra Nath Bose. This period witnessed a lively exchange of letters in the journal, with participants questioning the completeness of existing proofs and exploring alternative explanations for observed quantum phenomena.

The initial spark came from Whittaker’s 1943 Guthrie Lecture, briefly reported in Nature and Bose’s subsequent detailed account, which Henry T. H. Piaggio framed as demonstrating the This framing, however, did not go unchallenged, as critics began to scrutinize the underlying assumptions of the arguments against hidden variables. Wilfred W. Barkas, for example, questioned how seemingly random individual photon outcomes could consistently produce predictable statistical results, prompting Whittaker to draw an analogy to classical coin tosses and the role of symmetry in generating statistical regularity without requiring a deterministic mechanism.

“Their statistical regularity follows from the ordinary theory of probability together with the symmetry of the individual trial, without any need to assume a crypto-deterministic mechanism,” Whittaker wrote in response. Ernest Gold later challenged Whittaker’s symmetry argument, pointing out that the mechanism generating the coin toss itself need not be random, while Jerzy Horzelski presented an experiment attempting to fix the conditions of a coin toss to further investigate the issue. These exchanges highlight the limitations of applying one-bit hidden variables to explain quantum behavior.

As the source notes, “while both Whittaker and Bose reinforced von Neumann’s case against hidden variables, their contributions also reignited a debate that ultimately renewed skepticism about the supposed impossibility of hidden-variable theories.” The ideological weight of the debate often overshadowed the scientific nuances, with participants aligning themselves with opposing camps and sometimes projecting their own views onto the work of others. Schrödinger, while not directly involved in the initial Nature exchange, engaged with the core question of hidden variables, ultimately concluding that existing proofs of impossibility were limited in scope.

He cautioned that “we know to day [sic] that our idea of a lawful sequence of events in Nature has been abstracted from events which are based directly on statistical law.” The debate over local hidden-variable models continued for decades, culminating in Bell’s theorem in 1964 and subsequent experimental tests that ruled out this specific class of theories.

However, the initial exchanges in Nature, sparked by Whittaker and Bose, served as a reminder of the complexities involved in definitively disproving any form of determinism, and the importance of continually re-evaluating the foundations of quantum mechanics. The discussion, even in its early stages, revealed the deep-seated philosophical and scientific challenges inherent in reconciling the seemingly probabilistic nature of quantum mechanics with the enduring appeal of a deterministic worldview.

Malus’ Law as a Test for Hidden-Variable Models

His work, responding to a debate in Nature, focused specifically on the implications of Malus’ Law for hidden-variable theories, identifying a simplification within existing arguments against them. He noted that the perceived impossibility stemmed from restricting parameters to just two discrete values, a limitation he believed undermined the conclusions drawn. The manuscript, largely forgotten until now, demonstrates a subtle position often missed in discussions of Schrödinger’s views.

The search for hidden-variable models largely stalled for over a decade following this exchange, due to the perceived authority of von Neumann’s work. However, the manuscript reveals a contemporaneous line of inquiry that challenged this consensus.

👉 More information
🗞 Schrödinger's unpublished hidden-variable model in the 1944-45 debate over the "Collapse of Determinism"
✍️ Flavio Del Santo (University of Geneva, Group of Applied Physics, Rue de l’École-de-Médecine 21, Geneva 1211, Switzerland)
🧠 ArXiv: https://arxiv.org/html/2610.20752v1

More like this