Schr"odinger's unpublished hidden-variable model in the 1944-45 debate over the "Collapse of Determinism"

The gist: Erwin Schrödinger conceived an explicit hidden-variable model in response to a 1944-1945 debate questioning whether quantum mechanics could be supplemented by deterministic hidden variables.

The historical context of the debate

In the mid-1940s, a debate resurfaced concerning whether quantum mechanics could be supplemented by deterministic hidden variables, with Schrödinger drafting an unpublished response to this controversy In the mid-1940s, a debate over classical and quantum indeterminism suddenly resurfaced despite foundational discussions of quantum mechanics having been stifled by the war This revival was sparked by two independent events: the twenty-seventh Guthrie Lecture delivered on May 18, 1943, at the Physical Society in London by Edmund Taylor Whittaker The second event was the presidential address “The Classical Determinism and the Quantum Theory”, delivered by Satyendra Nath Bose at the Indian Science Congress in Delhi on January 3, 1944 Both contributions independently put forward roughly the same fundamental thesis: quantum mechanics had provided enough grounds to finally question once and for all the classical dogma of determinism in physics

Schrödinger’s deterministic hidden-variable model of Malus’ law

Schrödinger presented a deterministic rule for transmission, T(λ, φ) = (1, λ ≥ sin2 φ, 0, otherwise) The number of transmitted photons is then NT = N Z 1 sin2 φ ρ(λ) dλ = N cos2 φ to recover Malus’ law A difficulty arose because after transmission through the first polarizer, only photons satisfying λ ≥ sin2 φ remain in the ensemble, changing the distribution of the hidden variable to ρ(λ T) = 1/ cos2 φ for sin2 φ ≤ λ ≤ 1 To maintain consistency, Schrödinger provided a deterministic update rule for the hidden variable: λ′ = λ − sin2 φ / cos2 φ This rule maps the interval [sin squared φ, 1] onto [0, 1], restoring a uniform distribution for the updated hidden variable For a second polarizer oriented at an angle φ', the transmission rule is T′(λ′, φ, φ′) = (1, λ′ ≥ sin 2(φ' − φ), 0, otherwise) This leads to NT′ = NT Z 1 sin 2(φ'−φ) dλ′ = NT cos 2(φ'−φ), which agrees with the prediction of quantum mechanics

Philosophical considerations and the status of determinism

Schrödinger stated that “even very elaborate and mathematically rigorous deductions, such as Von Neumann’s, are of limited value” He argued that “Experimental evidence + logical deduction can never prove that either the deterministic or the indeterministic view is impossible” He asserted that “the strongest plea in favour of indeterminism is older than quantum mechanics” Schrödinger maintained that “classical mechanics itself is indeterminate” and criticized determinism as the “fundamental dogma of classical physics, a ‘philosophic prejudice’, and an ‘a priori belief’” The manuscript should be understood not as an endorsement of a deterministic picture of nature, but as a proof of principle

Why Schrödinger did not publish his model

Schrödinger did not publish his model because Pelzer had already published a paper presenting a hidden-variable explanation of Whittaker’s example Pelzer's paper provided an explicit quantitative hidden-variable model, but unlike Schrödinger, he did not specify an explicit deterministic transformation law for the hidden variable of each individual photon It is plausible that Schrödinger became aware of Pelzer’s result through its citation in the Nature debate Schrödinger’s manuscript was considerably broader in scope and philosophically more ambitious, combining a concrete hidden-variable model with a reflection on the limits of impossibility proofs The debate ultimately ended with Piaggio offering an opinion that “the existence of causality has not been disproved”

Conclusion

The paper reconstructs a forgotten debate that reopened between 1943 and 1945 the question of whether quantum mechanics could admit a deterministic completion A central new result is that Schrödinger himself conceived a hidden-variable model in the context of this debate, although he never published it The debate between determinism and indeterminism has divided physicists for over a century and continues to do so today Schrödinger’s model should therefore be understood not as an endorsement of a deterministic picture of nature, but as a proof of principle 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 This paper shows that even figures like Einstein and Bohm devoted much of their intellectual lives to questions of causality and indeterminism The paradigmatic case is that of Einstein, Schrödinger, and Bohm, perhaps the three figures most commonly regarded in the popular culture of physicists as the main authorities on the side of determinism All three devised hidden-variable models, yet none can simply be characterized as an advocate of determinism Schrödinger’s model should therefore be understood not as an endorsement of a deterministic picture of nature, but as a proof of principle This paper shows that even figures like Einstein and Bohm devoted much of their intellectual lives to questions of causality and indeterminism The debate between determinism and indeterminism has divided physicists for over a century and continues to do so today From the outset, it has carried a strong ideological charge, often taking the form of a partisan confrontation between two opposing camps The paradigmatic case is that of Einstein, Schrödinger, and Bohm All three devised hidden-variable models, yet none can simply be characterized as an advocate of determinism Schrödinger’s model should therefore be understood not as an endorsement of a deterministic picture of nature, but as a proof of principle Such models could serve as proofs of principle against widespread beliefs or alleged impossibility claims This paper shows that even figures like Einstein and Bohm devoted much of their intellectual lives to questions of causality and indeterminism The debate between determinism and indeterminism has divided physicists for over a century and continues to do so today From the outset, it has carried a strong ideological charge, often taking the form of a partisan confrontation between two opposing camps The paradigmatic case is that of Einstein, Schrödinger, and Bohm All three devised hidden-variable models, yet none can simply be characterized as an advocate of determinism Schrödinger’s model should therefore be understood not as an endorsement of a deterministic picture of nature, but as a proof of principle Such models could serve as proofs of principle against widespread beliefs or alleged impossibility claims The paper reconstructs a largely forgotten debate that reopened between 1943 and 1945 the question of whether quantum mechanics could admit a deterministic completion A central new result is that Schrödinger himself conceived a hidden-variable model in the context of this debate, although he never published it The debate between determinism and indeterminism has

1. textbfFundamental Model Testing for Hidden Variables: Implement Schrödiger's Continuous Parameter Model for Quantum Optics Analysis. The improved AI can perform high-fidelity simulations of polarization experiments by explicitly incorporating Schrödiger’s deterministic rule for transmission, T(λ, φ) = (1, λ ≥ sin2 φ). This allows the system to test the viability of hidden variable models against quantum predictions for multiple successive polarizers using continuous hidden variables instead of discrete ones.</item

2. textbfHandling Continuous State Evolution in Quantum Systems. The AI will be capable of modeling photon state evolution by applying Schrödiger’s update rule, λ' = λ − sin2 φ / cos2 φ, to track the distribution of hidden variables after transmission through a polarizer. This enables the system to predict and verify quantum statistics for subsequent measurements based on the continuously updated hidden variable distribution.</item

3. textbfCategorical Analysis of Determinism vs. Indeterminism in Physics Interpretation. The AI can be programmed to evaluate physical theories by applying Schrödiger's philosophical conclusion: Experimental evidence + logical deduction can never prove that either the deterministic or the indeterministic view is impossible. This allows the system to assess competing interpretations, such as von Neumann’s proof versus Popper’s view, by flagging them as extremely inconvenient hypotheses rather than absolute truths.</item

4. textbfBeyond Single-Bit Constraints in Parameter Encoding. The improved AI can recognize that One bit is therefore no longer sufficient: the outcome must depend on the interplay between a continuous hidden variable and the setting of the apparatus, as demonstrated by Schrödiger’s model for Malus’ law. This capability allows the system to identify situations where classical or quantum systems require continuous variables, moving beyond simplistic binary encoding assumptions.</item

5. textbf Historical Contextual Reasoning for Theoretical Development. The AI can analyze historical scientific debates, recognizing that the strongest plea in favour of indeterminism is older than quantum mechanics. This enables the system to contextualize modern physical theories by linking them to pre-quantum philosophical commitments, such as Schrödiger's earlier embrace of statistical character in nature.

In the mid-1940s, the question of whether quantum mechanics could be supplemented by deterministic hidden variables resurfaced in a debate in textit Nature. I show that Erwin Schrödinger drafted a hitherto unpublished response to this controversy. In it, Schrödinger provided an explicit hidden-variable model, thereby showing that such models could not be ruled out. I reconstruct the historical context of this forgotten manuscript and publish here the first transcription and analysis. Importantly, Schrödinger's proposal should be understood as a proof of principle rather than as evidence of his commitment to determinism.