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Schrödinger’s Unpublished Hidden-Variable Model Surfaces
A preprint posted on October 14 transcribes Erwin Schrödinger’s unpublished response to a 1944–45 debate over determinism. The manuscript describes a hidden-variable model that reproduces Malus’ law for polarized photons, but the paper fram
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“Importantly, Schrödinger’s proposal should be understood as a proof of principle rather than as evidence of his commitment to determinism.”
Flavio Del Santo, author of the paper
A missing reply in the “Collapse of Determinism” debate
A preprint posted on October 14 brings to light a response Erwin Schrödinger drafted during a 1944–45 dispute over whether quantum mechanics could be supplemented by hidden variables. The manuscript never appeared in print. In it, Schrödinger sets out a deterministic model for how polarized photons pass through a polarizer, challenging an argument that such a model could not reproduce quantum probabilities.
Flavio Del Santo, the paper’s author, reconstructs the debate and publishes the first transcription and analysis of Schrödinger’s draft. The exchange began after physicist and mathematician Edmund Whittaker argued that hidden parameters could not account for Malus’ law, which gives the probability that polarized light passes through a polarizer as a function of the angle between their orientations. Del Santo counts eleven published contributions to the debate; Schrödinger’s manuscript would have been a twelfth.
A continuous variable replaces Whittaker’s one-bit model
Whittaker’s argument considered a photon carrying one of two hidden values, each fixing whether it would be transmitted or reflected. Schrödinger’s response replaces that binary parameter with a real number, λ, uniformly distributed between zero and one. For a polarizer set at angle φ, his rule transmits a photon when λ is at least sin²φ.
That threshold gives a transmitted fraction of cos²φ, matching Malus’ law. Schrödinger also specifies how to update the hidden variable after transmission: the surviving photons’ values are remapped to a uniform distribution, allowing the model to reproduce the predicted transmission through a second polarizer. Del Santo notes that this update matters: matching a single measurement is not enough if the model is also meant to account for successive measurements.
The manuscript does not settle determinism
Del Santo cautions against reading the construction as a declaration of faith in a deterministic universe. Schrödinger’s draft also argues that neither deterministic nor indeterministic accounts can be ruled out by logic and experimental evidence alone. The paper places the model in the context of Schrödinger’s longstanding views, which, it says, leaned toward indeterminism.
The result has a narrow historical and mathematical scope. It answers Whittaker’s particular argument about photon transmission through polarizers; it does not overturn Bell’s later results about local hidden-variable models or show that every quantum prediction admits a deterministic explanation. Del Santo distinguishes Schrödinger’s demonstration from a general theory of quantum mechanics.
Why Schrödinger left it unpublished remains uncertain
The manuscript survives as an undated handwritten document titled “Collapse of Determinism” in the Erwin Schrödinger Archive at the Austrian Central Library for Physics, University of Vienna. Its references show Schrödinger had followed the debate at least through a November 1944 contribution. Del Santo suggests he may have learned of Heinrich Pelzer’s related published work and concluded that the central point had already been made, but presents that explanation as a possibility, not a known reason.
Pelzer had also described a continuous hidden parameter for reproducing the single-polarizer probability, though Del Santo says his account did not provide the explicit deterministic update rule Schrödinger supplied for successive polarizers. The newly transcribed pages therefore add a specific piece to the history of the debate: Schrödinger tested a construction against a claimed impossibility, without treating that exercise as proof that nature is deterministic.