Schr"odinger's unpublished hidden-variable model in the 1944-45 debate over the "Collapse of Determinism"

summary

Video file (mp4)

The gist

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.

In short

Schrödinger conceived a deterministic hidden-variable model for Malus' law during a 1944-1945 debate questioning quantum mechanics' determinism. He developed rules for photon transmission and updated the hidden variable distribution after each polarizer, showing how the model reproduces quantum predictions while philosophically arguing that experimental evidence cannot disprove either determinism or indeterminism.

Key concepts

Deterministic Hidden-Variable Model
This is a theory suggesting that quantum randomness is not fundamental but arises from our incomplete knowledge of underlying, deterministic variables. Schrödinger created a specific mathematical rule for how photons should be transmitted through polarizers based on these hidden variables.
Update Rule for Hidden Variables
When a photon passes through the first polarizer, only certain hidden variables remain possible. Schrödinger developed a specific mathematical transformation (λ' = λ − sin2 φ / cos2 φ) to map these remaining possibilities back onto a uniform distribution, ensuring the model remains consistent with subsequent measurements.
Proof of Principle
Schrödinger used his model not to prove determinism, but as an argument against the idea that it is impossible. He argued that neither quantum indeterminism nor classical determinism can be definitively ruled out by logic or experiment alone, making the model a 'proof of principle' against absolute claims.

Terminology used across episodes

This episode discusses

The paper

Schr"odinger's unpublished hidden-variable model in the 1944-45 debate over the "Collapse of Determinism" · Read on arXiv

Flavio Del Santo, Erwin Schr"odinger

University of Geneva · Constructor University, Bremen, Germany

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 I show that Erwin Schrödinger drafted a hitherto unpublished response to this controversy.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.

Transcript

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Schr"odinger's unpublished hidden-variable model in the 1944-45 debate over the "Collapse of Determinism"".

Mira: 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.

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So we're looking at "Schr"odinger's unpublished hidden-variable model in the one thousand nine hundred forty-four-forty-five debate over the 'Collapse of Determinism' <ref:2610.20752#pg1,Schr\"odinger's unpublished hidden-variable model in the 1944-45 debate>." It sounds like a really deep dive into how Schrödinger actually thought about this issue back then.

Mira: Yeah, it’s interesting because it’s not just about finding a new solution; it’s about reconstructing a lost response to that historical argument between determinism and indeterminism in quantum mechanics.

Lev: From an engineering standpoint, the fact that Schrödinger drafted this model means we need to look at what he actually proposed, not just the philosophical debate itself.

Kai: Right. We have to figure out what his explicit hidden-variable model actually says about how photons behave when they pass through those polarizers.

Mira: Exactly. We want to get past the historical context and see if his specific mathematical rules for transmission are viable, or at least, what they tell us about the structure of quantum theory.

The paper's summary: Kai: So, basically, the paper explains that Schrödinger put forward a deterministic rule for photon transmission called T(λ, φ) equals one if lambda is greater than sin squared phi and zero otherwise.

Mira: That’s the core idea—it gives a specific way to transmit photons based on their hidden variable lambda and the polarizer angle phi >

Lev: And then it shows how he derived the number of transmitted photons NT equals N times one sin squared phi multiplied by rho of lambda dlambda, which is essentially trying to recover Malus' law.

Kai: But there’s a catch because after transmission through the first polarizer, only those photons with lambda greater than sin squared phi remain in our ensemble, which changes the distribution of that hidden variable to rho of lambda T equals one over cos squared phi for sin squared phi less than or equal to lambda less than or equal to one.

Mira: That update rule is critical because it has to be deterministic, so Schrödinger provides a rule for the hidden variable itself: lambda prime equals lambda minus sin squared phi divided by cos squared phi.

Lev: That transformation maps that interval from sin squared phi to one onto zero to one, which restores a uniform distribution for the updated hidden variable, which is what he needs to keep the math consistent.

The paper's improvements: Kai: What’s interesting about this paper isn't just that Schrödinger proposed a model at all, but how the paper suggests we can improve on it by testing it.

Mira: It points out that even though Schrödinger’s work is a proof of principle, we can use his idea to test the limits of these theories.

Lev: The paper outlines three key areas where this framework helps: first, high-fidelity simulations for polarization experiments using continuous hidden variables instead of just discrete ones.

Kai: So, it suggests we can simulate multiple successive polarizers by explicitly incorporating Schrödinger’s deterministic rule for transmission, T(λ, φ) equals one if lambda is greater than sin squared phi >

Mira: And second, the update rule lambda prime equals lambda minus sin squared phi divided by cos squared phi enables us to model the continuous state evolution of photon distribution after each measurement.

Lev: That means we can predict and verify quantum statistics for subsequent measurements based on those continuously updated hidden variable distributions.

Conclusion: Kai: So, wrapping up, the paper "Schr"odinger's unpublished hidden-variable model in the one thousand nine hundred forty-four-forty-five debate over the 'Collapse of Determinism'" shows that even if a deterministic picture isn't what we are looking for, these models can still be used as proofs of principle against impossibility claims <ref:2610.20752#pg1,Schr\"odinger's unpublished hidden-variable model in the 1944-45 debate>.

Mira: It really highlights Schrödinger’s view that experimental evidence plus logical deduction can never prove that either the deterministic or the indeterministic view is impossible, which is a very cautious philosophical stance.

Lev: For real hardware, what this means is that we have a rigorous framework to test these models against actual measurements of photon statistics to see if they hold up under those constraints.

Kai: So even though Schrödinger never published it, this paper shows that figures like Einstein and Bohm were deeply engaged in these questions of causality and indeterminism, and his model should be understood as a proof of principle rather than an endorsement of determinism.

More episodes

← Home