NOTE-279: The problem of hidden variables in quantum mechanics

Contribution

The paper gives a precise necessary condition for hidden variables. It is not merely a phase space Ω reproducing the statistics (condition 1), which it shows is trivially satisfiable by a product measure on ℝ^𝒪 (§1). It requires that f_{g(A)} = g(f_A) for every Borel g (condition 4). This is recast as an embedding of the partial algebra of observables into a commutative algebra (§2). The paper then proves that for the finite partial Boolean algebra D ⊂ B(E³) no such embedding (indeed no Z₂-homomorphism) exists (Theorem 1). It realises D by physical observables, the squared spin components of a spin-1 system, measurable together through a rhombic-symmetry perturbation (§4). Three companion results frame it:

• Theorem 2. Allowing observables to split into several (“homomorphic relations”) does not help.
• §6. A classical model exists for spin ½, and von Neumann’s criterion wrongly excludes it (Theorem 3).
• Theorem 4. Embeddability is equivalent to the validity of classical tautologies under “meaningful” (commeasurable) substitutions. So the logic of quantum mechanics differs from classical logic.

Key insight

Demand classical consistency only where quantum mechanics itself licenses joint measurement: sums and products of commuting observables. That weaker demand is still unsatisfiable. The reason is that the commuting families overlap: a line in E³ lies in many orthogonal triples. The “exactly one of three orthogonal lines gets value 1” rule then cannot be met on a cleverly chosen finite web of overlapping triples.

Assumptions

• Observables and states. A theory has observables 𝒪, states S, and probability measures P_Aψ on ℝ (§1). “Function of an observable” is defined by P_g(A)ψ(U) = P_Aψ(g⁻¹(U)) (eq. 3), assuming observables are determined by their statistics.
• Commeasurability. A_i are commeasurable if A_i = f_i(B) for one observable B. In QM this is taken to coincide with pairwise commutation, by the von Neumann/Neumark theorem cited as [12, Thm 6].
• Partial algebra (definition, §2). A set with a reflexive, symmetric relation ⚥ (commeasurability), with 1 ⚥ everything. It is closed under +, ·, and scalars on commeasurable elements, and any three pairwise-commeasurable elements generate a commutative algebra. Over Z₂ this gives a partial Boolean algebra. The projections B(U^α) of a Hilbert space form one.
• Homomorphism. It preserves ⚥, linear combinations and products of commeasurable elements, and sends 1 ↦ 1. An imbedding is an injective homomorphism.
• The physical premise of §4. The system is in the n = 2, j = 1 eigenspace (three-dimensional), and H_S = aS_x² + bS_y² + cS_z² with a, b, c distinct is a genuine observable (cited to Stevens and Pryce).

Key results

• The necessary condition (§2, italicised). “A necessary condition for the existence of hidden variables for quantum mechanics is the existence of an imbedding of the partial algebra Q of quantum mechanical observables into a commutative algebra.” Restricted to idempotents, this is an imbedding of the partial Boolean algebra of propositions into a Boolean algebra.
• Theorem 0 (p. 67, verbatim). “Let 𝔄 be a partial Boolean algebra. A necessary and sufficient condition that 𝔄 is imbeddable in a Boolean algebra B is that for every pair of distinct elements a, b in 𝔄 there is a homomorphism h: 𝔄 → Z₂ such that h(a) ≠ h(b).” Necessity is by semisimplicity of Boolean algebras (Halmos). Sufficiency is by mapping into Z₂^S, with S the set of all such homomorphisms.
• The key reduction (§3, eq. 6). A homomorphism h: D → Z₂ sends exactly one of every three mutually orthogonal lines to 1.
• Lemma 1. The 10-vertex graph Γ₁ is realisable on the unit sphere S whenever its end-points p, q satisfy 0 ≤ sin θ ≤ 1/3. The proof is an explicit parametrisation, with cos φ = xy/((1+x²+x²y²)(1+y²+x²y²))^{1/2}, maximised at 1/3. In Γ₁, h(a₀) = 1 forces h(a₉) = 1.
• Lemma 2. The graph Γ₂, built from fifteen copies of Γ₁ around a triangle p₀, q₀, r₀, is realisable. It uses P_k, Q_k, R_k at angles πk/10 and π/10 < arcsin(1/3). Its image T has 117 points on S.
• Theorem 1 (p. 70, verbatim). “The finite partial Boolean algebra D has no homomorphism onto Z₂.” The proof: exactly one of p₀, q₀, r₀ is 1, say p₀. Propagating through the Γ₁ copies forces q₀ = 1, a contradiction.
• The continuum remark (p. 70). A fortiori there is no homomorphism from all of B(E³) onto Z₂. This was “first stated in Specker [17]” (1960) and also follows from Gleason’s theorem. The authors stress that finiteness matters physically.
• Γ₃ (p. 70). A 17-point graph gives a partial Boolean algebra F with h(a) = h(b) for every homomorphism, so F is not imbeddable, though homomorphisms may exist.
• The embedding into QM (§4). In the spin-1 representation, [J_x², J_y²] = [J_y², J_z²] = [J_z², J_x²] = 0 (eq. 7). The paper proves this with the given matrices and by a coordinate-free argument credited to J. Chaiken, and notes it fails in higher representations. The map ψ(α) = P J_α² (for lines), ψ(β) = P − P J_α² (for planes, α = β^⊥), ψ(E³) = P embeds D in the partial Boolean algebra 𝔅 of QM propositions. Commeasurability is realised physically, since J_x², J_y², J_z² are polynomials in H_J = aJ_x² + bJ_y² + cJ_z² (eq. 8). A measurement of H_S’s energy shift (a+b, b+c or c+a) reads off all three at once.
• The intuitive statement (pp. 73–74). For any prediction by a hidden-variable physicist X of “is the spin component along α zero?” for the 117 directions, Theorem 1 yields an orthogonal triple where X contradicts QM’s “exactly one of S_x, S_y, S_z is zero” (since S_x² + S_y² + S_z² = 2ħ²). The paper notes that the experiment “has probably not been carried out” on helium. The nearest analogue is paramagnetic resonance in nickel Tutton salts.
• Theorem 2 (§5). If a partial algebra 𝔄 has a non-trivial homomorphic relation with domain 𝔄 into a commutative algebra, then 𝔄 has a homomorphism onto some commutative C′. So Q admits no such relation, and splitting observables does not rescue hidden variables.
• The spin-½ model (§6). Ω = S². Under conditions (a)–(d) (continuous density, isotropy, covariance, support in a hemisphere), the density is uniquely u(θ) = (1/π)cos θ on 0 ≤ θ ≤ π/2, solved via Abel’s integral equation. It is realised as a particle on a randomly shaken disc projected onto the sphere, “rain falling on a ball”. It reproduces all spin-½ statistics, with deterministic outcomes given the hidden state.
• The von Neumann Lemma and Corollary (§6). If ℰ satisfies (1) ℰ(I) = 1, (2) homogeneity, (3) ℰ(A²) = ℰ(A)² and (4)′ additivity only on commuting pairs, then ℰ is multiplicative on commuting pairs, and ℰ(A) lies in the spectrum of A.
• Theorem 3. There is no ℰ: H → ℝ satisfying (1)–(4) with full additivity (4). The proof takes A = diag(1,0), B = ½[[1,1],[1,1]], since A + B has eigenvalues 1 ± ½√2. This holds already on H₂. Hence von Neumann’s criterion excludes the spin-½ model the paper just built, which the authors take to show the criterion is unreasonable.
• Theorem 4 (§7). Let 𝔄 be a partial Boolean algebra. (1) It is imbeddable into a Boolean algebra iff every classical tautology of the form φ ≡ ψ is valid in 𝔄. (2) It is weakly imbeddable iff every classical tautology is valid. (3) It maps homomorphically into a Boolean algebra iff no classical tautology is refutable. The proof uses the Completeness Theorem for predicate calculus. Validity is defined over “meaningful” substitutions only: the recursively defined domain D_φ, in which only commeasurable elements are combined.
• The Corollary (§7). “There is a propositional formula φ which is a classical tautology but which is false under a (meaningful) substitution of quantum mechanical propositions.” The explicit φ = 1 − Π(x_i + x_j + x_k + x_ix_jx_k) runs over orthogonal triples of D. A more economical formula has 86 variables.

Claims

id	claim	strength	support
C1	The statistical condition (1) alone is always satisfiable, so it is no test of a hidden-variable theory	proof	§1, the product measure on ℝ^𝒪
C2	Hidden variables (condition 4) require an imbedding of the partial algebra of observables into a commutative algebra	argument (definitional reduction)	§2
C3	A partial Boolean algebra embeds in a Boolean algebra iff Z₂-homomorphisms separate its points	proof	Theorem 0
C4	The 117-point partial Boolean algebra D ⊂ B(E³) has no homomorphism onto Z₂	proof	Lemmas 1–2 and Theorem 1; Γ₁ bound checked
C5	D embeds in the propositions of a spin-1 system, with each orthogonal triple jointly measurable via H_S	proof (algebra) plus physics citation	§4, eqs. 7–8 (checked); spin-Hamiltonian form cited to Stevens and Pryce
C6	Hence QM has no hidden variables in the paper’s sense	proof, given C2 and C5	§§3–4. As written, ψ(E³) = P, so the argument covers the system confined to the three-dimensional space N (the paper’s stated assumption) and says nothing about general dimension
C7	Letting observables split (homomorphic relations) does not help	proof	Theorem 2
C8	Spin ½ has a deterministic classical model reproducing QM statistics	proof (construction)	§6, the Abel-equation solution u(θ) = cos θ/π
C9	von Neumann’s no-go rests on additivity for non-commuting observables, and so wrongly excludes C8	proof plus argument	Lemma, Corollary and Theorem 3, §6
C10	Quantum logic differs from classical logic: some classical tautology is refuted by a meaningful substitution	proof	Theorem 4 and Corollary, §7
C11	Unlike the uncertainty principle, this departure “cannot be refuted at a later date” by adding states	informal argument	§7, last paragraph

Method

Algebraic reformulation (partial algebras, homomorphisms); explicit orthogonality-graph gadgets on the sphere with a trigonometric realisability bound; physical realisation through a spin Hamiltonian; model-theoretic compactness for the logic theorem; an integral equation for the spin-½ model.

Concepts

• Commeasurable (⚥). Jointly functions of one observable.
• Partial algebra / partial Boolean algebra. Operations defined only on commeasurable elements. Every partial algebra is isomorphic to a union of commutative algebras C_i, glued under conditions (a) and (b), which the paper glosses as “a category in which the objects are commutative algebras and the maps are imbeddings” (p. 66). That is the germ of the later presheaf/topos formulation.
• Prediction function. A homomorphism Q → ℝ, i.e. a value assignment respecting functional relations.
• Meaningful substitution, valid, refutable (§7).
• Weak imbedding. Injective on commeasurable pairs.

Connections

• LIT-263 (Budroni et al., review). The modern review of KS contextuality. KS 1967 is its primary source.
• THEORY-012. Abramsky–Brandenburger’s “no global section” is the sheaf form of Theorem 1 for the possibilistic, deterministic case. A homomorphism D → Z₂ is exactly a global assignment compatible on every context (orthogonal triple). KS’s remark that a partial algebra is “a category of commutative algebras and imbeddings” anticipates the presheaf view.
• THEORY-015. “KS noncontextuality = measurement noncontextuality + outcome determinism for sharp measurements.” KS’s homomorphisms are deterministic (values in Z₂ or in the spectrum, by the §6 Corollary). The §6 spin-½ model shows determinism is available in dimension 2.
• THEORY-011. KS arguments do not exist in dimension 2. That matches §6, which builds the model on H₂, and §4, which notes the J² relations fail above the spin-1 representation, so dimension 3 is essential to the construction.
• LIT-306 (Birkhoff–von Neumann). Its lattice of subspaces is the total structure. KS keep only the partial operations on commuting elements, and show that even this partial structure is non-classical.
• LIT-325, LIT-299, LIT-335, LIT-343, LIT-310 (Isham–Butterfield, Döring–Isham, seeded). These recast KS as the absence of global elements of the spectral presheaf. §2 is the algebraic starting point.
• LIT-353 (Stone). Theorem 0 is Stone’s representation theorem, used as a criterion. A Boolean algebra embeds in the power set of its Z₂-homomorphisms (its ultrafilters/prime ideals), and KS extend that criterion to partial Boolean algebras.

Bearing on the record

• Map row 8, “the no-global-valuation theorem behind ‘no global ultrafilter/tope’”. The paper does own this, and in exactly the ultrafilter form.
  • The ultrafilter form. A homomorphism of a (partial) Boolean algebra onto Z₂ is an ultrafilter. Theorem 0 says embeddability in a Boolean algebra, i.e. being a field of sets, requires enough of them. Theorem 1 shows a finite quantum example with none. “No global ultrafilter on the partial Boolean algebra of projections” is a faithful paraphrase of Theorem 1.
  • Dimension. The paper proves this for the three-dimensional case only, via 117 real directions. Any claim “for every dimension ≥ 3” should cite a later source (e.g. LIT-263).
  • “Tope”. This is oriented-matroid vocabulary (LIT-332) and does not occur here. The identification of Xiong’s regions with topes (LIT-267) and of topes with ultrafilters is the owner’s step, not KS’s.
• The single-context / Boolean-shadow diagnosis. It fits KS well. Each context (a commuting family) is a Boolean algebra with plenty of ultrafilters. The obstruction is only in gluing contexts. So a single-context model, one Boolean algebra, cannot see it. That is the paper’s own structure: classical within each C_i, non-classical in the union (§2).
• What it does not supply. Nothing about probabilities, the contextual fraction (LIT-265), or behavioural data (THEORY-013). It is a state-independent, deterministic, finite-dimensional result.
• ML practice. It carries nothing, and does not belong in the Anthology.

Limitations

• Dimension three only, as written. The embedding into QM sends E³ to the projector P onto the spin-1 eigenspace, not to the identity. The argument is therefore complete for a system confined to that three-dimensional space, which is the paper’s stated assumption. The paper does not treat an arbitrary Hilbert space of dimension ≥ 3.
• A physical premise. The joint measurability in §4 relies on the spin-Hamiltonian form, cited not derived, and the experiment was not performed at the time.
• The necessary condition is a modelling choice. Condition (4) and the §2 reduction define what “hidden variables” must satisfy. Bohm-type contextual theories, which the paper cites, are excluded by definition, not refuted. The paper’s §4 parenthesis addresses this only briefly.
• Economy. 117 vectors is far from minimal; later constructions are much smaller.

Open questions

• The paper points to its companion works [10], [11] for the positive description of quantum logic. For the record, the open question is whether the owner’s “no global tope” claim for LLM embeddings is meant as KS-type (state-independent, deterministic) or as the probabilistic sheaf version (THEORY-012). They need different evidence.

Corrections to the seeded skim

• Seeded from metadata; the text shows the summary is right in spirit but states more than the paper does. (1) The paper proves the result for three-dimensional Euclidean space, B(E³), realised in the three-dimensional spin-1 eigenspace N of orthohelium. It does not state or prove “for Hilbert spaces of dimension ≥ 3”. The words “dimension ≥ 3” do not occur. (2) “Respects functional relations among commuting observables” is right. The formal condition is a homomorphism of partial algebras, preserving sums and products only of commeasurable elements (§2). The paper contrasts it explicitly with von Neumann’s requirement of additivity on all observables (§6).
• The masthead prints Journal of Mathematics and Mechanics, Vol. 17, No. 1 (1967). The paper was communicated by A. M. Gleason, and dated “Communicated: October 31, 1966”. The DOI’s “1968” is the IUMJ archival year; the 1967 in the seed’s published: is right.
• Typographical slips in the text: in Lemma 2’s proof, P_k = cos(πk/10) ī + sin(πk/19) j̄ should read sin(πk/10). The Introduction says the comparison with von Neumann is “in Section 5”, but it is §6. §3’s Remark says “satisfying condition (4)” where condition (6) is meant.
• The map of “a finite set of directions in ℝ³” is right: 117 points for Theorem 1, and a 17-point graph Γ₃ for mere non-embeddability. The paper also mentions an 86-variable formula from Γ₂ (§7).