Anton Lorenz Vrba Independent Researcher, UK vrba@iow.onl
Richard D. Gill Mathematical Institute, Leiden University gill@math.leidenuniv.nl
29 September 2026
Abstract
We examine the logical and mathematical structure of Sanctuary’s recent paper (Quantum Rep. 2026, 8, 96), which argues that spin observables in the Bivector Standard Model (BiSM) are not jointly definable on a single Kolmogorov probability space, so that the CHSH quadruple cannot be formed and Bell’s and Fine’s theorems do not apply. The paper contains two constructions: local response functions of the analyser setting and a shared source phase, and a separate prescription that produces the correlation . We show that the first construction, as specified, meets the hypotheses of Bell’s theorem. Its outcome maps are defined on a common source domain with a setting-independent distribution, so the non-existence claim (Lemma 3, Corollary 3) does not hold for it, its product correlations obey the CHSH bound, and the paper’s own simulation of it yields the Bell triangle. The cosine is obtained instead from a scalar computed jointly from both stations’ phases and then digitised; it is not shown to be the correlation of the local outcomes. Execution of the supplementary programs further shows that the four BiSM_v2 populations do not form the partition stated in the paper, and that the doubled-angle harmonic attributed to the raw coherence data appears only after a sign convention is applied on the second half-cycle.
Introduction
Sanctuary’s paper “Spin Helicity and the Disproof of Bell’s Theorem” [3], together with [2], claimed that a bivector (quaternion) model of spin disproves Bell’s theorem. The published Comment [4] examined that claim. It concluded that the construction lies outside the hypotheses of Bell’s theorem rather than refuting it, and that the correlation is obtained by steps that “are heuristic mathematical procedures, not formal probabilistic identities”. In particular, the Comment identified the assembly of the correlation from separately computed polarisation and coherence contributions as “a modeling postulate”, for which “no general probabilistic theorem guarantees that such an additive assembly reproduces Bell-type statistics”.
In his Reply [5] the author accepted the first conclusion: “We agree that Bell’s theorem is mathematically valid within its stated assumptions. Our position is that bivector spin does not belong to the class of models Bell considered and therefore lies outside the scope of the theorem.” The Reply also defends the contextual construction of measurement domains, the statistical reconstruction of the phase relation, and a geometric account of the quantum limit. It offers no explicit probabilistic derivation resolving the objection to the split-and-recombine step, stating that “several issues raised by Vrba, particularly concerning the quantum limit and measurement structure, deserve fuller treatment than is possible in a short reply.”
Paper [1] develops that response. It does not supply the missing probabilistic derivation. Instead it isolates joint definability as a separate premise, which the model is said to fail, and it gives explicit rules: a phase uniformly distributed on , carried by local rotors to each station, and deterministic outcome rules. The present critique concerns the relation between the paper’s algebra and detector-level statistics. The explicit rules of [1] allow a sharper separation than was possible in [4] between two constructions: the local response functions, and the separate prescription that produces . The review that follows examines whether the joint-definability argument holds for the local responses (Section A), whether the correlation is derived from them or assembled separately (Section B), what the simulations compute (Section C), and which further claims rest on assertion rather than derivation (Sections D and E).
Unless stated otherwise, equation, section, definition and lemma numbers refer to [1]; the points of the present critique are numbered A1, B1, and so on. The supplementary material of [1] contains the programs BiSM_v1 and BiSM_v2 with their output files. Statements about the programs below rest on source inspection and output files.
A. The joint-definability argument
A1. Lemma 3 does not hold for the paper’s own definitions. Definition 12 introduces a context map , and Eq. (52) defines the outcome as . The composition is a function on for every setting , and likewise . Hence for every the four values
are simultaneously defined, and is a common domain of the kind Lemma 3 says does not exist. The proof of Lemma 3, whose argument is first set out in [5], rests on the sentence “observables are defined only after contextual instantiation”. That concerns which experiment is physically performed, not which functions are mathematically defined. Counterfactual definability in Bell’s sense requires only that the function be specified, not that the measurement be made.
A2. The explicit outcome rules are Bell-type local functions. The paper does not leave the outcome functions abstract. Eq. (15) gives the local phase for every and ; Eqs. (64) and (66) take the quantum-domain outcome as ; and Eq. (85) gives the polarisation outcome as , with the value assigned at zero. Each is a function of the local setting and the shared alone, which is Bell’s form . The sign-of-cosine rule is essentially Bell’s own illustrative model [6]; with a uniform source distribution and Bob’s opposite sign it gives the linear “triangle” correlation, as Figure 3 and Section 4.2.2 of [1] confirm.
A3. Geometric reality does not exempt from Bell’s framework. The paper states: “In the BiSM long–range EPR correlation originates from a common relative rotor phase established at separation and subsequently carried along the worldlines of Alice and Bob” (Section 2.1); the same description is given in [5]. Bell placed no restriction on the nature of : “It is a matter of indifference in the following whether denotes a single variable or a set, or even a set of functions, and whether the variables are discrete or continuous” [6]. A real, local geometric object created at the source and carried to both stations is the paradigm case of , not an exception to it. The local-response component of the model is therefore of exactly the kind Bell’s theorem addresses, and the statement that the bivector is “not a hidden variable in Bell’s sense” [5] does not hold for that component.
A4. A single Kolmogorov space is used explicitly. The paper places on with density (Eqs. (16), (70)) and uses this same space for every analyser separation: “Every segment in the simulation has the same free–flight singlet state”. The supplementary programs use a common finite grid of values for all settings, which is likewise a single space. With measurable response maps, the pushforward of under is a joint distribution reproducing the pairwise distributions of these local responses. Corollary 3 (“no global joint probability distribution exists for the CHSH quadruple”) therefore does not hold for them. The pushforward need not reproduce the separately generated pair distributions of the cosine branch (Section C).
A5. The author’s statement of Fine’s theorem, applied to the local responses. Section 5.5 states correctly that a joint distribution for the four observables “is equivalent to a deterministic Local Hidden Variable (LHV) representation”. Definition 15 asserts: “Individual measurement outcomes are real, discrete, deterministic and local.” The label alone would not settle the matter, but points A1, A2 and A4 show that the paper supplies local response functions on a common source domain and a source distribution that does not depend on the settings. For correlations formed as averages of products of these local outcomes,
for every , and integration against gives the CHSH bound; by the equivalence the author quotes, a joint distribution exists. The paper does not address the resulting alternatives:
either the reported correlations are averages of products of these local outcomes, in which case CHSH is satisfied;
or they are obtained from another quantity, in which case the locality of the response functions does not carry over to them.
The simulations (Section C) show that alternative (ii) is what occurs.
A6. Assumption 3 is not independent of Assumption 2 in the paper’s representation. The paper lists “Joint Definability” separately from “Outcome Definiteness”. Assumption 2 (“For each hidden parameter , the outcomes , are well-defined”) is stated for the settings generally; in the functional representation the paper supplies, any four settings then yield four defined values for fixed . Rejecting 3 while retaining 2 requires restricting 2 to the setting actually used, a restriction the paper does not state.
A7. The scope of Fine’s theorem. Fine’s theorem [7] concerns the scenario with two settings per party and two outcomes per setting: pairwise distributions with consistent single-observable marginals admit a joint distribution if and only if the full family of Bell–CHSH inequalities holds. A family that violates those inequalities is an instance of the theorem’s conclusion, not a case in which the theorem does not apply. Conclusion 5 of [1] states that “Bell’s theorem and Fine’s characterization remain mathematically valid within their stated premises”, which is narrower than the “inapplicability” of the title.
A8. Notation and analogy do not alter the common domain. The semicolon in is said to indicate “that is not merely another argument”; the notation does not change the domain or existence of the function. The proof of Lemma 2 moves from distinct function values, , to “distinct domains”; that step is not justified. Distinct instantiated domains need not be denied: the decisive point is that the context maps pull the observables back to the common source domain (A1). The induced-dipole analogy (Remark 1) illustrates this: a polarisability tensor fixes the response to every field direction simultaneously, so the counterfactual dipoles are jointly defined.
A9. The common-domain structure is already present in Bell’s representation. The paper states that “Bell did not isolate this common probability–space structure as a separate premise under the name of joint definability; it entered tacitly through the hidden–variable representation”. Bell wrote , and explicitly [6]. Joint definability is a consequence of that representation rather than an additional assumption, which is why it required no separate statement.
B. The correlation derivation
B1. The cosine is the scalar part of a joint product. Of the ellipsis in Eq. (13) the paper states: “We added the dots to emphasize that the two rotors do not cancel but are carried as independent phases by the separated EPR pair. The dots have no mathematical meaning expressing only locality by the product.” Eq. (14) nevertheless evaluates as the scalar part of the product of Alice’s and Bob’s rotors, in which cancels and results. The paper’s own second line of Eq. (14) shows the alternative: the product of the separate scalar parts, , which it states “obeys the CHSH bound”. The cosine is thus obtained from the scalar part of a product formed from both stations’ rotors, not from the product of the two locally generated Boolean detector outcomes. Joint processing is not itself the issue: ordinary coincidence analysis also compares records at a common later stage. The issue is which function of the records is computed.
B2. The cosine holds event by event. Eq. (20) gives the relative phase for every , so the scalar of Eq. (71) equals for every pair. Section 4.2.2 states that the correlation “emerges only statistically from the ensemble of rotor phases ”. The source phase affects the individual local quantities, but averaging over it is not what produces the cosine in : that cosine already holds event by event. The required identification of this scalar with the correlation of local Boolean outcomes is not supplied by the cancellation.
B3. The singlet operator is the identity. Eqs. (9), (13) and (72) define . Eq. (73) then writes
Inserting the identity between and yields directly. The target correlation thus enters through this algebraic representation, and no argument is given that the representation describes the detector statistics. Eq. (73) also combines matrix transposition with geometric-algebra products and uses , both as angles and as vectors; a clarification of the representation would help.
B4. The plotted mustache is a residual. Eq. (71) is the trigonometric addition identity,
The legend of Figure 3 reads verbatim “Mustache = Quaternion – Bell”, and the caption calls the mustache “Their difference”. The plotted mustache is therefore the residual between the target and the triangle; it is neither the continuous coherence term of Eq. (71) nor an independently derived contribution to a detector correlation. The cosine itself enters through the identity applied to the jointly held phases (B1, C1). The Comment [4] had described the corresponding split-and-recombine step as “a modeling postulate”.
B5. The split-and-recombine step is not connected to detector statistics. The Comment [4] identified the assembly of the correlation as a sum of separately computed polarisation and coherence contributions, , as “a modeling postulate” for which “no general probabilistic theorem guarantees that such an additive assembly reproduces Bell-type statistics”. Neither the Reply [5] nor [1] supplies the missing connection. The operators do not appear in [1], and how they relate to the continuous terms of Eq. (71) and to the populations of Section 4.3 is not stated. In [1] the recombination appears as Eq. (71), , with the two terms described as “complementary projections of a single underlying quaternion rotor” (Section 4.1). The algebraic sum is well defined and follows from the chosen rotor product. What is not established is its identification with the product of the recorded local Boolean outcomes, or with that product’s expectation. Describing and as complementary sectors does not supply this measurement relation, and the subsequent digitisation (C2) samples a new coincidence variable with mean rather than showing that the local detector products have that mean.
B6. The term “polarisation” is used for three different quantities. The term is used for: (Eq. (14), second line), which is not rotation-invariant; (Eq. (71)), which averages over to ; and the digitised triangle (Figure 3, Section 4.3). Moreover, Section 4.3.3 states that “The symmetric combination Equation (71) reconstructs the scalar invariant and produces the piecewise linear correlation”, whereas Section 4.2 states that the same equation yields . These two statements contradict each other.
B7. The Boolean benchmark is one member of the Bell-local class. The second line of Eq. (14) is said to follow “from Equation (11) by dropping the off–diagonal terms in the scalar product, ”, which “represents complete removal of the bivector phase structure”. The paper does not say which physical operation this prescription represents, or how it relates to the detector rule it specifies. The resulting product, , is then used as the Boolean benchmark that obeys CHSH. It is one example of a Bell-local correlation. Bell’s theorem, however, concerns the whole class of local response functions under a common preparation, so a comparison with this single product does not establish how the model stands in relation to that class.
B8. The rotor scalar is not shown to be a detector expectation. For a binary coincidence variable , and normalisation give , so Eq. (22) is a legitimate identity once the rotor scalar is known to be such an expectation (with the paper’s convention ). That identification is the step not supplied. The accompanying statement (“Since they are constructed from a product state of two independently propagating local rotors their use is consistent with locality”) asserts the locality of the construction rather than showing it.
C. The simulations
C1. The abstract’s description of BiSM_v1 differs from its construction. The abstract states: “Numerical simulations reproduce this correlation using only local detector events.” Section 4.2 states of : “It is not a local detector click and is never used to generate either Alice’s or Bob’s local Boolean outcome”, and Section 4.2.2 adds: “The scalar is not Bell’s response function .” is computed from both stations’ continuous phases together (Eq. (71)). The quoted statement holds for the local clicks of the Bell control, but the per-station outcomes of the cosine branch are generated from (C3).
C2. The digitisation step reproduces any prescribed mean. Eqs. (74)–(75) set whenever , with one auxiliary uniform random number per pair. Then
exactly, for any function with values in . Any correlation function with is “reproduced” in this way, including one whose four-setting CHSH combination exceeds the Tsirelson bound . The procedure samples a prescribed mean; it is not a test of whether that mean has a Bell-local realisation. The source of BiSM_v1 states that “No Malus probability P_eq=(1+R)/2 is formed or used anywhere in event generation”; the uniform threshold is nonetheless mathematically identical to sampling with that probability. The reported maximum error of (Eq. (78)) verifies the trigonometric identity in floating point.
C3. In the cosine branch the station outcomes are assigned after pairing. The paper describes the digitisation of into the coincidence variable (Eqs. (74)–(76)), but not how outcomes for Alice and Bob are then obtained. The source of BiSM_v1 shows this step: after is drawn, a further random number assigns an equal event to or , and an unequal event to or (“The equal pair is ++ or – with equal marginal weight”). The station-resolved counts in the output file are generated in this way. Alice’s recorded outcome in this branch is therefore not a function of her setting and : whether it agrees with Bob’s is decided by , a function of both settings. The marginals remain uniform, but the joint law is not of Bell’s form ; the program’s own Bell control, which is of that form, gives the triangle. Since identically (Eq. (20)), the retained phases carry exactly the difference of the two settings. Any experiment that records its analyser settings, as all Bell tests do, could generate the same coincidence data by this procedure.
C4. The paper’s local-outcome analysis is consistent with Bell’s theorem. Section 4.2.2 states that analysing the local records gives “the conventional Bell result of the piecewise–linear correlation”; executing BiSM_v1 confirms this, its Bell control giving at and at . Eqs. (80)–(81) present the two computations as analyses of “the same experimental results”. Both use the same enlarged simulated record, but they apply different functions to it, and the cosine branch also generates additional random pair labels. The enlarged record contains continuous phases and “retained local rotor geometry” (Eqs. (67)–(68)), and the paper does not establish that these are accessible detector records. The statement that “The same digitized coincidence statistics observed experimentally are thereby recovered” therefore compares a simulation using additional, jointly processed quantities with experiments that record only outcomes.
C5. The BiSM_v2 rules are not derived, and Eq. (88) does not describe the program. Several event rules are given without derivation from the bivector dynamics: the phase offsets in Eq. (84), which the program describes as implementing singlet preparation by antipodal source orientations; the factor in Eq. (85); and the “larger signed contribution” rule of Eqs. (86)–(87). Eq. (88) states that “Each event contributes to one of the four populations”, whose counts sum to . The supplementary program instead evaluates both the polarisation and the coherence rule for every event; its own comments state that “EQP+NEQP = 1 and EQC+NEQC = 1 separately” and that “the four fractions are NOT intended to sum to one”. Execution confirms that the four fractions sum to 2 at every angle. Eq. (88) therefore does not describe the implemented algorithm.
As implemented, is an average of products of local coherence outcomes, additionally averaged over absolute analyser orientation. Being generated by local functions of under a common preparation, such averages are subject to the CHSH bound (A2, A5); a harmonic shape in this angular summary does not by itself indicate a violation, which would require four consistently defined setting pairs. The program’s run report compares with on the interval only; its behaviour over the full cycle is taken up in C6.
C6. The doubled-angle harmonic appears only after a sign convention. Section 4.3.4 states that “the coherence imbalance exhibits the smooth doubled–angle, two–period harmonic”, and Section 4.3.6 that it “exhibits a doubled angular dependence and therefore has period in the analyzer separation”, from which the double-cover interpretation is drawn (E8). The implemented coherence rule does not produce this. Under the doubled-angle terms that select the blade are unchanged, while both single-angle phases change sign; away from threshold ties the coherence outcome therefore reverses, and . Execution of BiSM_v2 confirms this: , whereas . Over the full cycle the Fourier coefficient of is ; its dominant component is the first harmonic (coefficient of about ). On the data do follow , consistent with Eq. (94), , when is read as the separation folded into ; that function is not -periodic in .
The program then defines, for its portrait, an “oriented” coordinate for , describing this as a convention that “does not alter EQC or NEQC”. Only this coordinate carries a component (coefficient ) and is -periodic. The paper describes the portrait as the ordered pair of Eq. (95) and does not mention the sign reversal. The figure-eight thus contains an imposed orientation convention, and the doubled harmonic on which Section 4.3.6 relies is a property of that convention rather than of the raw populations. The lemniscate of Gerono, , illustrates the shape obtained from these two harmonics; the implemented polarisation coordinate is approximately triangular rather than exactly . The experiment proposed in Section 5.1.2 would require an operational means of controlling , resolving the two sectors, and deriving their observable statistics under that controlled preparation; the paper does not yet supply these.
D. Framing of the result
The introduction states that “Section 3 questions Bell’s locality theorem” and that “Failure of either locality or joint definability leads to failure of Bell’s locality theorem”. It also states that the work “does not challenge Bell’s derivation”, and Conclusion 5, like [5], calls both theorems “mathematically valid”. These positions are compatible only if “failure” means that the theorem’s hypotheses are not met; a theorem whose hypotheses are not met does not thereby fail.
The introduction states that “The CHSH bound applies to correlations constructed solely from Boolean outcomes ”. The bound is ; is the per- value (Eq. (62)). The same passage attributes the violation to “additional geometric effects” before any argument is given.
The paper states that “Entanglement is not required since the quaternion carries their common phase”, and that the bivector singlet of Eq. (9) “is not entangled”. Whether such a construction yields a Bell-local model depends on the conditions set out in A5: local response functions, a common setting-independent distribution, and correlations formed from products of outcomes. For the model’s local responses these conditions are met (A1, A2, A4); for the cosine they are not shown (B1, C1).
E. Physical claims made without derivation
E1. follows from the chosen function. It is obtained by maximising , which the paper calls “the simplest symmetric measure”. The maximum at is a property of that chosen function, not a consequence of symmetry as such (the equally symmetric has a minimum there), and no evolution or stability analysis links the extremum to a dynamical bifurcation. The quantum limit thus remains what [4] called a “structural postulate”.
E2. The Dirac-equation step is asserted. The coupled equation of Eq. (42) does not by itself imply the separate equalities and of Eq. (43). These require an independent restriction on the operators or their domains, which is asserted on the grounds that and act in opposite parity sectors but is not shown; a parity label is not an annihilation property. The statement “This occurs, as we show, at ” is not supported by Eqs. (37)–(44), in which does not appear; any dependence on would need to be exhibited.
E3. and Noether. The identification of with the angular momentum of the spin-1 state is asserted, without stating whether a component or a magnitude is meant. The statement that spin “does not become classical as unlike continuous variables governed by Noether’s theorem” attributes to Noether’s theorem a content it does not have: the theorem connects continuous symmetries of the dynamics with conservation laws, and does not classify discrete observables as exempt from classical limits. A fixed-spin limit and a large-spin semiclassical limit are also different and should be distinguished.
E4. No quantitative account of beta decay. Section 5.2.1 states that “A cleaved and free blade is indistinguishable from a neutrino”; Section 5.2.3 states that “There can be no neutrinos produced in the BiSM with its intact blades”. The paper distinguishes cleaved from intact blades, so these statements need not contradict each other, and it states that “no detailed BiSM reaction or assignment of charge is proposed here”. Until such a model is given, the proposal does not account quantitatively for the continuous beta spectrum and the energy and angular-momentum balance of the decay, which the neutrino hypothesis was introduced to explain.
E5. Assigned parity does not establish conservation in the decay. “The BiSM electron is odd to party [sic] while the Fermi electron of the SM is even to parity. This difference demonstrates that parity is not violated in the BiSM.” Parity assigned to one state does not show that the decay dynamics conserve parity or reproduce the asymmetry measured in the Wu experiment.
E6. Rhetorical claims supply no evidence for the alternative. Section 5.5.7 states that developments in quantum information “have no rational explanation”. The paper elsewhere acknowledges the experimental success of quantum mechanics and questions its interpretation; the statement expresses a philosophical position and supplies no mathematical evidence for the proposed alternative. Similarly, the assertion that “control of these new parameters restores determinism” (Section 5.1.1) would require preparation procedures and detector-response predictions for and , which are not given.
E7. The coherence analogy does not establish Bell-local correlations. Section 5.1.4 likens EPR correlations to “lasers, superconductors, superfluids, and Bose–Einstein condensates where collective behaviour emerges through long–range phase coherence”, and Section 4.3.5 to the double-slit experiment. The ordinary double-slit pattern invoked there is a single-particle distribution, whereas Bell’s theorem concerns joint statistics at two space-like separated stations with freely chosen settings. Macroscopic coherent systems are described quantitatively by quantum mechanics, and the analogy does not by itself supply a local model of two-station correlations.
E8. The double-cover claim is not established by a frequency ratio. Section 4.3.6 states that the original blade configuration is restored only after “a total rotation of of the blades”, and calls this “a geometric realization of the double cover ”. Unit quaternions do carry the standard double cover: with acting on vectors by , and induce the same rotation, and a physical rotation through lifts from to . Establishing that the proposed blade dynamics realise this structure requires showing how rotations about different axes, and their compositions, lift; a fixed frequency ratio about one axis does not establish it. Moreover, the period- dependence offered as its statistical signature is not present in the raw coherence data (C6). The same claim appears in [5].
F. Procedural note
The paper cites the two Quantum Reports articles [2, 3] addressed in the published Comment [4] and the author’s Reply [5], but does not cite either, or explicitly identify its responses to their specific objections, although both precede it and bear directly on its Sections 3.5, 3.6 and 5.5. The legend of Figure 3 defines the plotted mustache as the difference between the quaternion and Bell curves, the construction the Comment described as a modeling postulate. The paper’s central joint-definability argument restates, in new terminology, the argument of the Reply. The difference is that the paper now specifies explicit local response functions and a common source distribution (Eqs. (15), (16), (85), Definition 12). It is these, rather than the labels “local” and “deterministic”, that bring the local-response component within the scope of Bell’s and Fine’s theorems (A5).
G. Summary
The analysis establishes three related findings:
The local response functions of the model are defined on a common source domain with a setting-independent distribution, so Lemma 3 and Corollary 3 do not hold for them (A1–A4, A8).
Their product correlations therefore satisfy the CHSH bound, and by the equivalence the author quotes a joint distribution exists; the paper’s own simulation of these outcomes yields the Bell triangle (A5, C4).
The correlation is obtained from a scalar computed jointly from both stations’ phases and then digitised; it is not shown to be the correlation of the local outcomes (B1, B5, C1–C3). The supplementary programs further show that the BiSM_v2 populations do not form the stated partition, and that the doubled-angle harmonic appears only after a sign convention (C5, C6).
Declaration on the use of generative AI
This document was written with the assistance of Anthropic’s Claude Opus 5.5, under the direction of the authors, using material from the cited publications and the supplementary programs in [1]. Drafts were also reviewed for consistency and accuracy using OpenAI models (ChatGPT and Codex), and resulting suggestions were incorporated following manual review. Every claim in this document has been independently verified by both authors.
References
B. Sanctuary, Joint Definability, Context–Instantiated Geometry, and the Inapplicability of Fine’s Theorem, Quantum Rep.2026, 8, 96. doi:10.3390/quantum8030096 (including Supplementary Data: quantumrep-08-00096-s001.zip).
B. Sanctuary, EPR Correlations Using Quaternion Spin, Quantum Rep.2024, 6, 409–425. doi:10.3390/quantum6030026
B. Sanctuary, Spin Helicity and the Disproof of Bell’s Theorem, Quantum Rep.2024, 6, 436–441. doi:10.3390/quantum6030028
A. L. Vrba, A Collective Comment on “Spin Helicity and the Disproof of Bell’s Theorem” and Sanctuary’s Bivector Spin Framework (2023–2025), Quantum Rep.2026, 8(2), 56. doi:10.3390/quantum8020056
B. Sanctuary, Reply to Vrba, A.L. A Collective Comment on “Sanctuary, B. ‘Spin Helicity and the Disproof of Bell’s Theorem’ and Sanctuary’s Bivector Spin Framework (2023–2025)”, Quantum Rep.2026, 8, 57. doi:10.3390/quantum8030057
J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics Physique Fizika1964, 1, 195–200.
A. Fine, Hidden variables, joint probability, and the Bell inequalities, Phys. Rev. Lett.1982, 48, 291.
It has long been realized that the mathematical core of Bell’s theorem is essentially a classical probabilistic proof that a certain distributed computing task is impossible: namely, the Monte Carlo simulation of certain iconic quantum correlations. I will present a new and simple proof of the theorem using Fourier methods (time series analysis) which should appeal to probabilists and statisticians. I call it Gull’s theorem since it was sketched in a conference talk many years ago by astrophysicist Steve Gull, but never published. Indeed, there was a gap in the proof.
The connection with the topic of this session [IS18 – Quantum Computing and Statistics – organiser Yazhen Wang, University of Wisconsin-Madison] is the following: though a useful quantum computer is perhaps still a dream, many believe that a useful quantum internet is very close indeed. The first application will be: creating shared secret random cryptographic keys which, due to the laws of physics, cannot possibly be known to any other agent. So-called loophole-free Bell experiments have already been used for this purpose.
Like other proofs of Bell’s theorem, the proof concerns a thought experiment, and the thought experiment could also in principle be carried out in the lab. This connects to the concept of functional Bell inequalities, whose application in the quantum research lab has not yet been explored. This is again a task for classical statisticians to explore. R.D. Gill (2022) Gull’s theorem revisited, Entropy 2022, 24(5), 679 (11pp.) https://www.mdpi.com/1099-4300/24/5/679 https://arxiv.org/abs/2012.00719
I have recently been involved in acrimonious discussions in a Google group, https://groups.google.com/g/bell_quantum_foundations, devoted to Bell’s theorem and the interpretation of quantum mechanics. One of the group members, Bryan Sanctuary, insists that two particles leaving a source cannot remain entangled. He claims in preprints in a sequence of recent blog posts that the EPR-B correlations cannot remain valid once the particles are separated. Here is a link to one of the posts, http://blog.mchmultimedia.com/2022/05/09/hyper-helicity-and-the-foundations-of-qm/. At the same time, he claims that the EPR-B correlations do have a local and realistic interpretation once one introduces a hidden, quantum property which he calls hyper-helicity. He believes that in this way all notions of non-local collapse and weirdness can be banished from quantum mechanics. Another participant, Alexey Nikulov, also thinks that conventional quantum mechanics is wrong, and disbelieves in non-locality and collapse.
A small problem consists in figuring out what actually is Bell’s theorem. A long essay by Sheldon Goldstein (2011) on “Scholarpedia”, http://www.scholarpedia.org/article/Bell%27s_theorem, makes it clear that Bell twice quite radically changed his thinking, so in fact, one could say that there are three Bell theorems. The essential mathematics changed from the first to the second, and the physical motivation behind his characterization of a “local hidden variables model” evolved both in the first change and in the second change. The theorem is always the same, that a local hidden variables model cannot reproduce certain quantum mechanics predictions, but its assumptions and interpretation have matured.
Spacetime diagram for EPR–Bell type experiment (Goldstein & Tausk, Scholarpedia)
I have started writing out some notes, maybe they will become a whole paper, to make it clear that “collapse of the wave-function” is an interpretational optional extra, not needed in order to apply quantum mechanics in practice. No interpretation of what is going on behind the scenes is necessary if one’s purpose is to describe nature. An interpretation might be useful if it helps one to understand nature. One must realize that it is possible that understanding nature is something which will forever go beyond our poor facilities. That is not to say that the drive to gain understanding isn’t the true drive behind doing science. Like democracy and justice, understanding is an ideal which we have to continually renew and re-affirm.
That does mean that in introductory expositions it should be clearly labelled as a lie for children.
The usual colourful language involving the non-local collapse of the wave function can be thought just to be a description of a useful computational tool, not a description of physical changes to something existing in physical reality. The only thing assumed to exist are measurement outcomes, and the theory allows us to compute probability distributions of their outcomes, also in complex, composite, sequential, experimental set-ups. One can compute what one needs to know by pretending that the wave function collapse as suggested by the von Neumann-Lüders extension of the Born law is somehow real. One gets the right answer, as directly as possible. There is, however, no need to think of wave function collapse as being something physical (and necessarily non-local). Such thinking is an optional extra. Some people find it distasteful. Tastes differ. I think it can be usefully thought of as one of those lies for children which need to be seen in a different light as one gains maturity and knowledge.
We are all children! Learning never ends!
The second purpose is to write out the mathematical content of Sanctuary’s claims concerning hyper-helicity. Sanctuary feels that his interpretation of his mathematical formalism will revolutionize our understanding of quantum entanglement. Clearly, he has a long way to go, but I hope my own struggle to understand what he is doing will be helpful to others, if not to him.
Poet and McGill University emeritus professor of chemistry Bryan Sanctuary (Google scholar: https://scholar.google.com/citations?user=iqR_MusAAAAJ&hl=en&oi=ao; personal blog: https://mchmultimedia.com/sanctuaryblog/) is betting me 5000 Euro that he can resolve the EPR-Bell paradox to the satisfaction of the majority of our peers. Moreover, he will do it by a publication (or at least, a pre-publication) within the year. That’s this calendar year, 2022. Naturally, he expects public (scientific) acclaim to follow “in no time”. I don’t expect that. We will settle the bet by consultation with our peers, and this consultation will be concluded by the end of the following year. So that’s by the end of the succeeding calendar year, 2023.
John S. Bell inspects the Christmas present which his friends the Bertlmanns have just given him
I, therefore, expect his gracious admission of defeat and a nice check for 5000 Euro, two years from now.
He expects the opposite. (Poor Bryan! It’s like taking candy from a baby…)
(He presumably thinks the same)
The small print
Small print item 1: Who are our peers? Like a jury, they will be determined by having our mutual approval. To begin with, we will invite the members of a couple of Google groups/internet seminars in which one or both of us already participate. Here are links to two of them: Jarek Duda’s (Krakow) “QM foundations & nature of time seminar”: https://groups.google.com/g/nature-of-time/about and http://th.if.uj.edu.pl/~dudaj/QMFNoT; and Alexandre de Castro’s Google group “Bell inequalities and quantum foundations”: https://groups.google.com/g/Bell_quantum_foundations.
Small print item 2: What does Bryan think he’s going to achieve? Restoration of locality and realism, and banning of weirdness and spookiness from quantum mechanics.
Small print item 3: What do I think about his putative theory? Personally, but it is not up to me to decide, I would accept that he has won if his theory (which he has not yet revealed to the world) would allow me to win my Bell game challenge https://gill1109.com/2021/12/22/the-bell-game-challenge/ “against myself”. i.e., it would allow me to write computer programs to simulate a successful loophole-free Bell experiment – thus satisfying the usual spatiotemporal constraints on inputs and outputs while preventing conspiracy, and reliably violating a suitable Bell inequality by an amount that is both statistically and physically significant. This means that, in my opinion, he should only win if he can convince the majority of our peers that those constraints are somehow unphysical. I mention that if experimenters voluntarily impose those constraints (to the best of their ability) in real experiments, then there cannot be a metaphysical reason to forbid them. However, the bet will be settled by a democratic vote of our peers! Clearly, this does constitute a loophole for me: a majority of our peers might still fall for superdeterminism or any other craziness.
I suspect that Bryan believes he can now resurrect his previous attempt https://arxiv.org/abs/0908.3219. I think it is very brave of him but doomed to failure, because I don’t think he will come up with a theory that will catch on. (I even believe that such a theory is not even possible, but that’s my personal belief).
To reiterate: our peers will determine who has won our bet. Bryan is betting that a year from now he will have revolutionised quantum mechanics, restoring locality and realism and that his then appearing paper will rapidly force Zeilinger, Gisin, me, and a host of others, to retract our papers on quantum teleportation, quantum non-locality, and all that. I am betting that the world will not be impressed. Our peers will vote whether or not they believe that Bryan has achieved his goal.
Since 2015, Bell-type experiments designed to test local realism have the following format: the format of a so-called “loophole-free Bell test”. There is a fixed sequence of N time-slots, or more precisely, paired time-slots. These are time-slots in two distant labs owned by two scientists Alice and Bob. The time-slots are paired such that a signal sent at the start of one of Alice’s time slots from Alice’s to Bob’s lab, travelling at the speed of light, would only reach Bob’s lab after the end of Bob’s corresponding time-slot; and vice versa. Just after the start of each time-slot, each inserts a binary setting into an experimental device. Something goes on inside that apparatus, and before the time-slot is over, a binary outcome is produced. Each instance with two inputs and two outputs is called a trial.
From Bell’s “Bertlmann’s socks” paper. Inputs are shown below and outputs above the long horizontal box which encloses Alice and Bob’s devices and what is in between
Actually, many experiments require a slightly more elaborate protocol involving a third lab, which you may think of as a source of “pairs of particles”. Charlie’s lab is located somewhere between Alice and Bob’s. Charlie’s device outputs the message “ready” or “not ready” before the end of his time-slot (its length is irrelevant). The message however could only arrive at Alice and Bob’s lab after they have already input their input settings, so could not directly influence their choices. Outcomes get delivered anyway. After the experiment, one looks only at the inputs and outputs of each trial in which Charlie saw the output “ready”. The experiment continues long enough that there are N trials labelled by Charlie’s apparatus as “ready”. From now on, I will forget about this “post-selection” of N trials: the first N which went off to a good start. (The word “post-selection” is a misnomer. It is performed after the whole experiment is complete, but the selection is determined in advance of the introduction of the settings).
Space-time disposition of the time-slots of one trial. The sloping arrows are the boundaries of future light-cones with vertices at the start of Alice, Bob, and Charlie’s time-slots.
The settings are typically chosen to resemble sequences of outcomes of independent fair coin tosses. Sometimes they are generated by physical random number generators using physical noise sources, sometimes they are created using pseudo random number generators (RNGs). Sometimes they are generated on the fly, sometimes created in advance. The idea is that the settings are inputs which come from the outside world, outside the experimental devices, and the outcomes are outputs delivered by the devices to the outside world.
Below is a graphical model specified in the language of the present-day theory of causality based on directed acyclic graphs (DAGs), describing the dependence structure of what is observed in terms of “hidden variables”. There is no assumption that the hidden parts of the structure are classical, nor that they are located in classical space-time. The node “psi” stands for the state of all experimental apparatus in the three labs including transmission lines between them before one trial of the experiment starts, as far as is directly relevant in the causal process leading from experimental inputs to experimental outputs. The node “phi” consists of the state of external devices which generate the settings. The graphical model says that as far as the settings and the outputs are concerned, “phi” and “psi” can be taken to be independent. It says that Bob’s setting is not in the causal pathway to Alice’s outcome.
At the end of the experiment, we have N quadruples of binary bits (a, b, x, y). Here, a and b are the settings and x and y are the outcomes in one of the N “trials”. We can now count the number z of trialsin which x = yandneithera or b = 1, together with trials in which x ≠ yandbotha and b = 1. Those two kinds of trials are both considered trials having the result “success”. The trials remaining have the result “fail”.
Now, let B(p) denote a random variable distributed according to the binomial distribution with parameters N and p. Think of the number of successes z to be the outcome of a random variable Z. According to local realism, and taking p = 0.75, it can be proved that for all z > N p, Prob( Z ≥ z ) ≤ Prob( B(p) ≥ z ). According to quantum mechanics, and with q = 0.85, it appears possible to arrange that for all z, Prob( Z ≤ z ) = Prob( B(q) ≤ z ). Let’s see what those binomial tail probabilities are with z = 0.80 N, using the statistical programming language “R“.
N <- 1000 p <- 0.75 z <- 0.8 * N q <- 0.85 pbinom(z, N, p, lower.tail = FALSE) [1] 8.029329e-05 pbinom(z, N, q, lower.tail = TRUE) [1] 1.22203e-05
We see that an experiment with N = 1000 time-slots should be plenty to decide whether the experimental results are the result of local realism with a success rate of maximally 75%, or of quantum mechanics with a success rate of 85% (close to the theoretical maximum under quantum mechanics). The winning theory is decided by seeing if the observed success rate is above or below 80%.
Challenge: show by a computer simulation that my claims are wrong. ie, simulate a “loophole-free” Bell experiment with a success rate reliably exceeding 80% when the number of trials is 1000 or more. Rules of the game: you must allow me to supply the “fair coin tosses”. Your computer simulation may use an RNG (called a fixed number of times per trial) to create its own randomness, but it must have “set seed” and “restore seed” facilities in order to make each run exactly reproducible if required. For each n, Alice’s nth output x may depend only on Alice’s nth input a, together with (if desired) all the preceding inputs and outputs. Similarly, Bob’s nth output y may depend only on Bob’s input b, together with (if desired) all the preceding inputs and outputs
Here is a different version of the challenge using the classical Bell-CHSH inequality instead of the more modern martingale inequality. Another version could be specified using the original Bell inequality, for which one would also demand that at equal settings, outcomes are always equal and opposite. After all, the original Bell inequality also assumes perfect anti-correlation, so one must check that that assumption holds.
The whole point of a computer simulation is that an independent judge is unnecessary: your code is written in a widely and freely available language suitable for scientific computing, and anyone with basic computing skills can check that the programming team is not cheating (whether deliberately or inadvertently). The independent judge is the entire scientific community. If you are successful, the simulation will actually be an example of a classical physical system producing what has been thought to be a unique signature of quantum entanglement. You, the lead scientist, will get the Nobel Prize because you and your team (I imagine that you are a theoretician who might need the assistance of a programmer) will have disproved quantum theory by a reproducible and rigorous experiment. No establishment conspiracy will be able to suppress the incredible and earth-shaking news.
Here are my stipulations on the program. I am assuming that it uses a built-in pseudo-random number generator. I assume that it includes “set.seed” and “save.seed” facilities. Otherwise, it is not useful for scientific work and not eligible for my challenge.
From now on, the phrases “photon pair”, “time slot”, and “trial” are taken to be interchangeable. After all, we are talking about a computer simulation, so the actual evocative natural language words which we use as names for variables and functions are irrelevant.
The program must accept as input a number of trials N, a seed setting the RNG, and two lists of setting labels “1” and “2” of length N. It must generate as output two lists of outcomes +/–1, also of length N. For all n, Alice’s n‘th output depends only on Alice’s n‘th input, as well (if you like) on the inputs and outputs on both sides of earlier trials. And similarly for Bob. I will check this constraint by doing many random spot checks. This is where the rule concerning the RNG comes in.
Let’s take N = 10,000. You will win if the CHSH quantity S exceeds 2.4 in a few repeats with different RNG seeds and varying the lists of inputs. In other words, the violation of the Bell-CHSH inequality is reproducible, and reproducible by independent verifiers. I will supply the lists of inputs after you have published your code. The inputs will be the result of a simulation of independent fair coin tosses using standard scientific computing tools. If you don’t trust me, we can ask a trusted third party to make them for us.
At the 8th MaxEnt conference in 1998, held in Cambridge UK, Ed Jaynes was the star of the show. His opening lecture has the following abstract: “We show how the character of a scientific theory depends on one’s attitude toward probability. Many circumstances seem mysterious or paradoxical to one who thinks that probabilities are real physical properties existing in Nature. But when we adopt the “Bayesian Inference” viewpoint of Harold Jeffreys, paradoxes often become simple platitudes and we have a more powerful tool for useful calculations. This is illustrated by three examples from widely different fields: diffusion in kinetic theory, the Einstein–Podolsky–Rosen (EPR) paradox in quantum theory [he refers here to Bell’s theorem and Bell’s inequalities], and the second law of thermodynamics in biology.”
Unfortunately Jaynes was completely wrong in believing that John Bell had merely muddled up his conditional probabilities in proving the famous Bell inequalities and deriving the famous Bell theorem. At the conference, astrophysicist Steve Gull presented a three line proof of Bell’s theorem using some well known facts from Fourier analysis. The proof sketch can be found in a scan of four smudged overhead sheets on Gull’s personal webpages at Cambridge University.
Together with Dilara Karakozak I believe I have managed to decode Gull’s proof, https://arxiv.org/abs/2012.00719, though this did require quite some inventiveness. I have given a talk presenting our solution and point out further open problems. I have the feeling progress could be made on interesting generalisations using newer probability inequalities for functions of Rademacher variables.
Lorentz Center workshop proposal, Leiden, 6–10 September 2021
As quantum computing and quantum information technology moves from a wild dream into engineering and possibly even mass production and consumer products, the foundational aspects of quantum mechanics are more and more hotly discussed. Whether or not various quantum technologies can fulfil their theoretical promise depends on the fact that quantum mechanical phenomena cannot be merely emergent phenomena, emerging from a more fundamental physical framework of a more classical nature. At least, that is what Bell’s theorem is usually understood to say: any underlying mathematical physical framework which is able, to a reasonable approximation, to reproduce the statistical predictions made by quantum mechanics, cannot be local and realist. These words have nowadays precise mathematical meanings, but they stand for the general world view of physicists like Einstein, and in fact they stand for the general world view of the educated public. Quantum physics is understood to be weird, and perhaps even beyond understanding. “Shut up and calculate”, say many physicists.
Since the 2015 “loophole-free” Bell experiments of Delft, Munich, Vienna and at NIST, one can say even more: laboratory reality cannot be explained by a classical-like underlying theory. Those experiments were essentially watertight, at least as far as experimentally enforceable conditions are concerned. (Of course, here is heated discussion and criticism, too).
Since then however it seems that even more energy than ever before is being put into serious mathematical physics which somehow gets around Bell’s theorem. A more careful formulation of the theorem is that the statistical predictions of quantum mechanics cannot be reproduced by a theory having three key properties: locality, realism, and no-conspiracy. What is meant by no-conspiracy? It means that experimenters are free to choose settings of their experimental devices, independently of the underlying properties of the physical systems which they are investigating. In the case of a Bell-type experiment, a laser aimed at a crystal which emanates a pair of photons which arrive at two distant polarising photodectors, ie detectors which can measure the polarisation of a photon in directions chosen freely by the experimenter. If the universe actually evolves in a completely deterministic manner, then everything that goes on in those labs (housing the source and the detectors and all the cables or whatever in between) was determined already at the time of the big bang, the photons can in principle “know in advance” how they are going to be measured.
At the present time, highly respectable physicists are working on building a classical-like model for these experiments using superdeterminism. Gerard ’t Hooft used to be a lonely voice arguing for such models but he is no longer quite so alone (cf. Tim Palmer, Oxford, UK). Other physicists are using a concept called retro-causality: the future influences the past. This leads to “interpretations of quantum mechanics” in which the probabilistic predictions of quantum mechanics, which seem to have a built in arrow of time, do follow from a time symmetric physics (cf. Jaroslav Duda, Krakow, Poland).
Yet other physicists dismiss “realism” altogether. The wave function is the reality, the branching of many possible outcomes when quantum systems interact with macroscopic systems is an illusion. The Many Worlds Interpretation is still very alive. Then there is QBism, where the “B” probably was meant to stand for Bayesian (subjectivist) probability, in which one goes to an almost solipsistic view of physics; the only task of physics is to tell an agent what are the probabilities of what the agent is going to experience in the future; the agent is rational and uses the laws of quantum mechanics and standard Bayesian probability (the only rational way to express uncertainty or degrees of belief, according to this school) to update probabilities as new information is obtained. So there only is information. Information about what? This never needs to be decided.
On the right, interference patterns of waves of future quantum possibilities. On the left, the frozen actually materialised past. At the boundary, the waves break, and briefly shining fluorescent dots of light on the beach represent the consciousness of sentient beings. Take your seat and enjoy. Artist: A.C. Gill
Yet another serious escape route from Bell is to suppose that mathematics is wrong. This route is not taken seriously by many, though at the moment, Nicolas Gisin (Geneva), an outstanding experimentalist and theoretician, is exploring the possibility that an intuitionistic approach to the real numbers could actually be the right way to set up the physics of time. Klaas Landsman (Nijmegen) seems to be following a similar hunch.
Finally, many physicists do take “non-locality” as the serious way to go; and explore, with fascinating new experiments (a few years ago in China, Anton Zeilinger and Jian-Wei Pan; this year Donadi e al.), hypotheses concerning the idea that gravity itself leads to non-linearity in the basic equations of quantum mechanics, leading to the “collapse of the wave function”, by a definitely non-local process.
At the same time, public interest in quantum mechanics is bigger than ever, and non-academic physicists are doing original and interesting work, “outside of the mainstream”. Independent researchers can and do challenge orthodoxy, and it is good that someone is doing that. There is a feeling that the mainstream has reached an impasse. In our opinion, the outreach from academia to the public has also to some extent failed. Again and again, science supplements publish articles about amazing new experiments, showing ever more weird aspects of quantum mechanics, but it is often clear that the university publicity department and the science journalists involved did not understand a thing, and the newspaper articles are extraordinarily misleading if not palpably nonsense.
In the Netherlands there has long been a powerful interest in foundational aspects of quantum mechanics and also, of course, in the most daring experimental aspects. The Delft experiment of 2015 was already mentioned. At CWI, Amsterdam, there is an outstanding group led by Harry Buhrman in quantum computation; Delft has a large group of outstanding experimentalists and theoreticians, in many other universities there are small groups and also outstanding individuals. In particular one must mention Klaas Landsman and Hans Maassen in Nijmegen; and one must mention the groups working in the foundations of physics in Utrecht and in Rotterdam (Fred Muller). Earlier we had of course Gerard ’t Hooft, Dennis Dieks and Jos Uffinck in Utrecht; some of them retired but still active, others moved abroad. A new generation is picking up the baton.
The workshop will therefore bring a heterogeneous group of scientists together, many of whom disagree fundamentally on basic issues in physics. Is it an illusion to say that we can ever understand physical reality? All we can do is come up with sophisticated mathematics which amazingly gives the right answer. Yet there are conferences and Internet seminars where these disagreements are fought out, amicably, again and again. It seems that perhaps some of the disagreements are disagreements coming from different subcultures in physics, very different uses of the same words. It is certainly clear that many of those working on how to get around Bell’s theorem, actually have a picture of that theorem belonging to its early days. Our understanding has enormously developed over the decennia, and the latest experimentalists have perhaps a different theorem in mind, to the general picture held by theoretical physicists who come from relativity theory. Indubitably, the reverse is also true. We are certain that the meeting we want to organise will enable people from diverse backgrounds to understand one another more deeply and possibly “agree to differ” if the difference is a matter of taste; if however the difference has observable physical consequences then we must be able to figure out how to observe them.
The other aim of the workshop is to find better ways to communicate quantum mysteries to the public. A physical theory which basically overthrows our prior conceptions of time, space and reality, must impact culture, art, literature; it must become part of present day life; just as earlier scientific revolutions did. Copernicus, Galileo, Descartes, Newton taught us that the universe evolves in a deterministic (even if chaotic) way. Schrödinger, Bohr and all the rest told us this was not the case. The quantum nature of the universe certainly did impact popular culture but somehow it did not really impact the way that most physicists and engineers think about the world.
Illustration from Wikipedia, article on Bell’s Theorem. The best possible local realist imitation (red) for the quantum correlation of two spins in the singlet state (blue), insisting on perfect anti-correlation at 0°, perfect correlation at 180°. Many other possibilities exist for the classical correlation subject to these side conditions, but all are characterized by sharp peaks (and valleys) at 0°, 180°, and 360°, and none has more extreme values (±0.5) at 45°, 135°, 225°, and 315°. These values are marked by stars in the graph, and are the values measured in a standard Bell-CHSH type experiment: QM allows ±1/√2 = ±0.7071…, local realism predicts ±0.5 or less.
Don’t be so impatient. All will be explained, in due time. In fact, time, and associations in space and time, is what this posting is all about.
Firstly, the *image* is the album art of the eponymous Joy Division album [I so love using the word “eponymous”!]. If you really do want to listen to it, here’s a YouTube link: https://www.youtube.com/watch?v=3UYnyiL8-VI
I warn you, it’s not everyone’s cup of tea.
Secondly, I have to tell you that last Monday I gave a Zoom talk at the dept. of physics at the Jagellionian University, Kraków; in a seminar series, hosted by my friend Jarek Duda. The announcement said that the talk (on quantum foundations, and in particular on the issues of time in Bell’s theorem) would start at 17:00 hours Warsaw time and for some days I was under the misapprehension that I would give (and later, had given) a virtual talk in Warsaw. Kraków, Warsaw, … I have wonderful memories of a number of fascinating Polish cities.
While preparing my slides I belatedly learnt that two or three months previously Boris Tsirelson (Tel Aviv) , one of my greatest scientific heros, had passed away in Basel, aged 70. One year older than me. (His family originally came from Bessarabia – nowadays more of less Moldavia. More holocaust connections here). Boris’ whole approach to Bell’s theorem, and not just his famous inequality (the “Tsirelson bound”), had always deeply resonated with me. I felt devastated, but also inspired.
Actually when I was asked if I would like to make a contribution to the J U Kraków seminar, the provisional title of my talk, and its initial “abstract”, referred to “Bell-denialists”. Of course I was thinking of one of my current Bell-denialist friends (recently referred to as my “nemesis” by another one of my friends, but I think of him more as an inspiring sparring partner) Joy Christian. So there comes the word “Joy” again. Those who are not fans of English post-punk of the late 70’s and early 80’s might like to confer with Wikipedia, to find out what historical organisation was alluded to in the name of the band https://en.wikipedia.org/wiki/Joy_Division. The lead singer, Ian Curtis, famously committed suicide at the very young age (for suicidal rock stars) of 23. He certainly was a “troubled young man” … . See the very beautiful movie “Control” directed by the Dutch photographer Anton Corbijn https://en.wikipedia.org/wiki/Control_(2007_film).
Coincidentally, today I saw the announcement of a new paper by my quantum friend Sascha Vongehr, “Many Worlds/minds Ethics and Argument Against Suicide: for Emergencies and Evaluation in Long Term Suicide Prevention and Mental Health Outcome”, on viXra, https://vixra.org/abs/2004.0158. There are actually some very fine papers on viXra!
Perhaps, but maybe that will be on another day, and maybe even another posting, I will explain what my talks finally decided to be about.
In the meantime, thinking of requiems and Warsaw made me think of a piece by one of my favourite composers Alfred Schnittke, dedicated to the memory of the victims of the bombing of Belgrade by the nazi’s. I will add a link to a suitable YouTube performance, if I can find it. If this piece of music indeed exists anywhere, apart from in my mind. Google search is not giving me any help. I have to locate my CD collection…
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