quantum-foundations.
101 of 1538 NOTE documents. Back to the full index.
| # | Title | Summary | Status |
|---|---|---|---|
| NOTE-003 v2 | QM as stochastic optimization on spacetimes | The paper derives a Stueckelberg-type equation i∂φ/∂τ = (1/2m)□φ − Vφ (Eq. 36) from an HJB equation, but only after putting the imaginary unit in twice by hand. The first i is a factor √g = i multiplying the Lagrangian; standard relativity uses √(−g) = 1. The second is a complex “variance” σ² = i/m (Eq. 29), chosen so that the Hopf–Cole substitution J = log φ linearizes the equation. The linearization itself is the standard λ = σ²·(control cost) condition. The Schrödinger equation (Eq. 43) then comes from c → ∞ plus complex conjugation. The Dirac equation is not derived, and the “Klein–Gordon equation” (Eq. 40) has 2mV where m² should be. | Read |
| NOTE-011 v2 | Quantum Thermodynamics | A 2015–16 review with no new results, whose one synthesis (§5.2, §7) is that the field used four inequivalent notions of work: ensemble ⟨W⟩ = ∫tr[ρḢ]dt (Eq. 2), two-point-measurement W = E_m^(τ) − E_n^(0) (Eq. 24), ε-deterministic single-shot A^ε ≈ −kT ln(Z_Λ*/Z) (Eq. 49) and resource-theory w^max_ε = F^min_ε(ρ_S) − F(τ_S) (Eq. 50), of which only the last two coincide. It also judges that the field had “only made a small number of experimentally checkable predictions of new thermodynamic effects yet” (§7, p. 39). | Read |
| NOTE-013 v2 | Quasi-quantization | Take a classical statistical theory over a phase space Ω = R^{2n} or (Z_d)^{2n}, and allow only uniform distributions over V^⊥ + v, where V is an isotropic subspace (a Poisson-commuting set of linear “quadrature” variables) and v ∈ V gives their values. The result is operationally identical to the quadrature quantum subtheory (the stabilizer formalism for odd prime d) for continuous variables and for odd prime d; the proof is that the Wigner functions of that subtheory are exactly these distributions (§IV.A). For d = 2 the two theories are isomorphic element by element but make different predictions, because the qubit stabilizer subtheory is contextual (§IV.B). | Read |
| NOTE-016 v2 | The sheaf-theoretic structure of non-locality and contextuality | For finite measurement scenarios, an empirical model has a factorizable (local / non-contextual) hidden-variable realization iff its compatible family of distributions extends to a global section (Thm 8.1). Over the reals every no-signalling model has such a section (Thm 5.9: “negative-probability LHV” ⇔ no-signalling, because both span the same subspace, of dimension D = Σ_{U∈𝒰}(l−1)^{|U|}, Thm 5.4). Contextuality comes in three strictly nested strengths (probabilistic ⊋ possibilistic ⊋ strong), with the Bell, Hardy and GHZ(n≥3) models as witnesses of each level. | Read |
| NOTE-017 v2 | A structure theorem for generalized-noncontextual ontological models | Take a tomographically local GPT and any convex-linear, empirically adequate, diagram-preserving map M from it into FVect_R. Then M(T) = χ_B ∘ T ∘ χ_A⁻¹ for unique invertible linear maps χ_A (Thm 4.1). Consequences: - Every diagram-preserving quasiprobabilistic model is an exact (non-overcomplete) frame representation (Prop. 4.3). - Every diagram-preserving noncontextual ontological model has |Λ_A| = dim(A), the GPT dimension (Cor. 4.2, §4.3): exactly d² ontic states for a qudit and 4 for a qubit. - The paper also proves (Cor. 3.5) that three things are equivalent in arbitrary compositional scenarios, without needing tomographic locality: a noncontextual ontological model of the operational theory; an ontological model (simplex embedding) of its GPT; and a positive quasiprobabilistic model of its GPT. | Read |
| NOTE-018 v2 | Why interference phenomena do not capture the essence of quantum theory | The paper models each mode of the interferometer as a classical bit pair (occupation N ∈ Z_2, phase Φ ∈ Z_2). The agent may know only one of N, Φ or N⊕Φ and must be maximally ignorant of the rest. Under that restriction, local and deterministic dynamics (beamsplitter: swap N_L with Φ_L⊕Φ_R) reproduce exactly the 50-50, ϕ ∈ {0, π} Mach–Zehnder statistics (output port L with probability 1 or 0; ½/½ with a which-way detector). The same holds for the Elitzur–Vaidman bomb tester (success probability ¼), delayed choice and the quantum eraser (interference within each P_A outcome class, none within each Q_A class). So this “TRAP” slice of interference phenomenology cannot, on its own, force wave–particle complementarity, observer-dependence or nonlocal/retrocausal influence. The paper leaves aside the phenomenology it concedes is nonclassical (unbalanced beamsplitters, the functional form of the duality relation, Hardy’s interferometer). | Read |
| NOTE-021 v2 | Distinguishability and Accessible Information in Quantum Theory | Minimizing the Bhattacharyya overlap Σ_b √(tr ρ0E_b)√(tr ρ1E_b) over all POVMs gives exactly F(ρ0,ρ1) = tr√(ρ1^{1/2}ρ0ρ1^{1/2}) (Eq. 3.28). The minimum is attained by a projective measurement in the eigenbasis of M = ρ1^{-1/2}√(ρ1^{1/2}ρ0ρ1^{1/2})ρ1^{-1/2} (Eq. 3.67). For binary ensembles, the same Schwarz-inequality technique gives three results: a short proof of the Holevo bound I(t) ≤ S(t), with equality iff [ρ0,ρ1] = 0 (Eq. 3.149); a tighter upper bound L(t) (Eq. 3.187, integrated in Eqs. 3.230/3.244; closed form for qubits in Eq. 3.269); and a measurement-attained lower bound M(t) = tr[(1−t)ρ0 ln L_ρ(ρ0) + tρ1 ln L_ρ(ρ1)] (Eq. 3.203). The thesis also proves that two states can be broadcast iff they commute, and cloned iff they are identical or orthogonal (§4.3, proof written for invertible ρ0, ρ1). | Read |
| NOTE-030 v2 | Pusey, Barrett & Rudolph, reality of the quantum state (PBR) | Take pure states |ψ0⟩, |ψ1⟩ with |⟨ψ0|ψ1⟩| = cos θ. Assume (a) a model assigns each preparation a distribution µ_i(λ) over physical states, (b) n independent preparations yield the product distribution µ_{x1}(λ1)⋯µ_{xn}(λn), and (c) outcome probabilities depend only on (λ1, …, λn). If the model’s predictions are within ε of quantum theory, then D(µ0, µ1) ≥ 1 − 2ε^{1/n} for any n with 2^{1/n} − 1 ≤ tan(θ/2) (Eq. 7, App. B). At ε = 0 the distributions for distinct pure states cannot overlap on a set of non-zero measure, so Harrigan–Spekkens ψ-epistemic models are ruled out. | Read |
| NOTE-035 v2 | Plávala, General probabilistic theories: An introduction | A self-contained textbook on finite-dimensional GPTs whose central results are that state spaces and effects are dual (K = {φ ∈ A(K)*+ : ⟨φ, 1_K⟩ = 1}, Thm 3.19) and that optimal two-state discrimination succeeds with probability ½(1 + ‖λx₀ − (1−λ)x₁‖) in base norm (Thm 3.43). Every composite lies between the minimal and maximal tensor products (Prop. 5.6), and four statements are equivalent: id_K is measure-and-prepare, id_K is self-compatible, K admits a universal broadcasting channel, and K is a simplex (Thm 7.7); minimal = maximal iff one factor is a simplex is cited to [134], not proved. | Read |
| NOTE-038 v2 | Structural Realism (Stanford Encyclopedia of Philosophy) | The entry presents structural realism (Worrall 1989) as the attempt to keep both the no-miracles argument and the lesson of theory change. What is retained, and what we are entitled to believe, is structure, understood by ontic structuralists as modal structure. It sorts the view into an epistemic form, which falls to the Newman/Demopoulos–Friedman triviality result when stated as a Ramsey sentence (§3.2), and an ontic form with seven distinguishable theses, from eliminativism about individuals to “objects are constructs” (§4). The ontic form is motivated by quantum non-individuality and entanglement, gauge symmetry and the hole argument. The entry closes with seven objections, each answered in the author’s own voice (§5). | Read |
| NOTE-043 v2 | Observational entropy meets maximum entropy | The paper defines S^τ_M(ρ) = S(τ) − D_M(ρ‖τ) = −Σ_x p_x log(p_x/V_x), with V_x = Tr(τM_x)e^{S(τ)} (Eqs. 10–12). It is bounded as S(τ) ≥ S^τ_M(ρ) ≥ S(ρ) when ρ obeys the constraint S(ρ;τ) ≤ S(τ) (Eqs. 17–18). It reduces to 24 listed entropies, and it removes traditional observational entropy’s infinite-dimensional pathologies whenever S(τ) < ∞ (§VIII.7). Its “second laws” are bounds D_M(ρ̄‖τ) ≤ ε on the time-averaged state, split as equilibration plus prior-fit (Eq. 27). The paper proves the first term small in two regimes (Eqs. 28–29) and the second only under the ETH (Eq. 31). It states that in its own numerical examples the smallness “was [not] proved here analytically” (p. 12). | Read |
| NOTE-044 v2 | Grangier & Auffèves, What is quantum in quantum randomness? | The paper argues informally, with no equations anywhere in it, that an “elementary” system with a fixed finite number N of exclusive modalities, placed in contexts that can change, must behave unpredictably when the context changes: a return to C1 after C2 cannot be certain, or there would be N² > N joint modalities (p. 4). It asserts that adding one condition, that the context can change continuously, makes the N×N modality-transition matrix unistochastic, i.e. Born-rule, but defers that proof to refs [14, 15] (p. 5). The anti-reductionist conclusion follows from Postulate 0, which declares contexts non-contextual by definition, not from an argument against decoherence. | Read |
| NOTE-066 v2 | Backens & Duman, graphical calculus for Spekkens’ toy bit | Take the maximal-knowledge fragment of Spekkens’ toy bit theory with post-selected measurements, viewed as relations over {1,2,3,4}ⁿ. It has a ZX-style red–green graphical calculus that is universal (§4.3), sound (§4.4) and complete (Thm 17). Completeness goes through a proof that the toy theory’s binary “check-matrix” formalism coincides with the qubit stabilizer one (valid states SᵀJS = 0, valid transformations QᵀJQ = J, Lemma 3). With it come graph-state/local-complementation results for toy states (Thms 4–5), map–state duality (§5.2) and an rGS-LO normal form (Thms 10–11). The rewrite rules differ from the stabilizer ZX-calculus only through the phase group: Z₂×Z₂ against Z₄. That difference appears as an “11-commutation” rule in place of π-commutation (§4.5). | Read |
| NOTE-072 v2 | Hance et al., wavefunctions as knowledge and reality | The paper argues, informally and without formal results, that Harrigan–Spekkens’ ψ-ontic/ψ-epistemic are contradictories only by stipulation. On the informal senses, “represents reality” and “represents knowledge”, a wavefunction can be both. It offers a weaker sufficient condition for “epistemic” (Eq. 6: for some ψ, A, S, the response function A(S|λ) is not equal to the Born probability p^ψ_A(S) throughout Λψ), which non-overlapping models can meet. It concludes that PBR “shows that wavefunctions cannot overlap on state space, but it says nothing about whether they are epistemic in the broader sense”, so it “does not rule out all epistemic models” (§V). | Read |
| NOTE-077 v2 | Cuffaro & Hartmann, The Open Systems View | The paper argues philosophically, not formally, that the quantum theory of open systems “GT” is more fundamental than standard closed-system quantum theory “ST”. GT treats states as density operators that evolve, in general, non-unitarily, and it admits not-completely-positive maps. The ontic and epistemic cases each win only on a second, interpretation- or practice-laden criterion: OntFund-2 and EpFund-2. On the first, formal criteria (OntFund-1, EpFund-1) the two frameworks tie, and the authors say so because Stinespring dilation and dropping the Lindblad terms translate in both directions (§4.1, §4.2.1). Explanatory fundamentality is granted only deflationarily (ExFund-1), and the authors recommend dropping the notion (§4.2.2). The step from there to “the open systems view is fundamental tout court” (§4.3) rests on the special sciences and on “there are no other options”. | Read |
| NOTE-080 v2 | Barandes, The Stochastic-Quantum Correspondence | Take any N×N stochastic transition matrix Γ(t←0) on a configuration space. Writing it entrywise as Γ_ij = |Θ_ij|² = tr(Θ†P_iΘP_j) (Eq. 15) is always possible, and the paper calls it the stochastic-quantum dictionary. Choosing Θ unitary (by Stinespring dilation to ≤N³ configurations, §3.4) makes Γ unistochastic. The Born rule in the configuration basis (Eq. 23), the Schrödinger equation (Eq. 35, with H ≡ iħ(∂U/∂t)U†) and interference (Eq. 43, the gap between Γ(t←0) and |U(t←t′)|²|U(t′←0)|²) then follow by definition or short algebra. The Born rule for non-diagonal observables (Eq. 78) is derived by reading a measuring device’s configuration, given a composite unitary that produces Eq. (73). | Read |
| NOTE-082 v2 | Barnum, Gaebler & Wilce, ensemble steering and self-duality | A bipartite state ω steers its B-marginal iff ω̂ : [0,u_A] → [0,ω_B] is a strong quotient map of ordered sets (Thm 5.4). When ω̂ is injective and the marginal is interior, a steering state is an order-isomorphism (Cor. 5.5). With Thm 4.1/Cor. 4.2, it follows that universal self-steering makes every irreducible state space homogeneous and weakly self-dual (Prop. 5.8). The step from there to C*-algebraic quantum theory needs strong self-duality, which the paper does not derive, and a Hanche-Olsen correspondence it defers (“as we will establish elsewhere”, §6). | Read |
| NOTE-088 | The Homological Nature of Entropy | On a finite probability space, take an information structure S (a set of partitions closed under joins that have a common refinement in S and containing the trivial partition) and a probability family Q complete for it. The information cohomology H*(S;Q) is computed by a Hochschild-type bar complex whose action is Shannon’s averaged conditioning. If S is connected and “sufficiently rich” and cochains are measurable, then H¹ is one-dimensional and spanned by Shannon entropy (Thm 1). This makes entropy the unique measurable 1-cocycle, up to one multiplicative constant per connected component. In the unitary-invariant quantum case H¹_U = 0: Shannon entropy is the coboundary of minus the von Neumann entropy, which is the unique invariant 0-cochain with that property up to an additive constant (Thm 3). The tree (“arborescent”) version again gives H¹ spanned by entropy (Thm 4). | Read |
| NOTE-095 v2 | Morganti — From OSR to metaphysical coherentism | Morganti argues from a survey rather than proving anything. Strong, moderate and contextualist OSR each fail as metaphysics: particle kinds have intrinsic features no group-theoretic analysis reduces, and the argument from permutation invariance is inconclusive. What OSR gets right about symmetry, entanglement and particle kinds is better captured by “metaphysical coherentism”: object-like entities (in one domain) stand in symmetric partial ontological dependence on one another. Cycles of such dependence can be kept irreflexive, and so explanatorily non-trivial, by relaxing transitivity (quasi-transitivity) or by making only partial dependence symmetric. | Read |
| NOTE-105 v2 | Dorato — Events and the Ontology of Quantum Mechanics | Dorato argues, without formal results, that events are the metaphysical category best suited to quantum mechanics. An event reading fits GRW flashes, GRW matter density and Bohmian worldlines, and also the antirealist views of ψ held by Bohr, Rovelli, Haag and Pashby. Because it is the same across interpretations, he calls it “interpretation-independent” (p. 12). Where ψ needs a realist reading, it should be read as intrinsic dispositions of events. Esfeld et al.’s view has the opposite structure: bare matter points plus a single holistic power of the universal configuration. Dorato argues that view is unintelligible, and that it must change its ontology when it passes from the universal wave function to the effective one (pp. 13–15). | Read |
| NOTE-110 v2 | Belot — Quantum states for primitive ontologists | Inside primitive-ontology Bohmian mechanics, Belot compares four readings of the quantum state: a field on configuration space, a multi-field on 3-space, a law, and a history of velocity dispositions of the particle system. Every reading except the multi-field carries a hard open question: why the world looks 3-dimensional, whether laws can be time-indexed, and whether dispositions lose content. The last is shown by the particle in a box, where every energy eigenstate φn = sin(πnx) induces the same at-rest disposition. The orthodox should prefer the multi-field reading; Belot, on van Fraassen’s view of interpretation, prefers to keep the others alive. | Read |
| NOTE-112 v3 | Poltorak — Wave-function realism and causal closure | A closure principle has two tacit parameters, its domain and its completeness relation (causal, dynamical or explanatory). Wave-function realism falsifies spatiotemporal-domain closure (STC) but is compatible with fundamental-physical closure (FPC), so there is no “trans-domain causal breach”. The physicalist is left with a dilemma: STC is too narrow, and FPC needs an independent criterion of physicality plus a dynamical-to-causal bridge. The paper tests this across primitive-ontology, spacetime-state, multi-field, Everettian, nomological and dispositional readings. | Read |
| NOTE-121 v2 | Elshatlawy et al. — Generalized theory of observers | The paper defines a “minimal observer” as a finite-state transducer with a boundary: a tuple O = (X, Y, Z, f, g, B) with |X| > 1, |Y| ≥ 1, |Z| ≥ 1 and feedback closure (Def. 4.1, p. 57). It then asserts, without derivation, that measurement outcomes, reference frames, coarse-graining and the quantum–classical cut are all defined by such observers. The paper’s only formal results are that isomorphism of these machines is an equivalence relation (“Theorem 2”, p. 93), and a complexity bound C(O) ≥ log 2 (Prop. 4.1, p. 94) whose correction term Λ is never defined and whose proof supports only C(O) ≥ 0. | Read |
| NOTE-127 v2 | Holik et al. — Quasi-set theory for a quantum ontology | The paper argues, with no new theorem, that the Lombardi–da Costa ontology of properties for quantum mechanics can be stated in Krause’s quasi-set theory Q. On that ontology a system is a bundle of I-type-properties (instances of universal type-properties, represented by the algebra of observables) with no principle of individuality. For each property instance, the indiscernible instances form a qset U([Π]). An atomic bundle is any qset x with qc(x ∩ U) = 1 for each U; the Axiom of Choice in Q guarantees one exists. The bundle’s kind is Q(B) = {x ∈ P(Q) | R(x)}, whose members are all indiscernible. N particles are a subqset N(B) ⊂ Q(B) with qc = N, a definite number with no labels (§6, pp. 19–22). | Read |
| NOTE-129 v2 | Effective ontic structural realism | Ladyman and Lorenzetti argue that three recent “effective realisms” lack a metaphysics that makes them realist rather than constructive-empiricist, and that the Ladyman–Ross OSR package supplies one: realism about modal structure, “to be is to be a real pattern”, and scale-relative ontology. The three are Fraser & Vickers (and Egg) on non-relativistic QM, Williams and Fraser on QFT, and Robertson & Wilson on “theoretical relicts”. Scale relativity also dissolves Saatsi’s inconsistency objection, so Newtonian gravitational forces and GR gravitation “both exist at different scales”. Egg’s 2019 anthropocentrism objection is answered by grounding measurement in decoherence. | Read |
| NOTE-134 v3 | Carroll — Reality as a vector in Hilbert space | A programmatic essay, not a result paper. Carroll argues that a quantum theory is fixed, up to unitary equivalence, by the Hamiltonian’s eigenvalue list {E_n} on a finite-dimensional factor (he assumes a non-degenerate spectrum), and that parts are not given. Subsystems (tensor factorisations H = H_S ⊗ H_E or ⨂_α H_α), classicality and space are to be selected afterwards as rare structure that suits the dynamics. The only uniqueness he reports is for local factorisations, and it rests on a cited theorem (Cotler et al. 2019). The system/environment criterion rests on “simple numerical examples” in a companion paper, the metric and Einstein’s equation on “optimistic assumptions”, and emergent fields on a hope. | Read |
| NOTE-140 v2 | Gao — The Meaning of the Wave Function | Gao argues first that ψ is real. His case uses protective measurements, which yield ⟨A⟩ on a single system with certainty, in three versions (ch. 4): one in the ontological-models framework using preparation non-contextuality, two he says need no auxiliary assumption, and one that uses a new criterion of reality. He then argues that ψ describes random discontinuous motion (RDM) of N point particles in 3-space: |ψ|² is the density with which each particle is present at each instant, and ψ is a single-case propensity. The main support is that the charge density |ψ|²Q, which protective measurement shows to exist, has no self-interaction (chs. 6–7). If the Born probabilities come from RDM, collapse is real. Gao proposes an energy-conserving, Planck-time-discrete collapse model: Pᵢ(t+t_P) = Pᵢ(t) + (ΔE/E_P)[δ_{EsEi} − Pᵢ(t)], giving τ_c ≈ ħE_P/(ΔE)² (ch. 8). Combined with relativity, it requires a detectable preferred Lorentz frame (ch. 9). | Read |
| NOTE-152 v2 | Le Bihan & Read — Duality and ontology | A survey that sorts realist responses to dualities along two axes. The first is which of the naively interpreted dual worlds count as candidates for actuality: all of them (underdetermination), one (discrimination) or none (nihilism), each with or without the common-core world. That gives options (1)–(6), with the common-core approach being (1), (3) and (5). The second axis is what replaces the common core: an overarching theory, or “pluralism”, on which all dual structures are co-instantiated as parts of one world. Its verdicts are conditional: no option breaks the underdetermination without a further metaphysical principle, such as the PII, for which the authors see no independent support. | Read |
| NOTE-154 v2 | Allori — Wave-functionalism | Allori proposes that the wave function be defined by three roles — empirical, nomological and explanatory. On this view it is an ingredient that “partially realizes” the law, not a law and not a material field. So its time-dependence and non-uniqueness (a phase shift, a density matrix) are harmless, it counts as non-epistemic and ontic, and it can take part in constructive explanation. Each of these is argued informally, and the escape from PBR is by relabelling the ontic/epistemic distinction, not by engaging the theorem. | Read |
| NOTE-162 v2 | French — Unitary inequivalence and structural realism | French answers Howard’s objection that QFT’s unitarily inequivalent representations leave the structural realist with non-isomorphic, competing structures. He offers two routes. The first is to adopt cut-off “conventional” QFT with Wallace, on which the inequivalence is an artefact or harmless. The second is to follow Ruetsche’s Swiss Army Knife and treat the algebra as a modal structure spanning all representations. On that reading spontaneous symmetry breaking preserves the symmetry across the degenerate vacua, and the DHR analysis derives quantum statistics from the algebra. Adding Baker’s criterion, on which a symmetry mapping between representations suffices for intertranslatability, removes the assumption behind the objection, and French ends with a third option that combines the two routes. Every step is an informal argument layered on others’ technical results; the paper proves nothing itself. | Read |
| NOTE-174 v2 | Rovelli — The relational interpretation (RQM) | Rovelli restates RQM as an ontology of sparse relative facts, which arise in interactions and are labelled by the interacting systems. Its one technical rule is that an amplitude W(b,a) gives the probability P = |W|² only when a and b are relative to the same system. That rule removes the clash between P_collapse = Σᵢ|W(c,bᵢ)|²|W(bᵢ,a)|² and P_unitary = |ΣᵢW(c,bᵢ)W(bᵢ,a)|² (eqs. 2–4). Decoherence makes some facts approximately label-independent (“stable”). ψ is a relative-state bookkeeping device, like a Hamilton–Jacobi function S. The price, which Rovelli states, is giving up “strong realism”: values are not held at all times, may be held at different times for different systems, and there is no coherent global view and no quantum state of the universe. | Read |
| NOTE-175 v2 | Chen — Realism about the Wave Function | A survey that sorts realist readings of the non-relativistic universal wave function into four ontological readings (a field on 3N-dimensional space, a multi-field on 3-space, properties of systems as in spacetime state realism, a Hilbert-space vector), two nomological ones (strong: time-independent and simple like the classical Hamiltonian; weak: a Humean best-system variable) and Maudlin’s sui generis option. It tabulates their compatibility with BM, GRW0/m/f and S0/Sm and with high- versus low-dimensional fundamental space (Table 1, 10 of 56 cells ”?”), and declares no winner. | Read |
| NOTE-178 v2 | De Haro & Butterfield — Philosophy and Physics of Duality | The book gives a formal Schema: a duality is an isomorphism between the “model roots” (state space, quantities, dynamics) of two models of a common-core bare theory, with values and dynamics matching. It pairs this with an interpretative condition: duals are physically equivalent iff (A) they are duals and (B) i1 = i2∘d. On that basis it argues four things. First, there are no genuine cases of in-principle underdetermination in the examples, and a cautious realist can commit to the common core. Second, dualities do not favour structural realism. Third, duality is formally incompatible with ontological emergence, except “beside” it (AdS/CFT’s RG direction), via effective dualities, or epistemically. Fourth, quasi-dualities motivate a “geometric view” on which models are charts on a successor theory’s moduli space. | Read |
| NOTE-192 | Science, Metaphysics and Structural Realism | Ladyman argues that structural realism should be metaphysically, not epistemically, revisionary. He gives three grounds. Quantum statistics leaves the individuality of particles underdetermined: permutation invariance gives 3 two-particle states, not 4, so probabilities are 1/3 rather than 1/4 (p. 62). Entangled states give non-supervenient relations that refute Humean supervenience (pp. 71–73). The hole argument blocks point-based substantivalism (pp. 68–69). He proposes that science describes “the modal structure of the phenomena” (p. 73), with modal realism supplying the no-miracles intuition. The paper ends non-eliminatively: “the phenomena have structure but they are not structure” (p. 74). | Read |
| NOTE-204 v2 | GNS construction (Wikipedia) | Any state (positive normalized linear functional) on a C*-algebra determines, essentially uniquely, a Hilbert space and a cyclic *-representation in which that state becomes a vector expectation ρ(a)=⟨π(a)ξ,ξ⟩, and the representation is irreducible exactly when the state is pure. | Superseded — by NOTE-857 |
| NOTE-215 v2 | Bra–ket notation (Wikipedia) | Dirac’s bra–ket notation writes vectors as kets and continuous linear functionals as bras, and on a Hilbert space the Riesz representation theorem is what licenses treating every bra as the conjugate of a ket, so that ⟨φ|ψ⟩ is at once a functional applied to a vector and an inner product. | Superseded — by NOTE-855 |
| NOTE-220 | Jorgensen & Tian, Noncommutative Analysis (GNS, RKHS) | The textbook puts the kernel construction and the GNS construction in one frame. It calls building a Hilbert space from a positive definite function “the GNS construction”, and proves (Cor. 1.35) that the feature map realising a kernel as inner products is unique up to a unitary among minimal realisations. It also states that in the abelian case states are exactly probability measures, so that “the GNS construction is non-commutative measure theory”. | Skimmed |
| NOTE-236 v2 | Abramsky, Barbosa & Mansfield, contextual fraction | For any no-signalling empirical model, the contextual fraction CF (one minus the largest weight of a noncontextual sub-model) is a linear programme. Its dual gives a Bell inequality whose normalised violation equals CF, and no Bell inequality is violated more (Theorem 1). CF is invariant under relabelling, non-increasing under translation of measurements, coarse-graining and mixing, and NCF is multiplicative under products (Theorem 2). It lower-bounds the failure probability of Z2-linear MBQC (Theorem 3) and of k-consistent games (Theorem 4). | Read |
| NOTE-237 v2 | Budroni et al., Kochen–Specker contextuality (RMP review) | A review that fixes a minimal definition of Kochen–Specker contextuality as the impossibility of a global joint distribution whose marginals are the context distributions (Fine’s theorem; the sheaf-theoretic framework is one of several equivalent formulations), and that treats Spekkens’s generalized contextuality as a different notion, covered only to mark the difference. | Superseded — by NOTE-844 |
| NOTE-238 v2 | Spekkens, generalized contextuality (2005) | Defines a noncontextual ontological model as one in which operationally equivalent procedures (preparations, measurements, transformations) get identical representations; shows the traditional Kochen–Specker notion is the special case of measurement noncontextuality for sharp measurements plus outcome determinism, proves that preparation noncontextuality implies that outcome determinism, and proves preparation contextuality already for a qubit. | Superseded — by NOTE-748 |
| NOTE-239 | The Geometry of Information Retrieval | The book recasts IR’s vector-space, probabilistic and logical models in one Hilbert-space language. Documents are state vectors. Relevance and aboutness are Hermitian observables that need not commute. Subspaces carry a non-distributive logic whose conditional is shown to be a Stalnaker conditional equal to the Sasaki hook (ch. 5). Gleason’s theorem makes tr(ρP) the probability of a subspace, which ch. 6 uses to rewrite cosine matching, cluster representatives, relevance and pseudo-relevance feedback, dynamic clustering and ostensive retrieval. It is a proposed language, not a tested model: no experiment is reported, and its “content hypothesis” that the probability of a concept is cos²θ is argued, not tested. | Read |
| NOTE-247 | A new description of orthogonal bases | Theorem 5.1 proves that on a finite-dimensional complex Hilbert space, commutative †-Frobenius monoids in FdHilb and orthogonal bases are in bijection: the basis is the monoid’s copyable elements, and the monoid is the linear extension of copying and uniformly deleting that basis. §6 adds that the monoid is special (m∘δ = id) exactly when the basis is orthonormal. Corollaries 7.1 and 7.2 lift this to categories: with fully structure-preserving maps the category is equivalent to the groupoid of finite sets labelled by positive reals (the basis norms), and with comonoid homomorphisms it is equivalent to FinSet. | Read |
| NOTE-248 | H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics | In the category Hilb of complex Hilbert spaces of arbitrary dimension, a Frobenius algebra with a unit exists only in finite dimension (Lemma 3). So the paper drops the unit, and Theorem 22 proves that a nonunital commutative special †-Frobenius algebra corresponds to an orthonormal basis (the basis being the algebra’s copyable vectors) if and only if it is semisimple, if and only if it satisfies Ambrose’s H*-axiom (H), and if and only if it is a directed colimit of finite-dimensional unital ones. Whether (H) is automatic, i.e. whether a nonzero “radical” Frobenius algebra exists (Proposition 23), is left open. In categories of relations and of nonnegative ℓ²-matrices, Frobenius algebras decompose as disjoint unions of abelian groups and do satisfy (H) (Theorems 30–38). | Read |
| NOTE-249 v2 | Measurement as Sheafification: Context, Logic, and Truth after Quantum Mechanics | A programmatic essay, with no theorem, model or computation of its own. It restates Abramsky & Brandenburger’s criterion (contextuality and non-locality as the absence of a global section of a presheaf over measurement contexts) and adds three things. It declares measurement to be “sheafification” of that presheaf, without specifying a site, a topology or any sheafified object. It asserts that the Ghose–Patra seven-valued logic is a finite Heyting algebra, without giving an order or operations. It proposes that a quantum potential scaled by 0 ≤ λ ≤ 1 (λ = 1 quantum, λ = 0 classical Hamilton–Jacobi) carries the presheaf continuously to a sheaf, which is asserted, not modelled. Its one worked formula, v2’s Čech “obstruction class” [δs], is a coboundary and so is zero by construction. | Read |
| NOTE-250 v2 | The Cohomology of Non-Locality and Contextuality | For an empirical model e on a measurement cover 𝒰, a section s in the support of context C₁ gets a class γ(s) ∈ Ȟ¹(𝒰, F_{C̄₁}), relative Čech cohomology of the presheaf F = F_ℤ S_e of formal ℤ-linear combinations of support sections. γ(s) = 0 iff s belongs to a compatible family of such ℤ-combinations (Prop 4.2), so a non-vanishing γ(s) is a sufficient but not necessary witness of contextuality (Prop 4.3). γ is non-zero for every section of the PR box, GHZ, the triangle, the 18-vector Kochen–Specker set and the Peres–Mermin square, but it vanishes on the Hardy model and on one strongly contextual Kochen–Specker cover (§8). | Read |
| NOTE-251 | Contextuality, Cohomology and Paradox | For a model whose outcomes live in a commutative ring R, “All-vs-Nothing” (AvN_R) means the R-linear equations satisfied by every support section of every context have no global solution. Theorem 21 proves AvN_R(S) ⇒ SC(Aff S) ⇒ CSC_R(S) ⇒ CSC_ℤ(S) ⇒ SC(S), so every AvN model is strongly contextual with a non-vanishing Čech obstruction on every section. Theorem 4 shows that every state stabilised by an “AvN triple” of n-qubit Pauli operators yields such a model. The “paradox” half proves much less: an informal identification of Liar cycles with strongly contextual support tables, one exact case (the length-4 Liar cycle is the PR box), and the cohomology of paradox deferred to future work. | Read |
| NOTE-252 | On the Cohomology of Contextuality | Carù gives a possibilistic model on the two-party, two-setting, four-outcome Bell cover that is strongly contextual, yet whose Čech obstruction vanishes on all 22 support sections (Fig 4, Table 1). That refutes Abramsky–Mansfield–Barbosa’s Conjecture 8.1 for any symmetry or connectedness condition the Bell cover meets. I confirmed the refutation over ℤ and ℤ/2. The rest is weaker. Odd-degree “higher obstructions” form a hierarchy (Prop 5.5), but every level above the first is empty for every no-signalling model (Prop 5.6). Prop 6.3 restates the known torsor description of Ȟ¹. Prop 6.1 is false in the “only if” direction; the PR box refutes it. | Read |
| NOTE-253 | Towards a complete cohomology invariant for non-locality and contextuality | For cyclic scenarios, where the contexts’ intersection graph is a single chordless N-cycle, Carù proves that ℤ/2 Čech cohomology of the (N−1)-th iterated “joint model” is a complete invariant for logical and strong contextuality (Thm 7.7). I confirmed this on the Hardy model, on his own 2017 counterexample (detected on all 22 sections at level 3) and on 659 random contextual cycle models. The extension beyond cycles does not hold as stated. Prop 8.2 is false on the paper’s own Table 5 model. Worse, the one non-cyclic case the paper claims to repair, the §8 Kochen–Specker cover of Abramsky–Mansfield–Barbosa, is never detected at any level under the paper’s definitions (my proof and computation). That refutes the paper’s closing Conjecture 9.1. | Read |
| NOTE-254 | Logical and Topological Contextuality in Quantum Mechanics and Beyond | The thesis’s cohomology chapters republish the 2017 and 2018 papers with the “joint model” renamed the “line model”. The definitions, Thm IV.32 (complete on cyclic scenarios at level |M|−1), Prop IV.34, Thm IV.36 and Conjecture IV.41 carry over unchanged. So does every defect found in the preprint. Under the thesis’s own Def IV.26, the §8 Kochen–Specker cover of Abramsky–Mansfield–Barbosa is still detected on only 6 of 15 sections at levels 0–3 (my recomputation), and a lifted ℤ/2 family shows it is never fully detected at any level. The context-wise reading the worked examples use detects all 15 sections at level 2. The genuinely new material lies outside the cohomology programme: a probabilistic line-model construction (§IV.7), a valuation-algebra “theory of disagreement” with inference algorithms (Ch V, bound O(k^n·l^{k(n−1)+1}) on (n,k,l) scenarios), and the AvN-triple characterisation of stabiliser states (Ch VI). | Read |
| NOTE-271 | A topos perspective on the Kochen–Specker theorem: I. Quantum states as generalized valuations | Isham & Butterfield show that, for bounded self-adjoint operators with discrete spectrum, a global section of the spectral presheaf Σ(Â) = σ(Â) on the category O_d, whose arrows are functional relations B̂ = f(Â), is literally a valuation obeying the value rule and FUNC. A global section of the dual presheaf D(W) = Hom(W, {0,1}) on the poset W of Boolean subalgebras of projectors is literally a Kochen–Specker 0/1 assignment. The Kochen–Specker theorem (cited, not reproved) is therefore equivalent to “no global sections” when dim H > 2 (§§2.2–2.3). In place of the missing global valuations, they construct contextual truth values in the Heyting algebra of sieves: ν^ρ(A ∈ Δ) = {f: B̂ →  | tr(ρ Ê[B ∈ f(Δ)]) = 1}. For every density matrix these satisfy functional composition, null, monotonicity, exclusivity and unit conditions (§4.4, §5.3, Theorem 5.1). | Read |
| NOTE-275 v2 | The logic of quantum mechanics | The paper argues heuristically that the experimental propositions of a quantum system correspond to closed linear subspaces of Hilbert space. Meet is intersection, join is closed linear span, and negation is orthogonal complement. The lattice satisfies L1–L4 and complementation L71–L73 but not the distributive law L6. The paper proposes the modular law L5 as its replacement. Under a finite-dimension assumption, it concludes that the propositional calculus of QM is an abstract projective geometry. The Appendix proves that orthocomplementations on P_{n−1}(F), n ≥ 4, are exactly the polarities of definite diagonal Hermitian forms with respect to an involutory anti-automorphism of F. | Read |
| NOTE-277 v2 | A topos foundation for theories of physics: I. Formal languages for physics | The programme paper. Its contention is that a theory of a physical system S is a representation of a typed “local language” L(S) (ground types Σ and R, function symbols A : Σ → R) in a topos. Physical quantities become arrows Σ_φ → R_φ, and propositions become sub-objects of Σ_φ, which form a Heyting algebra. Classical physics is the representation in Sets. The paper proves nothing. Its one argument is that the axiom ¬(A ε Δ) ⇔ A ε ℝ∖Δ is not validated by non-Boolean Heyting representations (§3.2.1). | Read |
| NOTE-279 | The problem of hidden variables in quantum mechanics | Theorem 1 exhibits a finite partial Boolean algebra D, generated by 117 directions in E³, that admits no homomorphism onto Z₂. §4 embeds D in the projections of the spin-1 space of orthohelium’s 2³S₁ state, via α ↦ P J_α². So no assignment of 0/1 values to those quantum propositions respects all the Boolean relations within each commeasurable set. Theorem 0 makes the target precise: a partial Boolean algebra embeds in a Boolean algebra iff its Z₂-homomorphisms separate points. That is, hidden variables in the paper’s sense need enough “global valuations”, and quantum mechanics in dimension 3 has none. The same paper builds a classical hidden-variable model for spin ½ (§6). It argues that von Neumann’s no-go needed additivity on non-commuting observables (Theorem 3). | Read |
| NOTE-285 | A topos foundation for theories of physics: IV. Categories of systems | Proposes axioms for a category Sys of physical systems, symmetric-monoidal under composite ⋄ and under disjoint sum ⊔, whose arrows are translations of local languages. Each arrow j : S₁ → S is realised by a left-exact functor τ(j) between the systems’ topoi, together with arrows φ(j) : Σ_{S₁} → τ(j)Σ_S and β(j) : τ(j)R_S → R_{S₁}. Classical physics satisfies the scheme exactly. In quantum theory the disjoint sum H₁⊕H₂ satisfies it (the pulled-back arrow of Â₁⊕Â₂ is δ̆(A₁)), but the composite H₁⊗H₂ does not: the natural pull-back of δ̆(A₁) differs from δ̆(A₁⊗1̂) off the image of V(H₁), which the authors attribute to “operator entanglement”. | Read |
| NOTE-290 | A topos foundation for theories of physics: III. The representation of physical quantities with arrows δ̆°(A): Σ̲ → ℝ̲≽ | Each bounded self-adjoint  becomes an arrow δ̆^o(A) : Σ → ℝ^⪰, where ℝ^⪰ is the presheaf of real-valued order-reversing functions on ↓V. At λ ∈ Σ_V it sends V′ ⊆ V to λ(δ^o(Â)_{V′}), and δ^o(Â)_V is the spectral-order-least operator in V above Â. The map  ↦ δ̆^o(A) is injective, but ℝ^⪰ is only an additive monoid, not the real-number object. Its Grothendieck completion k(ℝ^⪰) is an abelian-group object whose global elements are exactly the functions of bounded variation on V(H). | Read |
| NOTE-292 v2 | A topos foundation for theories of physics: II. Daseinisation and the liberation of quantum theory | A projector P̂ is mapped to the clopen sub-object δ(P̂) of the spectral presheaf Σ over the poset V(H) of abelian von Neumann subalgebras, by “daseinisation” (de Groote’s V-support): δ(P̂)V = the least projector in V above P̂. The map is injective and preserves ∨, but only δ(P̂∧Q̂) ⪯ δ(P̂)∧δ(Q̂). This represents the propositional language PL(S) in the Heyting algebra Sub_cl(Σ). A pure state gives the truth object T^ψ, and “A ε Δ” gets the sieve-valued truth value {V′ ⊆ V | ⟨ψ|δ(Ê[A∈Δ]){V′}|ψ⟩ = 1}. | Read |
| NOTE-294 v2 | A theory of concepts and their combinations I: The structure of the sets of contexts and properties | Aerts & Gabora model a concept as an entity with states that change under contexts, a “state context property system” (Σ, M, L, µ, ν). They report an 81-subject e-mail study in which 7 contexts shift the rated frequency of 14 exemplars of “pet” (for example dog 0.50 under “The pet is chewing a bone” against 0.12 in the unit context) and the applicability of 14 properties. They then postulate that contexts and properties each form a complete orthocomplemented lattice, and call the structure quantum-like because a state can be an eigenstate of e ∨ f without being one of e or of f (eq. 19). No theorem is proved, the lattice is not shown to be non-distributive or orthomodular, and the paper contains no Hilbert-space model and no treatment of combination; both are deferred to Part II. | Read |
| NOTE-299 | A theory of concepts and their combinations II: A Hilbert space representation | Aerts & Gabora embed the “pet” data in ℂ¹⁴⁰⁰. Each of 1,400 “basic contexts” is one vector of an orthonormal basis, the ground state is the uniform superposition, and each context is the projector onto the span of its basic contexts. The Born rule then returns the measured frequencies as counting ratios, for example cat under “chewing a bone” = 75/303 = 0.25. “Pet fish” is modelled as the entangled state Σ_{u∈E} |u⟩⊗|u⟩/√100 in ℂ¹⁴⁰⁰⊗ℂ⁴⁰⁸, so that collapsing “pet” to goldfish collapses “fish” to goldfish too. Every projector in the model is diagonal in one basis, so the fitted model is a classical probability space in quantum notation, which the authors half-concede (a “quantization … of this set theoretical model”, §3.3). Two of the chosen counts are infeasible: parrot and guppy. | Read |
| NOTE-303 | On the imbedding of normed rings into the ring of operators in Hilbert space | Theorem 1 is the first abstract characterisation of C*-algebras. Every normed -ring (a unital Banach -algebra with ‖xx‖ = ‖x‖·‖x‖, ‖x*‖ = ‖x‖, and e + x*x invertible) is isometrically *-isomorphic to a norm-closed -subalgebra of the bounded operators on some, not necessarily separable, Hilbert space. The proof builds, for each maximal left ideal M, a positive functional f vanishing on M. The inner product is (ξ, η) = f(yx) on R/M, and the construction takes the direct sum over all M. This is the GNS construction. Lemma 1 is the commutative case: such an algebra is isometrically *-isomorphic to C(𝔐) for its compact maximal-ideal space. §4 adds that the simple weakly closed rings on separable Hilbert space are exactly the factors of types I_n, II₁ and III (Theorem 3). | Read |
| NOTE-304 | The intrinsic parity of elementary particles | The paper defines a superselection rule: a decomposition of Hilbert space into orthogonal subspaces A, B, C, … between which there are no spontaneous transitions and no measurable quantities with non-zero matrix elements. Relative phases across subspaces, and hence relative parities, are then unobservable. It proves such a rule between integer and half-integer angular momentum, from time reversal with U_AKU_AK = +1 and U_BKU_BK = −1. It postulates, without conclusive evidence, a charge superselection rule. The consequence is that “all Hermitean operators represent measurable quantities” must be abandoned. | Read |
| NOTE-312 | Studies on the Foundation of Quantum Mechanics. I | Husimi derives Birkhoff and von Neumann’s lattice axioms from statistical premises. Implication is P̄ ≤ Q̄ in every state (and requires compatibility), negation is Q̄ = 1 − P̄, and compatible propositions obey classical logic. Meet is obtained from “Ehrenfest’s principle” (additivity of mean values) via the top eigenvalue of aP + bQ (pp. 778–779), and the modular law, equivalently the parallelogram law |P| + |Q| − |P∩Q| = |P∪Q|, is proved under a finite-chain assumption. The orthomodular identity is used but never named or stated as an axiom. It appears as the step “since we admit traditional logic to any classical part … S + P∩Q = R” for R ⊃ P∩Q (p. 784), and as the case “(Q − P) + P = Q provided Q ⊃ P” of a classical identity (p. 780). | Read |
| NOTE-326 | ’What is a Thing?’: Topos Theory in the Foundations of Physics | The Döring–Isham programme in one volume. A theory of a system is a representation of a typed language L(S) in a topos. In quantum theory, presheaves over the abelian von Neumann subalgebras V(H), daseinisation sends projectors injectively to clopen sub-objects of the spectral presheaf Σ, preserving ∨ but not ∧. A pure state gives sieve-valued truth values. Every bounded self-adjoint  becomes an arrow Σ → ℝ^↔ whose value at a point is a pair of monotone functions, an interval that widens as contexts coarsen. Most of it reprints papers I–IV. New are the inverse-image corollary linking PL(S) and L(S), values in a pseudo-state, and the algebra of the quantity-value candidates. The Heyting structure on Sub_cl(Σ) is still only sketched. | Read |
| NOTE-327 | A Topos Perspective on the Kochen-Specker Theorem: III. Von Neumann Algebras as the Base Category | Hamilton, Isham & Butterfield move the series’ base category from operators (O) and Boolean subalgebras (W) to the poset V of commutative von Neumann subalgebras of B(H). There they restate Kochen–Specker as “the spectral presheaf Σ(V) = Gelfand spectrum of V has no global elements” (dim H > 2, cited, §2.2). They identify the coarse-graining presheaf G(V) = L(V), G(i)(P̂) = inf{Q̂ ∈ L(V₂) | P̂ ≤ Q̂} (Def. 3.1), with the presheaf CloΣ of clopen subsets of the spectra; the isomorphism is asserted, and part IV proves it. A state ρ yields the sieve-valued valuation ν^ρ_V(P̂) = {V′ ⊆ V | ρ(G(P̂)) = 1} (Def. 3.4) and the support projectors inf{P̂ ∈ L(V) | ρ(P̂) = 1}. These form a global element of G and hence an “interval valuation”, a subobject of Σ (§§4.2–4.4). The probability-r valuations give subobjects of G but not global elements. | Read |
| NOTE-328 | A Topos Perspective on the Kochen-Specker Theorem: IV. Interval Valuations | Butterfield & Isham show, over the poset V of commutative von Neumann subalgebras, that sieve-valued valuations and “interval valuations” (assignments of support projectors, or of subsets of Gelfand spectra) determine each other under stated conditions. If supports s(α, V) = inf{P̂ | α_V(P̂) = true_V} form a global element of the coarse-graining presheaf G, then α_{V₁}(P̂) = {V₂ ⊆ V₁ | G(s(α, V₁)) ≤ G(P̂)} is a sieve-valued valuation obeying FUNC (Theorem 3.1). There is an analogue for “tight” subobjects of Σ (Theorem 3.2). Both are satisfied by every state’s valuation ν^ρ. The ordering ≤ (or ⊆) is shown to be the natural relation securing sievehood, null, monotonicity, exclusivity and unit (§4). The review §2.4 also supplies the proof, via state extension, that G ≅ CloΣ: restricting the clopen set of P̂ to V₂ gives the clopen set of its coarse-graining. Part III had only asserted this. | Read |
| NOTE-329 | A Topos Perspective on the Kochen-Specker Theorem: II. Conceptual Aspects, and Classical Analogues | Butterfield & Isham argue that the sieve-valued “generalised valuations” of part I are not a quantum peculiarity but the natural valuations for any presheaf of propositions on any small category. Three supports are offered. (1) Classical physics: the value presheaf Υ(Ā) = Ā(S) on the category M of measurable functions has global sections (one per microstate, eq. 3.7), yet a macrostate R ⊆ S yields a sieve-valued valuation ν^R(A ∈ Δ) = {f_M | B̄(R) ⊆ f(Δ)} (eq. 3.10) with the same FUNC property as the quantum ν^ρ. (2) A proof that a sieve-valued valuation on any presheaf G obeys generalised FUNC, ν(B, G(f)(d)) = f*(ν(A, d)), iff it is a natural transformation G → Ω, i.e. a subobject of G (Theorem 4.1). (3) An informal argument, explicitly “not a genuine deduction” (§5.2), that if partial truth is fixed by which weakenings of a proposition are totally true, the truth value is a sieve. | Read |
| NOTE-333 | Topos Theory and Consistent Histories: The Internal Logic of the Set of All Consistent Sets | Isham takes the poset B of all Boolean subalgebras (“windows”) of the history orthoalgebra UP, ordered by coarse-graining, and shows that the presheaf A_d(W) = {α | some d-consistent W′ ⊇ W contains α} is a subobject of the constant presheaf ΔUP, whereas the naive “realisable in W” assignment is not a presheaf at all (§4.1–4.2). The probability claim ⟨α, p⟩ then gets the sieve-valued semantic value {W ⊆ W₀ | ∃ d-consistent W′ ⊇ W with α ∈ W′} if d(α, α) = p and ∅ otherwise (4.16), a global element 1 → Ω in Set^B, so the logic of “all consistent sets at once” is the distributive but non-Boolean Heyting algebra of sieves on B. | Read |
| NOTE-334 | Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity | Isham & Butterfield argue, without proving anything new, that presheaf toposes over a base of “contexts” suit physics because the subobject classifier Ω is fixed by the base. They say this answers the objection that many-valued logics are arbitrary (§2.2), and that it keeps the logic distributive (§1.1). They then sketch seven applications. Five are speculative one-paragraph proposals (frames, observers, causal sets, TQFT, synthetic-differential-geometry time in continuous histories). One (§3.7) recaps their Kochen–Specker results: KS ⇔ the dual presheaf D(Â) = Hom(W_A, {0,1}) on the operator category O has no global sections (dim H > 2), and every state ψ gives a FUNC-obeying sieve-valued valuation ν^ψ(A ∈ Δ) = {f_O : Â → B̂ | Ê[B ∈ f(Δ)]ψ = ψ}. | Read |
| NOTE-604 | Wang, Solloway, Shiffrin & Busemeyer, the QQ equality in 72 surveys | Across all 66 Pew surveys of 2001–2011 that varied the order of two questions, chosen without regard to their outcome, question order changes the answers (p = 0.0004 for the order-effect χ² distribution) while the probability of answering the two questions alike does not change (p = 0.4625 for the q values), as the QQ equality predicts. Paired context effects across 72 studies lie along the predicted line of slope −1 (r = −0.82). | Read |
| NOTE-612 | The Topology of Causality | Builds the sheaf-theoretic framework for causality over spaces of input histories: causal functions are free assignments of outputs to tip events, causality is continuity in the lowerset topology, and the presheaf of causal functions is a sheaf exactly when the space has no “solipsistic contextuality” (always when tight). Empirical models are compatible families on any open cover; standard empirical models on causal switch spaces, including total orders, are always local. | Read |
| NOTE-623 | Probabilistic Foundations of Contextuality | In Kolmogorovian probability, joint distributedness is transitive, so treating a measurement as the same random variable in overlapping contexts already forces an overall joint distribution. A Kochen–Specker or Bell contradiction therefore refutes that identification, not the joint distribution. Contextuality is restated as the impossibility of a coupling of context-indexed variables whose same-content copies satisfy a property C. For inconsistently connected binary systems, C is multimaximality, characterized by adjacent pairs (Theorem II.3). | Read |
| NOTE-624 v2 | The Geometry of Causality | Empirical models on any cover of a space of input histories are exactly the points of a causaltope, a product of simplices sliced by linear causality equations, which makes supported and causally separable fractions linear programs. Defines causal separability relative to an ambient space and shows, with entangled and contextually controlled quantum switches, that the causally separable fraction is bounded below by the separable local fraction and can equal it. | Skimmed |
| NOTE-629 | Sheaf-theoretic structure of definite causality | Extends the Abramsky–Brandenburger sheaf framework from non-locality (discrete order) to any finite causal order of events. Sections are causal functions over families of inputs on lower sets. Empirical models are causal conditional distributions forming a polytope. Locality is a global section, equivalently a mixture of deterministic causal functions. The authors withdrew the paper in 2024 because its “locale of inputs” is not fit for purpose. This reading finds that Proposition 5’s meet can leave the poset and that distributivity fails. | Read |
| NOTE-630 | Getting to the Bottom of Noether’s Theorem | Noether’s theorem in Hamiltonian form, “a generates symmetries of b iff b generates symmetries of a”, follows in Poisson algebras, complex *-algebras and Banach–Lie algebras from the antisymmetry of the bracket plus uniqueness of solutions; for a bilinear bracket antisymmetry is self-conservation, {a, a} = 0. What carries content is the map from observables (a Jordan algebra) to generators (a Lie algebra): i in complex quantum mechanics, none in the real or quaternionic case, and Alfsen–Shultz’s dynamical correspondence for JB-algebras. | Read |
| NOTE-635 | The Combinatorics of Causality | Replaces causal orders by spaces of input histories (join-prime sets of partial input assignments), which can express causal constraints that depend on inputs. Defines free choice, tip events, causal completeness and tightness, three kinds of composition, and a lattice of spaces; proves that the maximal causally complete spaces are exactly the causal switch spaces; and finds 2644 complete spaces on 3 binary-input events, against 19 from definite orders. | Read |
| NOTE-636 | Wang & Busemeyer, the QQ model and QQ equality | Derives the QQ equality from a Lüders-projection model of answering attitude questions: the probability of answering two questions differently is the same in both orders, for any belief state, any projectors and any dimension, provided only the first answer changes the state before the second. It holds in five of six data sets and fails, as the authors predicted, in the sixth, where new information came between the questions. The argument that classical models fail it is informal. | Read |
| NOTE-643 | Contextuality-by-Default 2.0 | Replaces CbD 1.0’s maximal coupling of each connection by a multimaximal one (every subset maximally coupled). For binary variables this coupling exists, is unique and has an explicit staircase form (Theorem 1), so a noncontextual system’s subsystems are noncontextual (Theorem 3) and partial and complete (non)contextuality coincide. Cyclic, consistently connected and two-copy systems are judged as before. For non-binary variables multimaximal couplings can fail to exist, can be non-unique, and are not stable under coarse-graining (Examples 1–3). | Read |
| NOTE-645 v2 | On the Quantum-like Contextuality of Ambiguous Phrases | Models meaning selection in two-word ambiguous phrases as a Bell-type measurement scenario. Hand-set supports give one possibilistically contextual (Hardy-type) model; every corpus-estimated model is signalling, so only Contextuality-by-Default applies, under which two noun–verb pairs read in both grammatical orders are contextual (1/30 and 7/30). The paper’s own bootstrap leaves both plausibly noncontextual (probabilities above .56 and .08), on very small counts. | Read |
| NOTE-646 | Combining contextuality and causality | Extends sheaf-theoretic contextuality to causal scenarios by an enabling relation on measurements and a game reading: Nature’s deterministic strategies, which may depend on causal history, replace the event sheaf, and the Abramsky–Brandenburger definitions then go through. Recovers flat contextuality and causal Bell scenarios, and expresses adaptive MBQC by Experimenter strategies. The strategy presheaf is not a sheaf: compatible deterministic local strategies can have no gluing or several. | Read |
| NOTE-647 | A comonadic view of simulation and quantum resources | Makes simulation between empirical models adaptive: measurement protocols form a comonoidal comonad MP, and a simulation of e by d is a deterministic map MP(d ⊗ c) → e with c non-contextual. Proves that such simulations are exactly the conversions by terms in the free operations of the resource theory of contextuality, which now include conditional measurement, with an equational theory and normal forms; the non-contextual fraction is monotone and contextual models cannot be cloned. | Read |
| NOTE-648 | Quantum-Like Contextuality in Large Language Models | Builds a three-sentence anaphora schema with the support of the PR prism, instantiates it 51,966,480 times from Simple English Wikipedia, and takes one referent probability per sentence from BERT: 0.148% of the models pass a signalling-corrected sheaf test and 71.1% the CbD test. Because the schema fixes the support, both verdicts depend only on how uncertain BERT is in the three sentences; the tie to embedding distance is a softmax identity with weak correlations. | Read |
| NOTE-650 | Closing Bell | Answers which maps between the empirical models of two scenarios are induced by classical, non-adaptive procedures: exactly those given by a non-contextual model of a hom scenario [S, T] whose outcomes are procedures, so deciding it is a linear program. A family of compatible local procedures glues into one global procedure exactly when that model is non-contextual; the rest are “contextual simulations”. The hom construction makes scenarios with predicates a closed category. | Read |
| NOTE-651 | Categories of Empirical Models | Gives empirical models on different measurement scenarios a category: a simulation answers each target measurement with a jointly measurable set of source measurements and a stochastic outcome map natural in the context. Non-contextuality becomes simulability from nothing, the non-contextual fraction becomes a functor (contextuality cannot grow along a simulation), strong contextuality is reflected, Graham reductions are simulations, and contextual models cannot be cloned. | Read |
| NOTE-654 | Analysing Ambiguous Nouns and Verbs with Quantum Contextuality Tools | Proves that in a binary cyclic system CbD’s signalling quantity Δ is twice the sum of each content’s minimal direct influence over Jones’s canonical causal models, which is the total-variation distance between its marginals. In 90 rank-2 noun–verb systems from British corpora, Δ is about 1.35 for every ambiguity class, and homonymous verbs carry about 70% of it against about 50% for polysemous verbs. That difference rests on 14 systems from six verbs and is marginal by a recomputed test (p ≈ 0.06 two-sided). No system is tested for contextuality. | Read |
| NOTE-737 v2 | Schmid, Selby & Spekkens — Unscrambling the omelette of causation and inference | In a process-theoretic framework that keeps causal and inferential structure apart, generalized noncontextuality becomes the demand that a realist representation preserve inferential equivalence. A Leibnizian classical realist representation implies a generalized-noncontextual one, so every noncontextuality no-go theorem is a no-go for Leibnizian representations; the converse is a conjecture. Bell-like theorems deny any classical realist representation, a weaker assumption, while noncontextuality theorems apply even to a prepare-measure scenario with no Bell-like constraint. | Skimmed |
| NOTE-748 | Spekkens — Contextuality for preparations, transformations, and unsharp measurements | Noncontextuality, redefined operationally so that operationally equivalent procedures get equivalent representations, fails for a qubit’s preparations, transformations and unsharp measurements, where no traditional Kochen–Specker proof exists. Preparation noncontextuality over the full set of quantum states implies outcome determinism for sharp measurements, so the old Kochen–Specker proofs refute universal noncontextuality. Measurement noncontextuality alone is consistent with quantum theory. | Read |
| NOTE-808 | Aerts — Quantum structure in cognition | Hampton’s membership weights for conjunctions and disjunctions of concepts fail a single Kolmogorovian model, item by item, in two ways: over- or underextension (∆-type) and a negative Kolmogorovian factor (k-type). A subspace lattice with meet and join can fit the k-type items, but provably never the ∆-type ones. Those need an emergent concept state in a second sector of Fock space. Negation appears only as the orthocomplement used to reach conjunction from disjunction, and is never measured. | Read |
| NOTE-825 | Catani et al. — What is nonclassical about uncertainty relations? | For any state whose four-state reflection orbit is realizable (equal predictabilities with flipped signs, and ½s1+½s3 = ½s2+½s4), preparation noncontextuality forces |⟨X⟩| + |⟨Z⟩| ≤ 1, tight, without perfectly distinguishable states. In a theory where every state has such an orbit, the bound constrains the uncertainty relation itself: the qubit’s circle violates it, up to √2, the stabilizer octahedron saturates it, and the simplicial theory escapes it because its jointly predictable states lack the orbit. | Read |
| NOTE-829 | Jorgensen & Tian — Noncommutative Analysis (GNS, RKHS) | The book calls the kernel-to-Hilbert-space construction “the GNS construction” (Remark 1.34) and states that minimal realizations ϕ(x,y) = ⟨Φ(x), Φ(y)⟩ are unique up to a unitary (Cor. 1.35). It does not prove the corollary; the proof is “left as an exercise”. The parallel uniqueness for states and cyclic representations is proved (Thm 4.35), and the argument transfers. The commutative GNS space L²(μ) is written out explicitly (Ex. 4.86, Lemma 3.28). | Skimmed |
| NOTE-844 | Budroni, Cabello, Gühne, Kleinmann & Larsson — Kochen-Specker Contextuality | A review of Kochen–Specker contextuality. It defines noncontextual hidden-variable models minimally, as a global distribution whose marginals are the context distributions (Fine’s theorem), with the sheaf, polytope, graph and hypergraph formulations called substantially equivalent. It argues that contexts need an operational definition stronger than joint measurability, which alone makes every nondisturbing correlation classically realizable. It surveys how disturbance is modelled in experiments, Kujala et al.’s maximally noncontextual models among them. Spekkens’s generalized contextuality is covered only to mark the difference, and the review nowhere derives the relation between the two notions. | Read |
| NOTE-855 | Bra–ket notation (Wikipedia) | Dirac’s notation writes vectors as kets and linear functionals as bras. On a Hilbert space the inner product, antilinear in its first slot, turns each ket into a bra, and the article cites the Riesz theorem for the reverse direction; on a Banach or untopologized space the bracket is only a functional applied to a vector, because Riesz does not apply. The article is a reference with internal inconsistencies, including an unsourced edit of 2026-09-30 to the date of the notation’s origin. | Read |
| NOTE-857 | Gelfand–Naimark–Segal construction (Wikipedia) | For a state ρ on a C*-algebra A, the semi-inner product ⟨a, b⟩ = ρ(b*a) on A, quotiented by its null left ideal and completed, carries A acting by left multiplication, with the class of the identity as a unit cyclic vector and ρ(a) = ⟨π(a)ξ, ξ⟩. Any two cyclic representations with cyclic vectors reproducing the same state are unitarily equivalent, by the unitary π(a)ξ ↦ π′(a)ξ′. The representation is irreducible exactly when the state is pure. The article says nothing about kernels. | Read |
| NOTE-952 | Pattee — the physics of symbols | The epistemic cut is the separation of subject from object, symbol from referent, controller from controlled, which dynamical laws cannot describe without losing the function across it (von Neumann’s regress). Measurement codes a rate-dependent dynamical state into quiescent symbols and control is its converse; both are executed by non-integrable (non-holonomic) constraints. The cut is epistemic, not ontological, and in life it arises first at the genotype–phenotype control interface: in Pattee’s account control of construction came first, not measurement. | Read |
| NOTE-955 | Fuchs, Mermin & Schack — an introduction to QBism | QBism in eight pages. Quantum states and all probabilities, 0 and 1 included, are an agent’s personal judgements; a measurement is “any action an agent takes to elicit a set of possible experiences”, and its outcome is the experience it elicits. The agent applies quantum mechanics to everything external to her, other agents and their reports included, and to nothing but her “own direct internal awareness of her own private experience”. Locality follows because one agent’s experiences are time-like separated; the Bell parameter λ is set outside physical science. No theorem or model: an interpretive essay. | Read |
| NOTE-1012 | Consciousness and the Collapse of the Wave Function | Chalmers and McQueen make consciousness-collapse precise as a super-resistance model (IIT’s Q-shape resisting superposition, with a continuous-collapse dynamics), find simple versions refuted by the quantum Zeno effect, and argue that collapse is the gap in causal closure that interactionist property dualism needs. Skimmed: the introduction, the objections and §8 were read, the formal §§2-7 were not. | Skimmed |
| NOTE-1214 | Di Biagio & Rovelli — Relative information, relative facts | A reformulation of relational quantum mechanics in Shannon terms. The information about A relative to B is the expected conditional information, I_{A|B} = I_A + I_{A:B}. A is a fact relative to B when that is maximal. Only commuting variables have a joint distribution, so perspectives belong to commutative subalgebras (“classical subsystems”) and not to quantum systems: a qubit has none. “Every fact is a relative fact”, and the joint distribution is itself relative to a further system, generally “us”. As a side result, in a von Neumann measurement the pointer’s information about the measured variable grows continuously, sin²(ωt)H_A in the model. The jump is only a threshold on what counts as a fact. Merging of perspectives, Wigner’s friend and the extended scenario are treated by worked examples, not theorems. | Read |
| NOTE-1219 | Deutsch & Marletto — Constructor theory of information | A programme paper, not a theorem paper. It defines information without distinguishability: a set of attributes is an information variable when every permutation of it and its cloning are possible tasks, and distinguishability is then defined from that, which removes the circularity of (1.1). It conjectures principles III–IX for classical information, and shows informally that one extra prohibition, a union of two disjoint information observables that is not one, yields indistinguishable states, no-cloning, complementarity, deterministic unpredictability, irreducible measurement disturbance and locally inaccessible information. It calls instantiating information “an exact, intrinsic property of the substrate”, so it supplies a modal, not a relational, account of what an information medium is. | Read |
| NOTE-1220 | Faye & Jaksland — Barad, Bohr, and quantum mechanics | Barad’s reading of Bohr is not Bohr’s. For Bohr a phenomenon is the observed effect under a given experimental arrangement, epistemic and derivative, with atomic object and instrument presupposed real; Bohr was an entity realist, his relational holism epistemic, and the subject–object, knower–known and word–world dualisms are presupposed by his account, not dissolved. Nor does quantum mechanics entail agential realism: Bohmian mechanics reproduces its predictions with an ontology of individual particles, and Rovelli’s relational quantum mechanics, though close, keeps systems prior to their interactions. And because Barad denies that measurement disentangles apparatus and object, she cannot borrow Rovelli’s answer to the measurement problem, so even agential realism’s consistency with quantum mechanics is unshown. Its merits as social theory, “if any”, do not stem from quantum mechanics. | Read |
| NOTE-1226 | Di Biagio & Rovelli — Stable facts, relative facts | A fact happens at every interaction and is relative to the system it affects. A fact relative to F is stable for another system W when W may use it as a conditional in the classical total-probability rule, eq. (3); interference is the failure of that rule. Decoherence gives stability: if F is entangled with an environment E whose branch states overlap by at most ε and W does not interact with E, eq. (3) holds for W up to O(ε). So stability is approximate (exact only as ε → 0) and relational (another system coupled differently to F–E may see interference), and a fact can be stable for W without being a fact relative to W. The authors base the ontology on relative facts, not stable ones, and read the Frauchiger–Renner contradiction and the Local Friendliness violation as the classical rule used across contexts where facts are not stable. | Read |
| NOTE-1227 | Rovelli — Relational quantum mechanics | Rovelli proposes that the incorrect notion behind the measurement problem is the observer-independent state of a system, as absolute simultaneity was behind the unease with the Lorentz transformations. Two descriptions of one measurement, by the measuring system O and by an external P who treats S+O unitarily, are both correct, so “q = 1” is elliptic for “q = 1 relative to O”, any physical system can be the observer, and the theory, taken as complete, is about the information (correlation) systems have about one another. The relation between two observers’ views is itself a quantum fact relative to a third, so no view can be deduced from another and there is no description in toto. Values are relational; systems are not: the world is assumed decomposed into systems, each characterized by its set of yes/no questions W(S) and an information capacity N. Two information postulates give an orthomodular lattice, and a third (superposition) postulate is needed for the Hilbert-space formalism; the reconstruction is, in the author’s words, “not fully successful”. | Read |
| NOTE-1229 | Zurek — Quantum Darwinism | A review. Decoherence singles out pointer states; quantum Darwinism adds that the environment is the channel observers read, and that decoherence leaves many copies of the pointer observable in it: the mutual information I(S : F_f) between the system and a fraction f of the environment rises steeply to a plateau at H_S, so R_δ = 1/f_δ fragments each hold nearly all of it, and redundancy is high only for the pointer observable. Zurek reads redundancy as objectivity. A clean unitarity argument (eq. 5) shows that only orthogonal states can be repeatably copied, without the Born rule, which he derives separately from envariance. The paper claims the measurement problem “largely settled”; it does not state the later caveat that mutual information alone does not guarantee independently accessible records. | Read |
| NOTE-1458 | Barad — Nature’s Queer Performativity | Barad argues that what classical ontology treats as separate things with fixed properties are phenomena, relations without prior relata, and that an intra-action within a phenomenon enacts an agential cut: “a ‘local’ resolution within the phenomenon of the inherent ontological indeterminacy”, giving agential separability, not absolute separation. The case is made by reading lightning, stingray receptor cells, Pfiesteria and the quantum-eraser experiment as failures of classical identity and causality. It is an essay; the physics is cited and interpreted, not argued from, and the term “apparatus” is used only of the two-slit set-up. | Read |