Quantum trajectory theories have not fully reconciled discrete quantum jumps with continuous unitary evolution. We address this challenge by developing a hidden variable formulation that reveals hidden correlations in individual trials. We represent jump sequences as logical propositions on energy eigenstate occupations, where logical variables describe their truth values and chain operators describe their probabilities. To circumvent established no-go theorems, we introduce a proposition selection rule that requires well-defined joint probabilities for a given density operator. This rule unifies compatibility criteria for quantum and hidden variable theories, eliminating incompatible propositions that lead to value assignment paradoxes. Unlike conventional criteria mandating operator commutativity, it permits propositions that become deterministic for specific initial conditions. This framework systematically captures deterministic relations in individual trials, including the exclusivity of state occupation implied by the observed jumps. This exclusivity, distinct from the Pauli exclusion principle, enables the formalization of a quantum version of Boolean logic as laws for quantum propositions. These assumptions circumvent Bell-type inequalities by prohibiting arithmetic operations between incompatible logical variables, while enabling classical descriptions within the complementarity. The resulting theory reconciles discrete jumps with continuous evolution, providing a direct description of individual trials while preserving standard probability predictions.
Quantum trajectory theories have not fully reconciled discrete quantum jumps with continuous unitary evolution. We address this challenge by developing a hidden variable formulation that reveals hidden correlations in individual trials. We represent jump sequences as logical propositions on energy eigenstate occupations, where logical variables describe their truth values and chain operators describe their probabilities. To circumvent established no-go theorems, we introduce a proposition selection rule that requires well-defined joint probabilities for a given density operator. This rule unifies compatibility criteria for quantum and hidden variable theories, eliminating incompatible propositions that lead to value assignment paradoxes. Unlike conventional criteria mandating operator commutativity, it permits propositions that become deterministic for specific initial conditions. This framework systematically captures deterministic relations in individual trials, including the exclusivity of state occupation implied by the observed jumps. This exclusivity, distinct from the Pauli exclusion principle, enables the formalization of a quantum version of Boolean logic as laws for quantum propositions. These assumptions circumvent Bell-type inequalities by prohibiting arithmetic operations between incompatible logical variables, while enabling classical descriptions within the complementarity. The resulting theory reconciles discrete jumps with continuous evolution, providing a direct description of individual trials while preserving standard probability predictions.
Quantum theory, despite its remarkable success, struggles to represent certain experimental data, particularly those involving integer functions and deterministic relations between quantum jumps. We address this limitation by proposing a hidden variable theory compatible with non-relativistic quantum electrodynamics (QED). Our approach introduces logical variables to describe propositions about the occupation of stationary states, based on three key assumptions: (i) the probabilities of propositions are predictable by quantum theory, (ii) the truth values of propositions conform to Boolean logic, and (iii) propositions satisfy a novel selection rule: Tr[K^(t)ρ^K^†(t)]=Tr[(PK^(t))ρ^(PK^(t))†]. Here, K^(t) is the product of projection operators for the given propositions, and PK^(t) denotes K^(t) rearranged arbitrarily. This selection rule extends the consistency condition in the Consistent Histories approach, relaxing constraints imposed by no-go theorems on logical variable representation, joint probability existence, and measurement contextuality. Consequently, our theory successfully describes the essential properties of individual trials, including the discreteness of quantum jumps, the continuity of classical trajectories, and deterministic relations between entangled subsystems. It achieves this without conflicting with quantum equations of motion, canonical commutation relations, or the operator ordering in quantum coherence functions. The direct description of individual trials made possible by this approach sheds new light on the discrete nature of quantum phenomena and the essence of reality.
Abstract We propose a novel class of local deterministic models to explain the correlations observed in entangled singlet quantum states. In these models, hidden variables influence both the entangled pair preparation and the measurement settings, thereby relaxing the measurement independence assumption used in Bell’s theorem. Our model accurately reproduces the expected quantum results and demonstrates that freely chosen measurement settings do not necessarily preclude the possibility of hidden variables inducing correlations between the settings and outcomes. This approach reveals an intricate relationship between local determinism and quantum mechanics, showing that local hidden variable theories can be consistent with the predictions of quantum theory. This result calls into question the conventional interpretation, based on Bell’s inequalities, that local theories are fundamentally incompatible with quantum mechanics.
The Statistical Interpretation of quantum theory is formulated for the purpose of providing a sound interpretation using a minimum of assumptions. Several arguments are advanced in favor of considering the quantum state description to apply only to an ensemble of similarily prepared systems, rather than supposing, as is often done, that it exhaustively represents an individual physical system. Most of the problems associated with the quantum theory of measurement are artifacts of the attempt to maintain the latter interpretation. The introduction of hidden variables to determine the outcome of individual events is fully compatible with the statistical predictions of quantum theory. However, a theorem due to Bell seems to require that any such hidden-variable theory which reproduces all of quantum mechanics exactly (i.e., not merely in some limiting case) must possess a rather pathological character with respect to correlated, but spacially separated, systems.
The thesis investigates the possibility of a classical semantics for quantum propositional structures. A classical semantics is defined as a set of mappings each of which is (i) bivalent, i.e., the value 1 (true) or 0 (false) is assigned to each proposition, and (ii) truth-functional, i.e., the logical operations are preserved. In addition, this set must be "full", i.e., any pair of distinct propositions is assigned different values by some mapping in the set. When the propositions make assertions about the properties of classical or of quantum systems, the mappings should also be (iii) "state-induced", i.e., values assigned by the semantics should accord with values assigned by classical or by quantum mechanics. In classical propositional logic, (equivalence classes of) propositions form a Boolean algebra, and each semantic mapping assigns the value 1 to the members of a certain subset of the algebra, namely, an ultrafilter, and assigns 0 to the members of the dual ultraideal, where the union of these two subsets is the entire algebra. The propositional structures of classical mechanics are likewise Boolean algebras, so one can straightforwardly provide a classical semantics, which also satisfies (iii). However, quantum propositional structures are non-Boolean, so it is an open question whether a semantics satisfying (i), (ii) and (iii) can be provided. Von Neumann first proposed (1932) that the algebraic structures of the subspaces (or projectors) of Hilbert space be regarded as the pro-positional structures PQM of quantum mechanics. These structures have been formalized in two ways: as orthomodular lattices which have the binary operations "and", "or", defined among all elements, compatible [symbol omitted] and incompatible [symbol omitted]; and as partial-Boolean algebras which have the binary operations defined among only compatible elements. In the thesis, two basic senses in which these structures are non-Boolean are discriminated. And two notions of truth-functionality are distinguished: truth-functionality [symbols omitted] applicable to the PQM lattices; and truth-functionality [symbol omitted] applicable to both the lattices and partial-Boolean algebras. Then it is shown how the lattice definitions of "and", "or", among incompatibles rule out a bivalent, truth-functional [symbols omitted] semantics for any lattice containing incompatible, elements. In contrast, the Gleason and Kochen-Specker proofs of the impossibility of hidden-variables for quantum mechanics show the impossibility of a bivalent, truth-functional [symbol omitted] semantics for three-or-higher dimensional Hilbert space structures; and the presence of incompatible elements is necessary but is not sufficient to rule out such a semantics. As for (iii), each quantum state-induced expectation-function on a PQM truth-functionally assigns 1 and 0 values to the elements in a ultrafilter and dual ultraideal of PQM³ where, in general the union of an ultrafilter and its dual ultraideal is smaller than the entire structure. Thus it is argued that each expectation-function is the quantum analog of a classical semantic mapping, even though the domain where each expectation-function is bivalent and truth-functional is usually a non-Boolean substructure of PQM. The final portion of the thesis surveys proposals for the introduction of hidden variables into quantum mechanics, proofs of the impossibility of such hidden-variable proposals, and criticisms of these impossibility proofs. And arguments in favour of the partial-Boolean algebra, rather than the orthomodular lattice, formalization of the quantum propositional structures are reviewed.
This article develops an expanded critical–propositional analysis of William T. M. Irvine, Juan F. Hodelin, Christoph Simon, and Dirk Bouwmeester’s 2005 experiment, “Realization of Hardy’s Thought Experiment with Photons,” published in Physical Review Letters, in systematic dialogue with the Theory of Objectivity (TO) formulated by Vidamor Cabannas and Denivaldo Silva. Irvine et al. experimentally implemented a photonic version of Lucien Hardy’s two-particle argument against local hidden-variable descriptions by employing two coupled Mach–Zehnder interferometers, nonmaximally entangled path states, photon bunching at a shared beam splitter, and coincidence detection. The experiment derived a Clauser–Horne-type inequality appropriate to its nonideal optical implementation and reported a violation of the local-hidden-variable bound by twelve standard deviations. Its photonic configuration relied crucially upon high two-photon indistinguishability; the measured distinguishability probability was approximately 7.8 percent, below the 12.5 percent threshold identified by the authors for observing the relevant violation. These results provide an experimentally controlled demonstration that the class of local hidden-variable models addressed by the inequality cannot reproduce the observed path correlations (Irvine et al. 2005). The present study asks a different question: whether, and in what restricted sense, Irvine et al.’s results can enter into empirical dialogue with the modal axioms and physical proposals of TO. Particular attention is devoted to the First Absolute Truth, according to which the pre-universal “Nothing” is not absolute nonbeing but an eternal logical-mathematical essence; the Second Absolute Truth, according to which every existing element possesses its own exclusive individualizing magnetic field; the Fourth Absolute Truth concerning boundaries or interfaces; the Fifth Absolute Truth, according to which logical existence requires an element to be observed by, or to lie within the frequency of, at least two other elements, where observation is physical and non-anthropocentric; and the Seventh Absolute Truth, in which the transcendent element is interpreted as knowledge or information produced in atomic relations and regarded by TO as equivalent to atomic radiation. The analysis distinguishes modal necessity from empirical confirmation. If an axiom is claimed to possess necessary status, no finite collection of observations can establish that necessity inductively. Experiments can nevertheless test operational bridge principles linking a modal ontology to measurable consequences. Under this distinction, Irvine et al. exhibit significant compatibility with TO’s emphasis on relationality, physically efficacious boundaries, non-anthropocentric registration, and objectively instantiated correlations. At the same time, the experiment generates a particularly important tension for the Second Absolute Truth: quantum interference requires photons to be operationally indistinguishable, whereas a strongly measurable individualizing field would constitute distinguishing information and could therefore degrade two-photon bunching. The article consequently proposes a distinction between ontological individuality and experimentally accessible distinguishability and formulates potential experimental bridges through which the individualizing-field thesis could become testable. The analysis further examines TO’s phenomenic elements, Inducing Effects, Cosmogonic Theorem, and Cosmological Eras, while refusing to infer cosmological confirmation from a laboratory-scale quantum-optical experiment. The conclusion is that Irvine et al. provide meaningful indirect corroborative contact with several relational and informational motifs of TO, but neither confirm TO uniquely nor establish the modal necessity of its axioms. The most scientifically productive consequence of the comparison is therefore not retrospective assimilation of quantum mechanics into TO, but the formulation of novel, risk-bearing, quantitative predictions capable of distinguishing TO from standard quantum theory. Keywords: Theory of Objectivity; Hardy paradox; quantum nonlocality; local hidden variables; photon bunching; quantum indistinguishability; physical information; non- anthropocentric observation; modal ontology; boundaries; empirical testability.
Bell nonlocality is an intriguing property of quantum mechanics with far reaching consequences for information processing, philosophy and our fundamental understanding of nature. However, nonlocality is a statement about static correlations only. It does not take into account dynamics, i.e. time evolution of those correlations. Consider a dynamic situation where the correlations remain local for all times. Then at each moment in time there exists a local hidden-variable (LHV) model reproducing the momentary correlations. Can the time evolution of the correlations then be captured by evolving the hidden variables? In this light, we define dynamical LHV models and motivate and discuss potential additional physical and mathematical assumptions. Based on a simple counter example we conjecture that such LHV dynamics does not always exist. This is further substantiated by a rigorous no-go theorem. Our results suggest a new type of nonlocality that can be deduced from the observed time evolution of measurement statistics and which generically occurs in interacting quantum systems.