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Logic Field Theory: A Foundational Program

Longmire, James (JD)

Published Jul 10, 2025• 0 citations
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Abstract

Logic Field Theory: A Foundational Program In this position paper, J. D. Longmire introduces Logic Field Theory (LFT), an ontological reconstruction of quantum mechanics rooted in the three classical laws of logic. By defining the physical state space as Ω = L(S) ,\Omega \;=\; L(S)\,, where SS is the Boolean algebra of all information configurations and LL enforces identity, non-contradiction, and excluded-middle, LFT demonstrates how orthomodular lattice structure, Hilbert-space representations, and unitary dynamics emerge naturally. Three central conjectures—(1) uniqueness of the complex Hilbert representation, (2) strong-continuity of the automorphism group yielding Schrödinger evolution, and (3) a pilot-wave ontology from logical non-distributivity—are verified in a finite C4\mathbb{C}^4 lattice model. A uniform counting measure over microstates reproduces the Born rule, and a perturbative “logic-scale” bound γ<4×10−17\gamma<4\times10^{-17} m from hydrogen spectroscopy (with a proposed muonic-hydrogen test to γ>10−20\gamma>10^{-20} m) offers a clear path to experimental falsification. LFT thus reframes quantum foundations as constraint-satisfaction in an information field, uniting measurement, non-locality, and the “unreasonable effectiveness” of mathematics under logical necessity.

DOI
10.5281/zenodo.15856800

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Logic Field Theory: A Foundational Program In this position paper, J. D. Longmire introduces Logic Field Theory (LFT), an ontological reconstruction of quantum mechanics rooted in the three classical laws of logic. By defining the physical state space as Ω = L(S) ,\Omega ;=; L(S),, where SS is the Boolean algebra of all information configurations and LL enforces identity, non-contradiction, and excluded-middle, LFT demonstrates how orthomodular lattice structure, Hilbert-space representations, and unitary dynamics emerge naturally. Three central conjectures—(1) uniqueness of the complex Hilbert representation, (2) strong-continuity of the automorphism group yielding Schrödinger evolution, and (3) a pilot-wave ontology from logical non-distributivity—are verified in a finite C4\mathbb{C}^4 lattice model. A uniform counting measure over microstates reproduces the Born rule, and a perturbative “logic-scale” bound γ<4×10−17\gamma<4\times10^{-17} m from hydrogen spectroscopy (with a proposed muonic-hydrogen test to γ>10−20\gamma>10^{-20} m) offers a clear path to experimental falsification. LFT thus reframes quantum foundations as constraint-satisfaction in an information field, uniting measurement, non-locality, and the “unreasonable effectiveness” of mathematics under logical necessity.
Logic Realism Theory (LRT) proposes that the Three Fundamental Laws of Logic (3FLL)—Identity, Non-Contradiction, and Excluded Middle—are not merely rules of reasoning but ontological constraints constitutive of physical distinguishability. The core evidence is an asymmetry: we can conceive of 3FLL violations, yet physical reality never produces them. This asymmetry indicates that 3FLL constrain reality, not merely cognition. This paper develops the philosophical foundations of this claim. We argue that: (1) 3FLL have a dual character, functioning both epistemically and ontologically; (2) quantum mechanics emerges as the unique "interface" structure mediating between non-Boolean possibility and Boolean actuality; (3) this framework resolves longstanding interpretive puzzles while maintaining scientific realism. The result is a form of structural realism grounded in logical necessity rather than contingent physical law.
Every physical measurement ever conducted has yielded exactly one outcome: self-identical, non-contradictory, determinate. This paper argues that the Three Fundamental Laws of Logic (Identity, Non-Contradiction, and Excluded Middle) are not cognitive conventions but ontological constraints constitutive of physical distinguishability. From this thesis, quantum mechanics emerges as the unique stable structure at the interface between non-Boolean possibility and Boolean actuality. The framework derives rather than postulates: complex Hilbert space structure follows from interface constraints combined with the physical requirement of compositional consistency (local tomography); the Born rule follows from Gleason's theorem; unitary dynamics follows from information preservation requirements. One structural prediction, that quantum mechanics requires complex amplitudes, has been experimentally confirmed (Renou et al., 2021). The framework specifies twelve explicit conditions that would falsify it. Logic Realism Theory provides unified explanation for seventeen quantum phenomena that other interpretations treat as independent postulates, while maintaining realism, locality, and parsimony.
…the quantum theory of fields is the contemporary locus of metaphysical research. (Stein 1970, p. 285) It is now generally recognized that Quantum Field Theory (QFT), after many vicissitudes and a period of almost total neglect, has emerged during the last ten years as a triumphant framework in which to do elementary particle physics. If philosophers are prepared to use current physical theory as a guide to resolving metaphysical questions (and that is a big “If”), then QFT is what philosophers should all be studying. But the subject is notoriously.recondite, and philosophers are busy people, so the objective of the present paper is to cut through all technicalities and try to tell in as straightforward a way as possible just what QFT has to say about “reality”.
Algebraic quantum field theory, or AQFT for short, is a rigorous analysis of the structure of relativistic quantum mechanics. It is formulated in terms of a net of operator algebras indexed by regions of a Lorentzian manifold. In several cases the mentioned net is represented by a family of von Neumann algebras, concretely, type III factors. Local quantum field logic arises as a logical system that captures the propositional structure encoded in the algebras of the net. In this framework, this work contributes to the solution of a family of open problems, emerged since the 30s, about the characterization of those logical systems which can be identified with the lattice of projectors arising from the Murray-von Neumann classification of factors. More precisely, based on physical requirements formally described in AQFT, an equational theory able to characterizethe type III condition in a factor is provided. This equational system motivates the study of a variety of algebras having an underlying orthomodular lattice structure. A Hilbert style calculus, algebraizable in the mentioned variety, is also introduced and a corresponding completeness theorem is established.
Quantum field theory is a powerful language for the description of the subatomic constituents of the physical world and the laws and principles that govern them. This book contains up-to-date in-depth analyses, by a group of eminent physicists and philosophers of science, of our present understanding of its conceptual foundations, of the reasons why this understanding has to be revised so that the theory can go further, and of possible directions in which revisions may be promising and productive. These analyses will be of interest to graduate students and research workers in physics who want to know about the foundational problems of their subject. The book will also be of interest to professional philosophers, historians and sociologists of science, because it contains much material for metaphysical and methodological reflections, for historical and cultural analyses, and for sociological analyses of the way in which various factors contribute to the way the foundations are revised.
We argue that limitations arise when using non-constructive mathematics (NCM) as a foundation for quantum mechanics, since the Boolean logical substrate of NCM fits a distributive propositional structure, but fits less naturally into the non-distributive lattice of quantum propositions. We distinguish three logical structures relevant to this discussion: Boolean algebras (classical, distributive, complemented), Heyting algebras (constructive, distributive, not necessarily complemented), and orthomodular lattices (quantum, non-distributive). The linear structure of quantum state space is compatible with NCM, but the propositional structure of quantum mechanics forms a non-distributive lattice rather than a Boolean algebra. Topos theory, one of the models of constructive mathematics, offers a way forward: the Döring–Isham framework provides an explicit bridge—the daseinization map—from quantum propositions to a topos whose internal logic is intuitionistic. Although some topoi carry Boolean logic, in general, topoi are not constrained to it. We argue that this bridged framework is more suitable for the mathematical foundation of quantum mechanics than one that retains the law of excluded middle (LEM) without such a bridge. Our discussion centers on the pivotal role of the LEM.