ACTION–THETA–ENTROPY (ATE) THEORY — IIIb: Relativistik Ayrım Dinamiğinin Kuantum Alan Teorisi Olarak İnşası Klein–Gordon Alanının Varyasyonel Kökeni ve Kanonik Kuantizasyonu
This document constitutes ATE–IIIb, the third stage of the relativistic development within the Action–Theta–Entropy (ATE) Theory series. It represents the second necessary step in the relativistic sector, following ATE–IIIa, where the Klein–Gordon equation was derived at the classical field level from a distinction-based variational principle. The purpose of ATE–IIIb is precise and deliberately limited: to demonstrate that the classical relativistic field obtained in ATE–IIIa, when subjected to standard canonical quantization, reproduces the complete and familiar structure of free relativistic scalar quantum field theory. No alternative quantization schemes are proposed, and no new quantum postulates are introduced. Rather, this volume serves as a compatibility and closure test of the ATE program against established quantum field theory. Within the ATE framework, quantization is not treated as a foundational axiom. Instead, it is applied only after a well-defined classical field structure has been established. ATE–IIIb follows this historical and conceptual ordering explicitly: first, the relativistic distinction dynamics is constructed at the classical level (ATE–IIIa); only then is canonical quantization applied. The result is the standard free Klein–Gordon quantum field theory, including its operator structure, Fock space, particle–antiparticle interpretation, conservation laws, and causal properties. A key contribution of this monograph lies not in altering quantum field theory mathematically, but in reframing its ontological interpretation. In ATE–IIIb, particles are not taken as primitive entities. Instead, they are understood as quantized modes of underlying distinction patterns. The field operators and the associated Fock space are treated as representational tools for the quantum structure of distinguishability, rather than as fundamental constituents of reality. This reinterpretation does not modify any experimental predictions, but it prepares the conceptual ground for a later, non-postulatory treatment of measurement, decoherence, and classical emergence within the ATE framework. The scope of this volume is intentionally restricted. It excludes interactions, gauge fields, screening mechanisms, and the Θ field, as well as renormalization and vacuum-energy issues. These topics are deferred to subsequent installments, where the full information-theoretic structure of ATE becomes essential. ATE–IIIb therefore completes the construction of free relativistic scalar quantum field theory, and nothing more—by design. The main text of this monograph is written in Turkish. An English version of the full manuscript is planned for a later release within the same series. It is nevertheless hoped that the universality of the canonical quantum-field-theoretic structure, together with the clear separation between classical construction and quantization, will allow interested researchers to engage with the core content despite the temporary language barrier. This Zenodo entry is intended to serve as: a definitive reference for the quantum-field-theoretic completion of the ATE relativistic program, a demonstration that ATE is fully compatible with standard free quantum field theory, and a conceptual bridge toward future volumes addressing interactions, Θ-field coupling, decoherence, and measurement.
This document constitutes ATE–IIIb, the third stage of the relativistic development within the Action–Theta–Entropy (ATE) Theory series. It represents the second necessary step in the relativistic sector, following ATE–IIIa, where the Klein–Gordon equation was derived at the classical field level from a distinction-based variational principle. The purpose of ATE–IIIb is precise and deliberately limited: to demonstrate that the classical relativistic field obtained in ATE–IIIa, when subjected to standard canonical quantization, reproduces the complete and familiar structure of free relativistic scalar quantum field theory. No alternative quantization schemes are proposed, and no new quantum postulates are introduced. Rather, this volume serves as a compatibility and closure test of the ATE program against established quantum field theory. Within the ATE framework, quantization is not treated as a foundational axiom. Instead, it is applied only after a well-defined classical field structure has been established. ATE–IIIb follows this historical and conceptual ordering explicitly: first, the relativistic distinction dynamics is constructed at the classical level (ATE–IIIa); only then is canonical quantization applied. The result is the standard free Klein–Gordon quantum field theory, including its operator structure, Fock space, particle–antiparticle interpretation, conservation laws, and causal properties. A key contribution of this monograph lies not in altering quantum field theory mathematically, but in reframing its ontological interpretation. In ATE–IIIb, particles are not taken as primitive entities. Instead, they are understood as quantized modes of underlying distinction patterns. The field operators and the associated Fock space are treated as representational tools for the quantum structure of distinguishability, rather than as fundamental constituents of reality. This reinterpretation does not modify any experimental predictions, but it prepares the conceptual ground for a later, non-postulatory treatment of measurement, decoherence, and classical emergence within the ATE framework. The scope of this volume is intentionally restricted. It excludes interactions, gauge fields, screening mechanisms, and the Θ field, as well as renormalization and vacuum-energy issues. These topics are deferred to subsequent installments, where the full information-theoretic structure of ATE becomes essential. ATE–IIIb therefore completes the construction of free relativistic scalar quantum field theory, and nothing more—by design. The main text of this monograph is written in Turkish. An English version of the full manuscript is planned for a later release within the same series. It is nevertheless hoped that the universality of the canonical quantum-field-theoretic structure, together with the clear separation between classical construction and quantization, will allow interested researchers to engage with the core content despite the temporary language barrier. This Zenodo entry is intended to serve as: a definitive reference for the quantum-field-theoretic completion of the ATE relativistic program, a demonstration that ATE is fully compatible with standard free quantum field theory, and a conceptual bridge toward future volumes addressing interactions, Θ-field coupling, decoherence, and measurement.
This document constitutes ATE–IIIa, a focused and self-contained monograph within the Action–Theta–Entropy (ATE) Theory series. While earlier volumes established the ontological foundations (ATE–I) and the full variational and parametric closure of the theory (ATE–II), the present work addresses a specific and structurally critical step: the unique Lorentz-covariant extension of the ATE kinematic principle and its consequence for relativistic dynamics. The central aim of ATE–IIIa is to demonstrate that the Fisher-type kinematic cost, originally formulated for non-relativistic distinction dynamics, admits one and only one consistent extension to a Lorentz-covariant form. This extension is not introduced by analogy with standard field theory, but follows from a minimal set of physical requirements: locality, covariance, positivity, and dimensional consistency. The monograph shows that, once these requirements are enforced, the resulting variational dynamics leads unavoidably to the Klein–Gordon equation, derived here without postulates, operator assumptions, or quantization rules. Importantly, this work is deliberately restricted to the classical field level. It does not introduce second quantization, field operators, particle creation or annihilation, nor spinor structures. This restriction is intentional and methodological: ATE–IIIa is concerned with establishing the classical relativistic core upon which any subsequent quantum field–theoretic construction must rest. In this sense, it is not a relativistic quantum field theory, but the necessary variational substrate for one. A key conceptual contribution of this volume is its ontological clarification. The fundamental physical quantities are the distinction density and its associated phase structure; the wave function appears only as a derived mathematical packaging. As a consequence, long-standing interpretational issues traditionally associated with the Klein–Gordon equation—most notably the so-called “negative probability problem”—do not arise at this level of description. Probability is not taken as a primitive concept; operational distinguishability is. ATE–IIIa thus occupies a precise position within the broader ATE program: it unifies the non-relativistic Schrödinger regime and the relativistic Klein–Gordon regime under a single kinematic principle, formulated entirely in terms of distinction cost. By doing so, it provides a closed, ontologically transparent, and Lorentz-consistent classical foundation for all subsequent developments, including quantization, Θ-field coupling, spinor extensions, and gauge structures, which are intentionally deferred to later monographs. The main text of this monograph is written in Turkish. An English version of the full manuscript is planned for a future release within the same series. It is nevertheless hoped that the universality of the mathematical and variational structure, together with the conceptual clarity of the construction, will allow researchers to engage with its core results despite the temporary language barrier. This Zenodo entry is intended to serve as: a stable reference for the relativistic variational core of the ATE framework, a clear bridge between non-relativistic quantum dynamics and relativistic field equations, and a rigorous starting point for future ATE volumes addressing quantization, spin, and gauge symmetries.
This document constitutes ATE–IIIc, the third and conceptually critical installment of the relativistic quantum field–theoretic stage within the Action–Theta–Entropy (ATE) Theory series. Building directly on the relativistic scalar core established in ATE–IIIa and its canonical quantization completed in ATE–IIIb, this monograph addresses the next unavoidable questions: fermions, interactions, and the quantum consistency of the Θ field. The purpose of ATE–IIIc is not to present a complete unified theory or a finished extension of the Standard Model. Instead, it aims to identify and construct the lowest-order, mathematically consistent, and physically controlled core by which interacting and fermionic quantum field theory can be embedded within the ATE framework. In this sense, the work is deliberately conservative in scope but foundational in intent. Three tightly connected problems are addressed: First, the monograph clarifies how spin-1/2 (fermionic) fields can be incorporated into a distinction-based ontology. While scalar distinction geometry admits a Fisher-type formulation, fermionic degrees of freedom require a fundamentally different geometric structure. ATE–IIIc shows that the natural setting for fermionic distinction dynamics is the projective geometry of Hilbert space, leading to a Fisher–Fubini–Study–type structure. Under minimal requirements of Lorentz covariance, locality, and positivity, this structure reduces uniquely to the Dirac core, which is interpreted here not as a postulate, but as the lowest-cost linear realization of relativistic fermionic distinction dynamics. Second, the Θ field—introduced earlier as the dynamical carrier of the distinction cost–entropy balance—is embedded consistently into relativistic quantum field theory. The Θ sector is treated explicitly as an effective field, with a covariant kinetic structure and an environment-dependent effective potential. No claim of ultraviolet completion is made at this stage; instead, the emphasis is on identifying the correct effective-field-theory (EFT) framework in which Θ can be meaningfully coupled to scalar and fermionic matter. Third, the monograph examines what happens when the Θ field is allowed to enter quantum loop corrections. Rather than seeking formal finiteness, ATE–IIIc adopts the standard modern perspective: renormalizability where possible, and controlled EFT behavior where screening and environment-dependent effects dominate. Issues of radiative stability, naturalness, and mass corrections are explicitly acknowledged as open but well-posed problems, to be addressed at later stages of the program. Throughout, the emphasis is not on introducing novel calculational machinery, but on conceptual consistency and structural necessity. Interactions are included only at the lowest meaningful order, fermions only at the minimal Dirac level, and Θ only in regimes where its role can be technically controlled. This disciplined approach ensures that ATE remains compatible with established relativistic quantum field theory, while offering a distinct ontological and information-theoretic interpretation. The main text of this monograph is written in Turkish. An English version of the full manuscript is planned for a subsequent release within the same series. It is nevertheless hoped that the universality of the field-theoretic language—covariance, symmetry, renormalization structure, and effective descriptions—will allow researchers to engage with the core results despite the temporary language barrier. This Zenodo entry is intended to serve as: a definitive reference for the fermionic and interacting extension of the ATE relativistic program, a clarification of how Dirac dynamics and interactions emerge naturally from distinction geometry, and a controlled starting point for future work on gauge fields, symmetry breaking, and ultraviolet completion within the ATE framework.
This document is part of an ongoing monograph series developing the Action–Theta–Entropy (ATE) Theory, a variational and information-theoretic reconstruction of fundamental physics based on the concept of distinction cost. The ATE framework is constructed step by step, beginning from an explicit ontological shift: physical reality is not grounded in particles or fields as primitives, but in operationally distinguishable states. From this starting point, the theory systematically derives quantum mechanics, thermodynamics, decoherence, measurement, and gravitation as different regimes of a single variational balance between kinematic distinction cost and entropic irreversibility. The present monograph represents a fully extended and mathematically explicit stage of this construction. It includes: a precise definition of distinction density as the fundamental physical variable, a dimensionally closed variational principle balancing Fisher-type kinematic cost against entropy production, the derivation of Schrödinger dynamics in the closed-system limit, a dynamical, energy-dimensioned Θ field governing information–energy conversion, a non-postulatory emergence of decoherence and the Born rule, and an information-theoretic extension of Einstein’s equations. This work does not claim to be a final or complete theory. Rather, it should be read as a coherent core formulation within a longer research program. Subsequent monographs in the series will continue the construction, address remaining open structures (e.g. full dissipative variational closure, parameter unification, and relativistic quantum field limits), and further refine the experimental and phenomenological consequences. The main text of the present monograph is written in Turkish. An English version of the full text is planned and will be uploaded in a later release as part of the same series. In the interim, it is hoped that the universality of the mathematical language—explicit equations, dimensional analysis, and variational structures—will allow interested researchers to engage with the core content despite the temporary language barrier. This repository entry is therefore intended as: a stable, citable record of the current state of the ATE framework, an invitation for critical scrutiny and conceptual discussion, and a reference point for future installments of the monograph series.
This work presents the finalized and dimensionally consistent formulation of the observer sector in the Action–Theta–Entropy (ATE) framework. In contrast to interpretation-dependent approaches to quantum measurement, ATE treats observation as an ordinary physical process characterized by irreversible informational distinctions. The observer’s influence is shown to enter the dynamics through two well-defined and measurable channels: local entropy production and the information distinction rate. Crucially, the theory resolves earlier dimensional ambiguities by introducing a time-integrated information density, rather than a raw information rate, as the quantity responsible for screening effects in the Θ field. This refinement ensures full dimensional closure in both SI and natural units, while preserving the original physical intuition of the model. The Θ field is formulated as an energy-dimensioned scalar field with a stabilizing potential and an entropy-driven source term. Observer activity modifies the effective mass of the Θ field through a dimensionally consistent screening mechanism that depends on information density, not instantaneous measurement strength. Decoherence emerges as a continuous dynamical crossover controlled by this information density, rather than as an ad hoc or instantaneous collapse. A concrete experimental protocol based on superconducting or trapped-ion qubits is presented. The protocol is feasible with current technology and yields falsifiable predictions, including a characteristic nonlinear scaling of decoherence with information density that distinguishes ATE from standard measurement backaction models. Explicit evaluation and falsification criteria are provided. This upload serves as a self-contained technical exposition of the observer–Θ coupling, suitable as a standalone reference or as a formal appendix to the broader ATE monograph series. The formulation is strictly effective-field-theoretic in nature: the predicted backreaction is local to the Θ sector and does not imply any modification of spacetime geometry or gravitational dynamics.
The Klein–Gordon equation, written down in 1926 by Oskar Klein and Walter Gordon (and earlier in unpublished form by Schrödinger himself), is the natural relativistic extension of quantum mechanics for spin‑zero particles: (□ + m²c²/ℏ²) Ψ = 0 where □ = (1/c²) ∂ₜ² − ∇². The equation is second‑order in both time and space, treats them symmetrically, and reduces to the free‑particle Schrödinger equation in the non‑relativistic limit. Its plane‑wave solutions satisfy the relativistic mass‑shell condition E² = p²c² + m²c⁴. Why is a relativistic version needed at all? The Schrödinger equation describes how quantum systems evolve when they’re moving slowly compared to the speed of light — fine for atoms, molecules, and everyday chemistry, where electrons move at less than 1% of c. But the Schrödinger equation is first‑order in time and second‑order in space, treating the two on different footings. Special relativity says space and time are unified into a single four‑dimensional structure where Lorentz transformations mix them, and any equation that privileges time over space breaks Lorentz invariance. A truly fundamental quantum equation must respect this symmetry. Klein–Gordon is the minimal scalar‑field equation that does. The standard derivation runs by analogy: start with the relativistic energy–momentum relation E² = p²c² + m²c⁴, replace E by iℏ∂ₜ and p by −iℏ∇, apply both sides to a wavefunction. This produces the right equation but it’s not really a derivation — it’s a guess motivated by the Schrödinger operator‑substitution procedure. This walkthrough shows how the Event Density framework derives the Klein–Gordon equation from substrate primitives — the same primitives that gave the Born rule, the Schrödinger equation, and the Bell–Tsirelson bound — plus the commitment to Lorentz covariance. The participation measure becomes Lorentz‑covariant. The unique second‑order Lorentz‑scalar differential operator on a scalar field is the d’Alembertian; the minimum‑order Lorentz‑scalar linear equation is therefore (□ + M²) Ψ = 0. Dimensional analysis fixes M² = m²c²/ℏ². Local U(1) gauge invariance forces the covariant derivative and the conserved four‑current. The non‑relativistic limit, via rest‑energy factorization, recovers the free‑particle Schrödinger equation exactly. The structural payoff: the Klein–Gordon equation is what you get when the substrate ontology is pushed through a Lorentz‑covariant reformulation. No new primitives are required. Relativistic and non‑relativistic quantum mechanics are not two theories glued together by analogy; they are the same substrate ontology applied at different velocity regimes. The walkthrough is honest about the known caveats — negative‑energy solutions and the non‑positive‑definite j₀ current — both of which resolve at the Dirac level and at QFT.
The diffculties of relativistic particle theories formulated my means of canonical quantization, such as Klein-Gordon and Dirac theories, ultimately led theoretical physicists to turn on quantum field theory to model elementary particle physics. The aim of the present work is to pursue a method alternative to canonical quantization that avoids these dfficulties. In order to guarantee this result, the present approach is constrained to be developed deductively from physical principles. The physical principles assumed for a free particle consist of the symmetry properties of the particle with respect to the Poincaré group and of the transformation properties of the position observable, expressed by means of a suitably conceived notion of quantum transformation. In so doing, the effectiveness of group theoretical methods is exploited. Our work has pointed out the necessity of new classes of irreducible representations of the Poincaré group the theory can be based on. For spin 0 particle, four inequivalent theories are completely determined, that differ from Klein-Gordon theory for the physical interpretation but also for the mathematical structures. In particular, two of these theories requires irreducible representations in the new classes. The present lack of the explicit transformation properties of position with respect to boosts forbids the complete determination of non zero spin particle theories. A particular form of these transformation properties was in past adopted by Jordan and Mukunda. We check its consistency within the present approach; it is found that for spin 1/2 particles there is only one consistent theory; it is unitarily related to Dirac theory, but, once again, it requires the new classes of irreducible representations.