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Q Quantum Theory II-C: Electrodynamics - Correspondence with Finite-Coherent and Local QED

Gordon Liu

Published Sep 30, 2026• 0 citations
Publisher

Abstract

This paper develops the representation-level correspondence between Q electrodynamics (QQED), its finite-coherent operator/Fock realization, and renormalized local QED. The primitive physical object remains the complete Q of Paper II-A; the purpose here is not to introduce a second ontology but to show how the same finite-response mother is represented by response-space operators, Fock sectors, Hamiltonian/TOPT amplitudes and, when the response is unresolved, the ordinary local-QED organization. The exact branch is followed from its cut-plane master response through an operator-valued response lift, cocycle-Fock dynamics, a balanced contractive interaction, coefficientwise optical and TOPT-Euclidean reconstruction, and the complete one-loop self-energy/vacuum-polarization/proper-vertex Ward family. The ultraviolet proofs remain in Paper II-B. The construction is then extended to the charged-Fock domain, relative infrared completion, non-Fock charged asymptotic sectors, dressed one- and multi-Q scattering, asymptotic photon fields and response-resonance sewing.

DOI
10.5281/zenodo.23061927

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Q Quantum Theory is formulated here at the foundational and quantum-mechanical level. Its basic postulate is that the physical quantum entity, denoted by Q, is the whole quantum state rather than an independently real material point whose unknown position is described by a secondary probability cloud. In a one-Q sector admitting a local amplitude representation,\begin{equation} Q(x)=\sqrt{\rho(x)}\,e^{i\theta(x)}Z(x),\end{equation}where $x$ is a spacetime point, $\rho$ is the Born response weight, $\theta$ is the phase, and $Z$ is a normalized internal structural direction. A dagger denotes the Hermitian adjoint and $\partial_\mu$ differentiates with respect to $x^\mu$. The decomposition has a local rephasing redundancy. This is a redundancy of the local factorization of the complete Q amplitude, not the physical electromagnetic $U(1)$ gauge symmetry. The associated decomposition-invariant phase-orientation one-form is\begin{equation} \kappa_\mu=\partial_\mu\theta-iZ^\dagger\partial_\mu Z =\frac{\operatorname{Im}\langle Q,\partial_\mu Q\rangle}{\rho}, \qquad \rho>0.\end{equation}In Schr\"odinger correspondence sectors this structure yields exact probability-current and momentum identities, including the standard Abelian gauge-covariant current. In that audit the electromagnetic connection is a correspondence variable; its Q-level interpretation as a local representation of inter-Q relational comparison geometry is developed in the companion manuscript Paper II-A. Klein-Gordon and Dirac audits show that relativistic currents and stress-energy depend on the Lorentz representation: $\kappa_\mu$ carries phase-orientation translation structure, while amplitude gradients and internal orientation can contribute independently to energy and stress. The standard Hilbert-space probability calculus is retained. Interference, tunnelling, entanglement, localized detection and cloud-chamber tracks are interpreted as properties or records of whole-Q states rather than as evidence for hidden point trajectories. Entanglement is represented by a nonfactorizable joint Q state; standard Bell correlations and no-signalling are retained without identifying correlation with a superluminal physical signal. Boundaryless state support is distinguished from the propagation of a newly generated disturbance, which is required to obey retarded relativistic causality. For stable relativistic sectors, invariant mass is treated as a derived characterization of the translation spectrum: timelike Q states possess a rest frame and rest energy, whereas null Q states do not. The foundational structure does not specify a unique interaction action or establish ultraviolet finiteness. Finite-response dynamics requires a separate construction.

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Quantum Field Theory and Statistical Mechanics: Expositions

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I Infinite Renormalization of the Hamiltonian Is Necessary.- II Quantum Field Theory Models.- I. The ?22n Model.- Fock space.- Q space.- The Hamiltonian H(g).- Removing the space cutoff.- Lorentz covariance and the Haag-Kastler axioms.- II. The Yukawa Model.- Preliminaries.- First and second order estimates.- Resolvent convergence and self adjointness.- The Heisenberg picture.- III Boson Quantum Field Models.- I. General Results.- Hermite operators.- Gaussian measures and the Schrodinger representation.- Hermite expansions and Fock space.- II. The Solution of Two-Dimensional Boson Models.- The interaction Hamiltonian.- The free Hamiltonian.- Self-adjointness of H(g).- The local algebras and the Lorentz group automorphisms.- IV Boson Quantum Field Models.- III. Further Developments.- Locally normal representations of the observables.- The construction of the physical vaccum.- Formal perturbation theory and models in three space-time dimensions.- V The Particle Structure of the Weakly Coupled P(?)2 Model and Other Applications of High Temperature Expansions.- I. Physics of Quantum Field Models.- Five years of models.- From estimates to physics.- Bound states and resonances.- Phase space localization and renormalization.- VI The Particle Structure of the Weakly Coupled P(?)2 Model and Other Applications of High Temperature Expansions.- II. The Cluster Expansion.- The main results.- The cluster expansion.- Clustering and analyticity: proof of the main results.- Convergence: the main ideas.- An equation of Kirkwood-Salsburg type.- Covariance operators.- Derivatives of covariance operators.- Gaussian integrals.- Convergence: the proof completed.- VII Particles and Bound States and Progess Toward Unitarity and Scaling.- VIII Critical Problems in Quantum Fields.- IX Existence of Phase Transitions for ?24 Quantum Fields.- X Critical Exponents and Renormalization in the ?4 Scaling Limit.- Formulation of the problem.- The scaling and critical point limits.- Renormalization of the ?2(x) field.- Existence of the scaling limit.- The Josephson inequality.- XI A Tutorial Course in Constructive Field Theory.- e?tH as a functional integral.- Examples.- Applications of the functional integral representation.- Ising, Gaussian and scaling limits.- Main results.- Correlation inequalities.- Absence of even bound states.- Bound on g.- Bound on dm2/d? and particles.- The conjecture ?(6) ? 0.- Cluster expansions.- The region of convergence.- The zeroth order expansion.- The primitive expansion.- Factorization and partial resummation.- Typical applications.

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Object-relative ultraviolet weighting of electromagnetic modes and one-loop ultraviolet finiteness of internal photon lines in quantum electrodynamics

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This work explores whether localized electromagnetic interactions can be modeled by a physically concluded object-relative ultraviolet weighting of photon modes. The proposal is motivated heuristically by a weak-field self-backreaction estimate for localized energy-carrying modes and by a three-dimensional overlap argument. In the resulting ansatz, the infrared sector remains unchanged up to a characteristic angular wavenumber k_c, while ultraviolet contributions are suppressed asymptotically by a factor of order k_c^3/k^3. The mode structure is object-relative: a real photon is weighted by its frequency u k in the rest frame of the interaction object with four-velocity u. The occupation of the modes and the amplitudes of the vacuum fluctuations, in contrast, follow the rules of special relativity, so that virtual photons in loops are weighted by the invariant k k, i.e. by the Euclidean photon momentum. The weighting applies to photon lines only; applied in the same way to the electron line, it conflicts with gauge invariance or with the measured running of the fine-structure constant. One-loop diagrams with internal photon lines become ultraviolet finite, and the weighting can be represented by a gauge-invariant kinetic term of the photon field, so that the Ward-Takahashi identities are preserved. Four test cases are discussed. A Bethe-type estimate of the Lamb shift places the characteristic scale near the electron Compton wavenumber. The anomalous magnetic moment and the relativistic part of the Lamb shift show, however, that a photon weighting at this scale requires additional structure on the electron line to remain compatible with high-precision measurements. This structure, the vacuum polarization, and the spectral consistency of the weighting define the tasks for future work.

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This paper develops the representation-level correspondence between Q electrodynamics (QQED), its finite-coherent operator/Fock realization, and renormalized local QED. The primitive physical object remains the complete Q of Paper II-A; the purpose here is not to introduce a second ontology but to show how the same finite-response mother is represented by response-space operators, Fock sectors, Hamiltonian/TOPT amplitudes and, when the response is unresolved, the ordinary local-QED organization. The exact branch is followed from its cut-plane master response through an operator-valued response lift, cocycle-Fock dynamics, a balanced contractive interaction, coefficientwise optical and TOPT-Euclidean reconstruction, and the complete one-loop self-energy/vacuum-polarization/proper-vertex Ward family. The ultraviolet proofs remain in Paper II-B. The construction is then extended to the charged-Fock domain, relative infrared completion, non-Fock charged asymptotic sectors, dressed one- and multi-Q scattering, asymptotic photon fields and response-resonance sewing.
Q Quantum Theory is formulated here at the foundational and quantum-mechanical level. Its basic postulate is that the physical quantum entity, denoted by Q, is the whole quantum state rather than an independently real material point whose unknown position is described by a secondary probability cloud. In a one-Q sector admitting a local amplitude representation,\begin{equation} Q(x)=\sqrt{\rho(x)}\,e^{i\theta(x)}Z(x),\end{equation}where xxx is a spacetime point, ρ\rhoρ is the Born response weight, θ\thetaθ is the phase, and ZZZ is a normalized internal structural direction. A dagger denotes the Hermitian adjoint and ∂μ\partial_\mu∂μ​ differentiates with respect to xμx^\muxμ. The decomposition has a local rephasing redundancy. This is a redundancy of the local factorization of the complete Q amplitude, not the physical electromagnetic U(1)U(1)U(1) gauge symmetry. The associated decomposition-invariant phase-orientation one-form is\begin{equation} \kappa_\mu=\partial_\mu\theta-iZ^\dagger\partial_\mu Z =\frac{\operatorname{Im}\langle Q,\partial_\mu Q\rangle}{\rho}, \qquad \rho>0.\end{equation}In Schr\"odinger correspondence sectors this structure yields exact probability-current and momentum identities, including the standard Abelian gauge-covariant current. In that audit the electromagnetic connection is a correspondence variable; its Q-level interpretation as a local representation of inter-Q relational comparison geometry is developed in the companion manuscript Paper II-A. Klein-Gordon and Dirac audits show that relativistic currents and stress-energy depend on the Lorentz representation: κμ\kappa_\muκμ​ carries phase-orientation translation structure, while amplitude gradients and internal orientation can contribute independently to energy and stress. The standard Hilbert-space probability calculus is retained. Interference, tunnelling, entanglement, localized detection and cloud-chamber tracks are interpreted as properties or records of whole-Q states rather than as evidence for hidden point trajectories. Entanglement is represented by a nonfactorizable joint Q state; standard Bell correlations and no-signalling are retained without identifying correlation with a superluminal physical signal. Boundaryless state support is distinguished from the propagation of a newly generated disturbance, which is required to obey retarded relativistic causality. For stable relativistic sectors, invariant mass is treated as a derived characterization of the translation spectrum: timelike Q states possess a rest frame and rest energy, whereas null Q states do not. The foundational structure does not specify a unique interaction action or establish ultraviolet finiteness. Finite-response dynamics requires a separate construction.
This manuscript establishes the absolute operator-theoretic closure of quantum electricity. It formally defines quantum electricity strictly as the gauge-covariant conserved charge transport generated by a self-adjoint matter Hamiltonian under electromagnetic minimal coupling, rather than treating it as a supplementary physical ontology. The paper rigorously derives the exact one-particle current, its operator-valued local density, its Heisenberg continuity law, and its variational identity as the functional derivative of the Hamiltonian with respect to the vector potential. The framework is then systematically extended to spin-1/2 Pauli matter—proving the exact decomposition of the conserved current into distinct transport and divergence-free magnetization sectors—and to second quantization, establishing the many-body continuity law. Furthermore, the manuscript demonstrates that classical electrical source fields emerge exactly as the macroscopic expectation values of the quantum current operators. This provides the rigorous source-side completion for Maxwell field evolution. The theory is also generalized to relativistic quantum electrodynamics (identifying the current as the Noether current of local U(1) gauge invariance), expressed through spectral representation, and applied to open-system transport via scattering theory and wave operators. Ultimately, the manuscript proves a unified closure: all admissible electrical behavior is completely exhausted by the combination of gauge-covariant matter current, Maxwell field evolution, constitutive response, and boundary data.
We develop a coherent-field formulation of quantum electrodynamics (QED) based on coherent-state boundary conditions, in which laser fields are represented by asymptotic coherent states rather than by prescribed classical background fields. Starting from coherent-state boundary conditions and the displacement-operator formalism, we construct operator and path-integral descriptions of transitions between distinct electromagnetic coherent states. This formulation provides a coherent-field extension of conventional background-field QED and incorporates the quantum dynamics of the coherent field itself. The corresponding path-integral representation contains a functional integral over coherent-field histories in addition to the usual functional integral over quantum fluctuations, thereby extending the conventional background-field description. The resulting path-integral representation naturally leads to an effective action for the stationary coherent-field configuration. The corresponding saddle-point condition yields an effective Maxwell equation containing contributions from vacuum polarization, photon fluctuations, and coherent-field fluctuations. In the limit where the latter two contributions vanish, the formalism reduces to the conventional Heisenberg-Euler description.
We develop a manifestly gauge and Lorentz invariant perturbative calculus to study the finite time and distance evolution of the electrically charged states created by Wilson-line-dressed scalar field operators in scalar quantum electrodynamics. By an Osterwalder-Schrader reflection and Euclidean gluing, the overlap of two dressed states is expressed as a Euclidean 2 point function of dressed operators, which is then evaluated in renormalized perturbation theory with a Stueckelberg photon mass as a gauge, BRST and Lorentz invariant infrared cutoff. This renders explicit and computable in closed form several structural features of the charged sector that have so far been accessible mainly through non-perturbative or algebraic arguments. We find that states dressed by non-parallel Wilson lines are ``cloud orthogonal". Their overlap vanishes as a power of the infrared regulator, with an exponent that depends only on the angle between the two dressings and is an infrared analogue of the cusp anomalous dimension with the difference that it is one-loop exact whenever the charged matter is mass-gapped. Charged states are thereby superselected by the orientation of their dressing. Within a superselection sector the 2 point function is infrared finite but exhibits the infraparticle. The mass shell pole is replaced by the edge of a branch cut. As a result, the asymptotic spread of a dressed charge is a power law in proper time with a one-loop-exact computable exponent depending on the dressing and on the direction of motion. We show that although cloud orthogonality and the infraparticle are each extremely sensitive to fundamental infrared cutoffs, the infraparticle scaling law persists across a wide window of proper time.
I Infinite Renormalization of the Hamiltonian Is Necessary.- II Quantum Field Theory Models.- I. The ?22n Model.- Fock space.- Q space.- The Hamiltonian H(g).- Removing the space cutoff.- Lorentz covariance and the Haag-Kastler axioms.- II. The Yukawa Model.- Preliminaries.- First and second order estimates.- Resolvent convergence and self adjointness.- The Heisenberg picture.- III Boson Quantum Field Models.- I. General Results.- Hermite operators.- Gaussian measures and the Schrodinger representation.- Hermite expansions and Fock space.- II. The Solution of Two-Dimensional Boson Models.- The interaction Hamiltonian.- The free Hamiltonian.- Self-adjointness of H(g).- The local algebras and the Lorentz group automorphisms.- IV Boson Quantum Field Models.- III. Further Developments.- Locally normal representations of the observables.- The construction of the physical vaccum.- Formal perturbation theory and models in three space-time dimensions.- V The Particle Structure of the Weakly Coupled P(?)2 Model and Other Applications of High Temperature Expansions.- I. Physics of Quantum Field Models.- Five years of models.- From estimates to physics.- Bound states and resonances.- Phase space localization and renormalization.- VI The Particle Structure of the Weakly Coupled P(?)2 Model and Other Applications of High Temperature Expansions.- II. The Cluster Expansion.- The main results.- The cluster expansion.- Clustering and analyticity: proof of the main results.- Convergence: the main ideas.- An equation of Kirkwood-Salsburg type.- Covariance operators.- Derivatives of covariance operators.- Gaussian integrals.- Convergence: the proof completed.- VII Particles and Bound States and Progess Toward Unitarity and Scaling.- VIII Critical Problems in Quantum Fields.- IX Existence of Phase Transitions for ?24 Quantum Fields.- X Critical Exponents and Renormalization in the ?4 Scaling Limit.- Formulation of the problem.- The scaling and critical point limits.- Renormalization of the ?2(x) field.- Existence of the scaling limit.- The Josephson inequality.- XI A Tutorial Course in Constructive Field Theory.- e?tH as a functional integral.- Examples.- Applications of the functional integral representation.- Ising, Gaussian and scaling limits.- Main results.- Correlation inequalities.- Absence of even bound states.- Bound on g.- Bound on dm2/d? and particles.- The conjecture ?(6) ? 0.- Cluster expansions.- The region of convergence.- The zeroth order expansion.- The primitive expansion.- Factorization and partial resummation.- Typical applications.
This work explores whether localized electromagnetic interactions can be modeled by a physically concluded object-relative ultraviolet weighting of photon modes. The proposal is motivated heuristically by a weak-field self-backreaction estimate for localized energy-carrying modes and by a three-dimensional overlap argument. In the resulting ansatz, the infrared sector remains unchanged up to a characteristic angular wavenumber k_c, while ultraviolet contributions are suppressed asymptotically by a factor of order k_c^3/k^3. The mode structure is object-relative: a real photon is weighted by its frequency u k in the rest frame of the interaction object with four-velocity u. The occupation of the modes and the amplitudes of the vacuum fluctuations, in contrast, follow the rules of special relativity, so that virtual photons in loops are weighted by the invariant k k, i.e. by the Euclidean photon momentum. The weighting applies to photon lines only; applied in the same way to the electron line, it conflicts with gauge invariance or with the measured running of the fine-structure constant. One-loop diagrams with internal photon lines become ultraviolet finite, and the weighting can be represented by a gauge-invariant kinetic term of the photon field, so that the Ward-Takahashi identities are preserved. Four test cases are discussed. A Bethe-type estimate of the Lamb shift places the characteristic scale near the electron Compton wavenumber. The anomalous magnetic moment and the relativistic part of the Lamb shift show, however, that a photon weighting at this scale requires additional structure on the electron line to remain compatible with high-precision measurements. This structure, the vacuum polarization, and the spectral consistency of the weighting define the tasks for future work.