We continue the program in which action is treated as a primary structure and the phase \[ \phi=\frac{S}{\hbar} \] as its associated operational variable. In previous works, the phase structure was connected to quantum dynamics and effective gravitational geometry. In the present article, we focus exclusively on the electromagnetic sector. We show that the electromagnetic coupling can be formulated as a local modulation of the phase by a 1-form \(A_\mu\), such that the minimal substitution \[ \partial_\mu \phi \;\longrightarrow\; \partial_\mu \phi - \frac{e}{\hbar}A_\mu \] emerges naturally as a way to preserve phase coherence in the presence of external interaction. In this language, the electromagnetic field acts as a phase connection, gauge invariance expresses the redundancy of the absolute phase reference, and the Aharonov-Bohm effect appears as a direct experimental manifestation of the physical reality of the connection's holonomy. The objective of this work is not to fully derive Maxwell's equations from a new microscopic model, but to make explicit that the formal structure of electromagnetism is compatible with a formulation based on phase, connection, and coherence. In this sense, the electromagnetic interaction can be reinterpreted as an active phase modulation, in contrast to the role of gravity as a passive propagation structure.
We continue the program in which action is treated as a primary structure and the phase [ \phi=\frac{S}{\hbar} ] as its associated operational variable. In previous works, the phase structure was connected to quantum dynamics and effective gravitational geometry. In the present article, we focus exclusively on the electromagnetic sector. We show that the electromagnetic coupling can be formulated as a local modulation of the phase by a 1-form (A_\mu), such that the minimal substitution [ \partial_\mu \phi ;\longrightarrow; \partial_\mu \phi - \frac{e}{\hbar}A_\mu ] emerges naturally as a way to preserve phase coherence in the presence of external interaction. In this language, the electromagnetic field acts as a phase connection, gauge invariance expresses the redundancy of the absolute phase reference, and the Aharonov-Bohm effect appears as a direct experimental manifestation of the physical reality of the connection's holonomy. The objective of this work is not to fully derive Maxwell's equations from a new microscopic model, but to make explicit that the formal structure of electromagnetism is compatible with a formulation based on phase, connection, and coherence. In this sense, the electromagnetic interaction can be reinterpreted as an active phase modulation, in contrast to the role of gravity as a passive propagation structure.
We propose a unified formulation, at a structural level, in which action occupies the primary role and the phase ϕ=S/ℏ functions as a universal intermediate variable. It is shown that, from this identification, two distinct formal directions emerge naturally. In the first, the phase structure leads to the wave description of matter, to the limits of spectral localization, and to the linear form of quantum dynamics. In the second, the phase determines eikonal propagation, geodesic structure, and, under minimal variational hypotheses, an effective geometry associated with gravity. The objective of this work is not to fully deduce quantum mechanics and general relativity from a single closed microscopic model, but to make explicit that both can be reinterpreted as complementary realizations of the same variational architecture based on action, phase coherence, and effective structure. In this sense, action ceases to be merely an auxiliary functional of dynamics and assumes the role of a primary structure common to the quantum and gravitational regimes.
Continuing the program in which the action constitutes the primary structure of dynamics and the phase ϕ=S/ℏ plays the role of the operational variable, we investigate the relationship between global variational frustration and the emergence of gauge connections in the structural operator of the network. In previous works, it was shown that local extremizing trajectories of the action are only physically realizable when compatible with a global spectral condition, formalized by the Laplacian operator associated with the admissible network, and that electromagnetism can be reinterpreted as a local modulation of the phase by a U(1) connection. The present article unifies these two directions. We show, first, that the Geometric Spectral Stability Principle (GSSP) does not replace the action as a fundamental structure, but acts as a superselection principle: it restricts the set of locally admissible variational configurations to those that remain globally stable. Next, we demonstrate that the incompatibility between local extrema and global coherence --- expressed by the inequality between the global minimum and the naive sum of local minima --- can be encoded as phase holonomy along cycles of the graph. This holonomy naturally leads to the promotion of the real adjacency matrix Aij to a complex Hermitian matrix Aij=Aijeiθij, in which θij represents a discrete U(1) connection. This results in a magnetic Laplacian K=D−A, whose global quadratic functional Sglobal[Ψ]=21∑(i,j)∈EAijψi−eiθijψj2 provides the minimal form compatible with gauge, self-adjointness, and spectral stability. We show that the spectral response of the eigenvalues λn to the holonomy defines the structural route for the emergence of charged states. Spin is not derived in this work; however, we argue that local frustrations incompatible with a scalar representation naturally point to a multicomponent extension of the formalism. The central result is that topological frustration should not be treated as a defect of the theory, but as the structural source that promotes the real combinatorial operator to a gauge covariant operator. Thus, mass and charge come to be seen as complementary manifestations of the same variational-spectral problem: mass associated with the spectrum of the admissible structural operator, and charge associated with the response of this spectrum to the holonomy connection.
In this work, we propose a structural reformulation of the Action--Phase program based on the interpretation of operators as unitary linear filters acting on the physical phase ϕ=S/ℏ. The central thesis is that the various effective sectors of physics can be reinterpreted as regimes of operator action on the phase and its gradient, rather than independent collections of final equations. We start from the chain already established in previous works --- action → phase → episodic compatibility → event network → structural operator --- and show that the emergent local linear dynamics admits a natural reading in terms of filter theory. In particular, the effective network operator can be seen as a transfer function, whose Cayley realization defines a local unitary all-pass filter of strictly unitary modulus. This structure provides a direct interpretation of unitarity as information conservation. We also show that the spatial variation of the local coefficients of the operator induces modulation of the phase response, leading to the definition of a local quasi-energy Ω(x,k). In the slowly varying regime, this structure reproduces an effective dynamics governed by Hamilton--Jacobi-type equations, recovering trajectories previously obtained via the eikonal approach. In this framework, gravity can be reinterpreted as the geometric optics of phase propagation in a structurally inhomogeneous network. As an operational illustration, we present a comparative numerical experiment between discrete unitary evolution (Cayley realization) and explicit integration schemes of the semidiscrete Schrödinger equation. The results show that the unitary formulation preserves the norm at machine-level precision and remains stable in intense potential regimes, while explicit schemes exhibit accumulated drift and more severe time-step restrictions. This evidences that dynamics can be implemented as the iterative application of local unitary operators. The results do not constitute a complete gravitational theory nor do they replace established continuous formulations, but they identify a unifying mathematical structure that organizes the propagation, modulation, and refraction of the phase in a non-ad hoc manner and without circular reasoning. This structure suggests an extension program based on derivations and direct numerical tests, aimed at the explicit recovery of known gravitational results from phase dynamics.
Building on the Harvey space framework—a five-dimensional complex-time manifold with coordinates (t, τ, x, y, z) whose imaginary-time coordinate τ is compactified onto a circle S¹ = U(1) gauge fibre—we give the electromagnetic sector a single narrative line: Global topological conservation in complex time → Quantum non-locality (after projecting out τ) → Gauge symmetry (localising the phase reference) → Maxwell’s equations We delimit precisely what this geometry can and cannot derive through three core pillars: 1. Quantum Non-Locality, Malus's Law, and the Born Rule In the full five-dimensional manifold, there is no action at a distance: an entangled pair shares one global holomorphic circulation closed on the τ fibre, whose conservation is a local statement. No-Signalling: The "spooky" correlation is merely the residue left on the τ = 0 Minkowski slice, where the symplectic form ω = 2c² dt ∧ dτ vanishes and H_ω = 0 strictly enforces no-signalling. Correlations: Numerically, the spinor double cover reproduces the single-particle Malus's law cos²(θ/2). The literal closure of the τ loop (acting as a forward × backward branch) yields the Born rule |·|² as a direct geometric consequence, recovering the full quantum correlations up to the Tsirelson bound 2√2 (without sliding to the PR-box value of 4). 2. Local Gauge Principle and Action Convergence The local gauge principle is not introduced as a new axiom, but arises naturally from the conjunction of the unobservability of the global U(1) phase and the locality of the per-node τ clock (no global simultaneity), forcing the introduction of a connection A_μ. Consequently, Berry holonomy, the Aharonov–Bohm phase, and the Born/ABL closed-time contour are welded into the holonomies of this single τ-fibre connection. Bypassing the two historical defects of the Kaluza–Klein ansatz, we show that the Maxwell action −(1/4)F² and the gravitational action both descend from a single node-stiffness principle—identified with the Dirichlet (harmonic) energy forced by the holomorphy of the complex-time metric: Gravity (Rate Part): Non-compact sector → |∇ω|² Gauge (Phase Part): Compact S¹ sector → (1 − cos) → (1/4)F² Any periodic, even, analytic node cost naturally yields (1/4)F² in the continuum limit, with the two-derivative truncation being a derived Planck-scale effective-field-theory consequence. 3. The Truth Boundary (Derivable vs. Non-Derivable) Finally, we draw a strict truth boundary: What is derivable: Topology dictates charge quantisation via the winding number π₁(S¹) = ℤ, and the coupling ratio g₂ = g₃ at the fibre scale emerges as a clean group-theoretic result. What is NOT derivable: The framework cannot derive the coupling strength itself; the fine-structure constant α is dictated by the dimensionful fibre modulus and is not derivable (exactly as the numerical value of G is not). This paper reproduces, and does not modify, electromagnetism, quantum mechanics, and the gauge structure of the Standard Model; it predicts no observable deviation.
The Aharonov-Bohm (AB) effect, wherein an electron acquires a phase shift in a field-free region due to electromagnetic potentials, poses a profound challenge to the ontology of quantum mechanics and gauge theories. This paper demonstrates that gauge-invariant explanations, which attribute the phase to measurable quantities, rely on nonlocal and discontinuous mechanism and fail to account for its continuous accumulation along the electron's path—a process evident in the generalized AB effect, where a time-varying magnetic flux induces a phase that builds gradually over time, as predicted by quantum mechanics. Through a new analysis spanning quantum mechanics and quantum electrodynamics, I argue that the electromagnetic potentials Aμ, fixed in the Lorenz gauge, emerge as the fundamental physical reality, offering a local, relativistically consistent account of the phase’s generation. This exclusion of gauge-invariant paradigms reverberates across gauge theories: it redefines the Higgs mechanism, favoring dynamic potentials over static invariants, and extends to general relativity, where gravitational potentials gμν may anchor spacetime’s substantival reality via a gravitational AB effect. By unraveling the AB phase’s continuous generation—locally mediated by potentials—this study not only addresses a long-standing conundrum but also bridges electromagnetic, gravitational, and Yang-Mills frameworks, offering a unified potential-centric perspective with novel implications for physics and philosophy.
Abstract Chapter 9 discusses the rigorous incorporation of electromagnetism in quantum mechanics via minimal substitution and the inclusion of spin, which leads to the Pauli equation. The gauge invariance of this formulation is inspected more closely, which leads us to understanding quantum mechanics as a gauge theory, and on to fascinating geometric effects such as the Aharonov-Bohm effect and the discovery of geometric phases such as the Berry phase and the Aharonov-Anandan phase. To give a better context to these effects, a compact introduction to the mathematical background of differential geometry is provided, introducing concepts such as differentiable manifolds, tangent and cotangent spaces, and maps between them, as well as from their elements to the real numbers, arriving at tensors and differential forms. With these tools at hand, we first define fibre bundles before formulating the previously discussed geometric phases as well as the so-called Dirac monopole in this language