Unified Gravitational, Electromagnetic, and Spin Dynamics from Fermi–Walker Geometrodynamics Constrained by the Particle’s S¹ Compton Clock Fiber

1. Introduction

Gauge interactions and spacetime geometry constitute the two fundamental pillars of modern theoretical physics, underlying respectively the Quantum Field Theory (QFT) and General Relativity. Despite their different nature, both frameworks rely on local symmetry principles and connection structures, suggesting the possibility of a deeper unifying mechanism. However, while gravity has a well-established geometrodynamical formulation within GR, gauge interactions are generally introduced by postulating internal local symmetries and the corresponding gauge connections, rather than derived from spacetime geometry, [1].

We identify a concrete realization of this idea within Elementary Cycles Theory (ECT), [2,3,4,5,6,7,8,9,10,11,12,13,14,15], where each particle of mass m carries an intrinsic proper-time Compton recurrence $s\in S_{T_C}^1$, implemented through covariant Periodic Boundary Conditions (PBCs) that constrain the field dynamics consistently with the variational principle. We refer to the resulting variational Hamilton–Jacobi (HJ) phase dynamics as “Phase Harmony” (PH). In the present work ECT is used in a deliberately restricted sense to select a single on-shell field mode and track the covariant transformation of its natural spacetime recurrence. We show that Local Lorentz Transformations (LLTs) of observer-adapted Fermi-Walker (FW) frames, [16,17], compatibly with PH, induce an effective Abelian holonomy, which in turn reproduces the Electromagnetic (EM) connection, the Lorentz force as a Coriolis-like internal effect, and the Maxwell dynamics. In this sense, the present framework provides a direct link between spacetime geometry and gauge interaction, namely geometro-EM, placing EM on a footing conceptually closer to gravitational geometrodynamics.

Since Newton’s Principia, interactions are introduced in physics as deviations from uniform motion, namely as accelerations of bodies. There is, however, a dual and equally classical description rooted in the HJ optomechanical dualism (HJ), [18,19,20]. Any Hamiltonian system admits an equivalent undulatory description in which the energy and the momentum are proportional to a frequency and wave-number, hence to the inverse of an instantaneous time periodicity and wavelength, T and $\lambda$, respectively. Its most familiar application is the undulatory mechanics underlying QM, and thus QFT, where the proportionality constant ℏ is the reduced Planck constant (we adopt $c=\hslash =1$). For a free scalar particle, the four-momentum $p_{\mu }$ (tangent covector) and the instantaneous spacetime periodicity ${\tau }^{\mu }=\{T,\lambda \}$ (tangent contravariant vector), form a relativistic PH invariant $p_{\mu }{\tau }^{\mu }=2\pi$; e.g.as under Lorentz transformations $\Lambda$, where $p_{\mu }\to p_{\mu }^{\prime }={\Lambda }_{\mu }^{\nu }{p}_{\nu }$ and ${\tau }^{\mu }\to {\tau }^{\prime \mu }={\Lambda }_{\nu }^{\mu }{\tau }^{\nu }$, so that $p_{\mu }{\tau }^{\mu }=p_{\mu }^{\prime }{\tau }^{\prime \mu }=2\pi$. Hence, in the rest frame, $p_{\mu }\to m={\Lambda }_0^{\nu }{p}_{\nu }$ and ${\tau }^{\mu }\to T_C={\Lambda }_{\nu }^0{\tau }^{\nu }$, the mass m naturally fixes an intrinsic recurrence of Compton period $T_C=2\pi /m$ in the proper-time s: $p_{\mu }{\tau }^{\mu }=m{T}_C=2\pi$. This is known as particle’s Compton (de Broglie) internal clock1. Hence, every elementary particle can be regarded as an elementary clock2 3, with (ultrafast) $S^1$-valued phase ${\theta }_{PH}$ advancing along its worldline. ECT promotes this proper-time recurrence to the cyclic condition $s\sim s+T_C$, implemented in terms of PBCs ($s\in S_{T_C}^1$), whose covariant transformations constrain particle dynamics, i.e.its PH. See also the relational interpretation of general physical dynamics suggested by this hypothesis [2] — a further justification is in Rovelli’s statistically motivated observation that generally covariant systems should admit “internal times”, [3,24].

Assuming global $U\left(1\right)$ symmetry, HJ prescribes that any local symplectomorphism (e.g., canonical transformation) makes the local changes $p_{\mu }\to p_{\mu }^{\prime }\left(x\right)$ and ${\tau }^{\mu }\to {\tau }^{\prime \mu }\left(x\right)$ co-modulate, [4]. In fact, for a scalar charged field or Dirac field, the scalar phase one-form transforms as $d{\theta }_{PH}=p_{\mu }d{x}^{\mu }\to {\alpha }_{PH}^{\prime }:=p_{\mu }^{\prime }\left(x\right)d{x}^{\mu }$. The associated symplectic two-form is ${\omega }_{PH}^{\prime }=d{\alpha }_{PH}^{\prime }=d{p}_{\mu }^{\prime }\wedge d{x}^{\mu }$, in agreement with Darboux’s theorem. Stokes’ lemma then implies that the phase accumulated after any fundamental spacetime recurrence interval $T^{\prime \mu }\left(x\right)$, i.e.after a local closed spacetime cycle ${\mathcal{V}}_x^{\prime }\left(T^{\prime \mu }\left(x\right)\right)$ based at the point x, is invariant along the symplectic flow:

$${\mathcal{S}}_{PH}\left[x\right]={\oint }_{{\mathcal{V}}_x^{\prime }\left(T^{\prime \mu }\left(x\right)\right)}{\alpha }_{PH}^{\prime }={\oint }_{{\mathcal{V}}_x^{\prime }\left(T^{\prime \mu }\left(x\right)\right)}{p}_{\mu }^{\prime }\left(y\right)d{y}^{\mu }=p_{\mu }^{\prime }\left(x\right){\tau }^{\prime \mu }\left(x\right)=p_{\mu }{\tau }^{\mu }=2\pi .$$

(1)

This expression summarizes the concept of PH. The instantaneous spacetime periodicity ${\tau }^{\mu }\left(x\right)$ — a tangent vector in general different from the spacetime recurrence interval ${\tau }^{\mu }\left(x\right)\ne T^{\mu }\left(x\right)$ — evolves along the symplectic flow, precisely mirroring the local deformation of the manifold chart, and thereby guarantees causality and locality in the undulatory description.

For example, in the linear approximation, gravitational interaction can be consistently derived in terms of PH — so that the gravitational dynamics can be related to Einstein’s equation by requiring general invariance [25]. In a weak gravitational field ${\rm Y}\left(x\right)$, the local energy shift $p_0^{\prime }\left(x\right)\simeq [1+{\rm Y}\left(x\right)]p_0$ implies gravitational redshift ${\tau }^{\prime 0}\left(x\right)\simeq [1-{\rm Y}\left(x\right)]{\tau }^0$ through PH. Likewise, the spatial contractions adjust with the local momentum so that $p_{\mu }^{\prime }\left(x\right){\tau }^{\prime \mu }\left(x\right)=2\pi$ is preserved along the motion. The curved spacetime manifold (e.g.Schwarzschild) directly encodes the local spacetime modulations of ruler and clocks as dual geometrodynamical manifestation of gravitational interaction.

The metric tensor, however, doesn’t encode all possible local modulations of ${\tau }^{\mu }\left(x\right)$. For instance, LLTs are position dependent isometries, yet they locally transform ${\tau }^{\prime \mu }\left(x\right)$ and $p_{\mu }^{\prime }\left(x\right)$, producing Fermi-Walker (FW) transport and precessions [16,26] naturally described by the Ehresmann connection on the tangent space, [27,28], whose holonomy can generate nontrivial covariant contributions to the phases.

ECT introduces a specific formalism to investigate PH under LLTs. In fact, ordinary point-particle Hamiltonian formalism does not explicitly track the recurrence structure of the associated matter wave. On the other hand, in ordinary field theory — where gauge invariance is postulated rather than inferred from geometry [1] — a field is a superposition over all momentum modes ${\varphi }_p$ — an integral over all global frames. LLTs thus merely reshuffle field modes, hiding the geometrodynamical origin of gauge interactions, as we will see.

Within the restricted use of ECT adopted here, sec.(Section 2), the on-shell field mode ${\varphi }_{\mathbf{p}}$ selected by PBCs is followed under LLTs through the covariant transformation of its recurrence cell, consistently with the variational principle: ${\varphi }_{\mathbf{p}}\stackrel{LLT}{\to }{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }$. The PH-admitted transformations are described by observer-adapted FW frames whose transport generator ${\Omega }^{EM}$ co-modulates ${\tau }^{\mu }\stackrel{LLT}{\to }{\tau }^{\prime \mu }\left(x\right)$ and $p_{\mu }\stackrel{LLT}{\to }{p}_{\mu }^{\prime }\left(x\right)$, forcing the compensating internal transformation of the field mode required by PH, ${\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }\left(x\right)\ne {\varphi }_{\mathbf{p}}\left(x\right)$,5]. The generator ${\Omega }^{EM}$ is defined modulo integrable contributions ${\Omega }^{gauge}$, which generates effective compact$U\left(1\right)$ gauge invariance, and an extra vorticity contribution $\Delta {\Omega }^{SR}$ relevant in the spin sector and associated with anomalous magnetic-moment effects. In this way, the covariant PBCs are here essentially used to reinforce the natural undulatory mechanics of the field modes. However, in this paper we keep only the fundamental mode allowed by the PBCs — while the full harmonic expansion may be relevant in extensions to QFT, second quantization, and other aspects of QM, as developed in previous works, [2,3,4,5,6,7,8,9,10,11,12,13,14,15].

To study the geometrodynamical origin of gauge interactions, we focus here on $U\left(1\right)$ holonomies, which amounts to considering fields with global $U\left(1\right)$ symmetry and thus Abelian scalar phase structure, such as charged scalars and Dirac fields. Indeed, after projection onto the scalar phase, the LLT-induced modulations of ${\tau }^{\prime \mu }\left(x\right)$, whose generators are in general non-compact and non-Abelian, describe an effective compact Abelian holonomy as precession of the internal clock, [29,30,31]. Then, the internal transformation identifies a $U\left(1\right)$ Abelian connection, identifying the standard EM gauge field $A_{\mu }\left(x\right)$. EM is therefore the bookkeeping connection that compensates the observer-adapted FW precession of the particle’s internal clock, whose $U\left(1\right)$ holonomy reproduces the Lorentz force (Coriolis-like), in analogy with gravito-EM, [32,33,34]. This study sets the basis for an extension to a geometrodynamical formulation of the electroweak $SU{\left(2\right)}_L\times U{\left(1\right)}_Y$ gauge sector and related gauge symmetry breaking, which will be the subject of a forthcoming paper [35].

In sec.(Section 3), we develop the geometrodynamical method based on PBCs encoding the “particle’s internal clock” $s\in S_{T_C}^1$. Then we derive the Lorentz force, gauge invariance, and Maxwell equations in the dynamical and Lagrangian descriptions, sec.(Section 4) and sec.(Section 5), respectively. The generalization to curved spacetime, where EM emerges alongside gravitational interaction, is given in sec.(Section 6). In sec.(Section 7), we provide a four-dimensional (4D) interpretation of the Kaluza–Klein (KK) mechanism, [12,13],showing that the Maxwell kinetic term emerges directly from spacetime geometry. Sec.(Section 8) extends the construction to Dirac fermions, where we will find that geometro-EM and spin transport originate from the same underlying geometrodynamics, yielding a generalized account of Bargmann–Michel–Telegdi (BMT) spin kinematics and dynamics. This also suggests a possible geometrodynamical contribution to the anomalous magnetic moment $g-2$. The possible interpretation of the higher harmonic modes, and their relation to second quantization, canonical quantization, scalar QED, the Feynman path integral, holography, and the AdS/CFT correspondence, has been developed in previous works [2,3,4,5,6,7,8,9,10,11,12,13,14,15] and faithfully recalled in sec.(Section 9).

From a conceptual viewpoint, the present framework is close in spirit to several geometrodynamical approaches to unification, including Rainich–Misner–Wheeler ideas, gauge-theoretic formulations of gravity, Hamiltonian/cotangent-bundle constructions, and double-copy inspired correspondences [36,37,38,39,40,41,44,45]. The crucial difference is that here the starting point is the intrinsic $S^1$ proper-time recurrence associated with the particle, e.g. [2]. PH then projects LLT onto a $S_{T_C}^1$ clock fiber, [46,47] inducing an effective Abelian $U\left(1\right)$ holonomy whose Ricci geometry yields the Maxwell kinetic term, while ordinary gravitational geometrodynamics remains encoded by Local Diffeomorphisms (LD) and the metric.

2. Periodic Boundary Conditions and Phase Harmony for Field Modes

Let us consider a scalar charged particle of mass m, free of any interaction, in a classical-relativistic Klein-Gordon (KG) field description — flat spacetime ${\eta }_{\mu \nu }=\mathrm{diag}(+,-,-,-)$. We select, in every point x, the single mode ${\varphi }_{\mathbf{p}}\left(x\right)$ of four-momentum $p_{\mu }$ by imposing related PBCs, [5],

$$\begin{array}{|c|}\hline {\varphi }_{\mathbf{p}}\left(x^{\mu }\right)={\varphi }_{\mathbf{p}}(x^{\mu }+T^{\mu })\\ \hline\end{array},$$

(2)

such that $p_{\mu }{T}^{\mu }=m{T}_C=2\pi$. Thus ${\varphi }_{\mathbf{p}}\left(x\right)$ is fundamental solution of the free KG action

$${\mathcal{S}}_{Free}[x,\mathbf{p};T_C]={\oint }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y\mathcal{L}({\partial }_{\mu }{\varphi }_{\mathbf{p}},{\varphi }_{\mathbf{p}};m),$$

(3)

where ${\oint }_{{\mathcal{V}}_x\left(T^{\mu }\right)}$ represents the integration with implicit PBCs, eq.(2), over the local fundamental recurrence cell anchored at the base point x,

$${\mathcal{V}}_x\left(T^{\mu }\right):={\displaystyle \prod _{\mu =0}^3}[x^{\mu },x^{\mu }+T^{\mu }].$$

(4)

The free KG Lagrangian density, of global $U\left(1\right)$ symmetry, is indeed

$$\mathcal{L}({\partial }_{\mu }{\varphi }_{\mathbf{p}},{\varphi }_{\mathbf{p}};m)={\partial }_{\mu }{\varphi }_{\mathbf{p}}^{†}{\partial }^{\mu }{\varphi }_{\mathbf{p}}-m^2{\varphi }_{\mathbf{p}}^{†}{\varphi }_{\mathbf{p}}.$$

(5)

Throughout all this work we however retain only the fundamental mode ${\varphi }_{\mathbf{p}}$ of the harmonic series resulting from the PBCs, eq.(2). This means that we investigate the geometric origin of gauge interaction at the level of the fundamental mode, before applying any explicit second-quantization procedure. The physical interpretation of the higher harmonics and their relationship with second quantization have been developed in [3,4], the generalization to scalar QED of our framework is given in [5,12,13] and briefly reviewed in sec.(Section 9), while general quantum aspects are investigated in all previous works [2,3,4,5,6,7,8,9,10,11,12,13,14,15].

In the free case the instantaneous spacetime periodicity ${\tau }^{\mu }$ is constant at every base point x of the system evolution, and therefore equal to the spacetime recurrence $T^{\mu }={\tau }^{\mu }$. Notice that the mass and the spacetime recurrences are determined by the proper-time recurrence through the Compton relation $T_C=2\pi /m$ and its covariant transformations, respectively.

The fundamental mode, solution of ${\mathcal{S}}_{Free}$, selected by the PBCs, eq.(2), is thus:

$${\varphi }_{\mathbf{p}}\left(x\right)={\varphi }_I{e}^{-i{p}_{\mu }{x}^{\mu }},$$

(6)

where ${\varphi }_I$ is a normalization factor. Eventually, the ordinary KG field ${\varphi }_{KG}\left(x\right)$ is reconstructed by integrating over all the possible fundamental on-shell modes ${\varphi }_{\mathbf{p}}\left(x\right)$, i.e.by a superposition over the spatial momentum $\mathbf{p}$, eq.(7), weighted by the classical coefficient $a\left(\mathbf{p}\right)$,

$${\varphi }_{KG}\left(x\right)=\int {\displaystyle \frac{d^3\mathbf{p}}{{\left(2\pi \right)}^3}}a\left(\mathbf{p}\right){\varphi }_{\mathbf{p}}\left(x\right).$$

(7)

The formalism above should be understood locally: the PBCs, eq.(2), identify the field mode at opposite points $x^{\mu }\sim x^{\mu }+T^{\mu }$ of the elementary recurrence cell, ${\mathcal{V}}_x\left(T^{\mu }\right)=\mathcal{V}\left(T^{\mu }\right)$ for the free case, in the fundamental domain based at x. Thus, the compact structure introduced here should not be interpreted as a global compactification of the macroscopic spacetime manifold. It should be viewed instead as the intrinsic cyclic support of the elementary on-shell mode, i.e.an effective local $S_{T_C}^1$ recurrence fiber, named here $S_{T_C}^1$ clock fiber, attached to each base point x. The ordinary non-compact spacetime chart remains the unfolded description of the propagation of the mode. Nevertheless, this intrinsically cyclic internal support provides a self consistent relational description of systems dynamics as illustrated in [2].

2.1. Phase Harmony Consistency and the Variational Principle

A rigorous way to introduce PH is through the variational principle applied to the action eq.(3). Its bulk variation gives the ordinary KG equation, while the boundary contribution is

$$\begin{array}{ccc}\hfill \delta {\mathcal{S}}_{Free}^{\partial }& =& {\displaystyle \frac{1}{2}}{\int }_{\partial {\mathcal{V}}_x\left(T^{\mu }\right)}d{\Sigma }_{\mu }\left[{\partial }^{\mu }{\varphi }_{\mathbf{p}}^{†}\delta {\varphi }_{\mathbf{p}}+{\partial }^{\mu }{\varphi }_{\mathbf{p}}\delta {\varphi }_{\mathbf{p}}^{†}\right]\hfill \\ & =& {\displaystyle \frac{1}{2}}{\left[{\partial }^{\mu }{\varphi }_{\mathbf{p}}^{†}\delta {\varphi }_{\mathbf{p}}+{\partial }^{\mu }{\varphi }_{\mathbf{p}}\delta {\varphi }_{\mathbf{p}}^{†}\right]}_{x^{\mu }}^{x^{\mu }+T^{\mu }}\equiv 0.\hfill \end{array}$$

(8)

This vanishes because of the PBCs, eq.(2), which identify the field and its admissible variations on opposite faces of the recurrence cell ${\mathcal{V}}_x\left(T^{\mu }\right)$. PBCs define a consistent variational boundary sector for the KG action, thus preserving locality and causality in the resulting theory. In fact, it is known from string and extra-dimensional (XD) theories [48,49,50,51] that PBCs, as well as combinations of Dirichlet and Neumann boundary conditions, are admissible choices for the stationary action principle.

In general, we refer to PH as the variational principle consistency of field mode phase dynamics ${\mathcal{S}}_{PH}$ with the covariant PBCs imposed on its local recurrence cell. PH is thus preserved under canonical transformations, including local transformations of spacetime, through the covariant transport of the phase one-form and Stokes’ theorem [3,4,5,6,7,8,9,10,11,12,13,14]. The scalar PH is summarized by eq.(1), reducing to $p_{\mu }{T}^{\mu }=2\pi$ in the free case. In sec.(Section 4.6) we will discuss the possible generalization to non-Abelian gauge theories and electroweak sector.

3. Spacetime Geometrodynamics of Accelerated Frames in the Flat Case

Now we switch on interaction, namely geometro-EM, by introducing in each base point x a generic infinitesimal LLT $x\to x^{\prime }\left(x\right)$,5], with related tetrad

$$\begin{array}{|c|}\hline e_{\mu }^a\left(x\right)={\displaystyle \frac{\partial x^{\prime a}\left(x\right)}{\partial x^{\mu }}}\\ \hline\end{array}.$$

(9)

This LLT is a position dependent isometry preserving the flat spacetime metric

$${\eta }_{\mu \nu }\stackrel{LLT}{\to }{\eta }_{\mu \nu }^{\prime }\left(x\right)=e_{\mu }^a\left(x\right)e_{\nu }^b\left(x\right){\eta }_{ab}={\eta }_{\mu \nu },$$

(10)

so that no gravitational interaction is introduced — it will be included in sec.(Section 6).

3.1. Congruence of Adapted Accelerated Frames

The resulting flat yet locally twisted spacetime identifies a congruence of particle’s worldlines ${\gamma }_x$, labeled by the base point x, with unit time-like field of velocity $u^{\prime \mu }=e_0^{\mu }$. In particular we assume that the full tetrad $e_{\mu }^a$ is transported along each worldline with related, not vanishing field of acceleration $a^{\prime \mu }=u^{\prime \rho }{\partial }_{\rho }{u}^{\prime \mu }$, where $u^{\prime }·u^{\prime }=1$ and $u^{\prime }·a^{\prime }=0$. To this congruence it is also naturally associated the vorticity ${\omega }_{ab}^V$ defined as antisymmetric transverse gradient through the projector $h_{ab}$,16]:

$${\omega }_{ab}^{\prime V}:={\displaystyle \frac{1}{2}}{h}_a^c{h}_b^d({\partial }_c{u}_d^{\prime }-{\partial }_d{u}_c^{\prime }),h_{ab}={\eta }_{ab}-u_a^{\prime }{u}_b^{\prime }.$$

(11)

The relativistic LLT transport law of this congruence is, [26,52]:

$$\begin{array}{|c|}\hline D_s{e^{\mu }}_a=-{{{\Omega }^{PH}}_a}^b{e^{\mu }}_b\\ \hline\end{array},D_s:=\left(u^{\prime \rho }{\partial }_{\rho }\right),$$

(12)

where ${{\Omega }^{PH}}_{ab}$ is the generator of the LLT flow. Then, PH selects subclasses of possible physical worldlines congruences, constraining the otherwise general field of velocity $u_a^{\prime }\left(x\right)$, and thus the Lorentz flow of the four-momentum $p_a^{\prime }\left(x\right)=m{u}_a^{\prime }\left(x\right)$ according to variational principle applied to the action with PBCs, eq.(19). In fact, the instantaneous four-vectors transport law under the LLT flow, ${\tau }^{\mu }\stackrel{LLT}{\to }{\tau }^{\prime a}={e^a}_{\mu }{\tau }^{\mu }$ and $p_{\mu }\stackrel{LLT}{\to }{p}_a^{\prime }={e_a}^{\mu }{p}_{\mu }$, is given by

$$D_s{{\tau }^{\prime }}^a=-{{\Omega }^{PH}}_b^a{{\tau }^{\prime }}^b,D_s{p}_a^{\prime }=-{{\Omega }^{PH}}_a^b{p}_b^{\prime },$$

(13)

so that PBCs/PH, eq.(1), implies antisymmetric generators, $D_s\left({{\tau }^{\prime }}^a{p}_a^{\prime }\right)\equiv 0$, for the flow:

$$\begin{array}{|c|}\hline {{\tau }^{\prime }}^a\left(x\right)p_a^{\prime }\left(x\right)=2\pi ⟹{{\Omega }^{PH}}_{ab}=-{{\Omega }^{PH}}_{ba}\\ \hline\end{array}.$$

(14)

We therefore distinguish three antisymmetric contributions to the PH-admitted local Lorentz generator ${\Omega }^{PH}$ whose physical implications will be investigated separately: i) the so called observer-adapted or geometro-EM generator ${\Omega }^{EM}={\Omega }^{FW}+{\Omega }^V$, sec.(Section 4), [16,17,32]; ii) the integrable contribution ${\Omega }^{gauge}$, sec.(Section 4.1); iii) the extra rotational contribution $\Delta {\Omega }^{SR}$, sec.(Section 8):

$$\begin{array}{|c|}\hline {\Omega }^{PH}={\Omega }^{EM}+{\Omega }^{Gauge}+\Delta {\Omega }^{SR}\\ \hline\end{array}.$$

(15)

The first antisymmetric contribution is the geometro-EM generator, [16,17,32,33]:

$$\begin{array}{|c|}\hline {{\Omega }^{EM}}_{ab}=\underset{{\Omega }^{FW}}{\underbrace{(u_a^{\prime }{a}_b^{\prime }-a_a^{\prime }{u}_b^{\prime })}}-\underset{{\Omega }^V}{\underbrace{2{{\omega }^V}_{ab}}}=-({\partial }_a{u}_b^{\prime }-{\partial }_b{u}_a^{\prime }).\\ \hline\end{array}$$

(16)

These form the subclass of LLT congruences selected by PH/PBC consistency, whose scalarized projection reproduces EM, where ${\Omega }^{FW}$ is the FW sector (local pure boost) and ${\Omega }^V$ is the natural vorticity of the congruence. They indeed originate the electric and magnetic fields, ${\Omega }^{FW}⇝\mathbf{E}$ and ${\Omega }^V⇝\mathbf{B}$. The geometro-EM generator ${\Omega }^{EM}={\Omega }^{FW}+{\Omega }^V$ is in general a non-integrable contribution to the LLT flow, because of its vorticity.

Compatibly with PH, we also assume integrable contributions ${\Omega }^{Gauge}$, namely pure endpoint Lorentz-frame transformations. Their scalarized image results in an exact $U\left(1\right)$ phase, as we will see. Rigid Killing flows provide a simple spacetime realization of this sector.

A further contribution, compatible with the variational principle applied on the boundary of eq.(3), is given by extra spatial rotations. The only PH-admitted rotational term4 is an extra vorticity contribution $\Delta {\Omega }^{SR}=a_e{\Omega }^V$ weighted by the coefficient $a_e$. This is a misalignment of the vorticity sector with respect to the natural vorticity of the congruence in the geometro-EM sector, ${\Omega }^V$. Since ${{\Omega }^V}_b^a{u}_b^{\prime }=0$, both vorticity contributions have no effect on resulting EM connection, which are obtained, as we will see, through projection on $u^{\prime }$. Only the natural vorticity ${\Omega }^V$ of the geometro-EM sector is indirectly recovered in the gauge dynamics, e.g., whenever the gradient of the congruence is involved (as in the field strength), according to eq.(16). However, in the spin sector of Dirac field modes, the full rotational contribution ${\Omega }^V+\Delta {\Omega }^{SR}=(1+a_e){\Omega }^V$ survives, giving an extra magnetic contribution $a_e$ with respect to the gauge sector, which results in an anomalous magnetic moment described by a parameter $a_e$.

Thus, the most physically relevant contribution to geometro-EM comes from the FW transport, ${\Omega }^{FW}$, so that this description in terms of accelerated frames allows one to describe the local modulation of the particle’s internal clock ${\tau }^{\mu }\left(s\right)$ in precise analogy with the relativistic gyroscope transport [53,54].

3.2. Induced Internal Transformation from Observer-Adapted Frames

We first investigate the effect of the geometro-EM generator ${\Omega }^{EM}$, eq.(16). Solving eq.(12) along the worldline ${\gamma }_x:x_0\to x$ anchored at base point x, and treating the LLT as infinitesimal with expansion parameter q in terms of the accumulated transport

$$q{{{\Theta }^{EM}}^a}_b\left(x\right):={\int }_{{\gamma }_x}(u^{\prime }·dy){{{\Omega }^{EM}}^a}_b\left(y\right).$$

(17)

In fact, at first order in q, the transported tetrad is

$$\begin{array}{c}\hfill {e^{\mu }}_a\left(x\right)=\left[{{\delta }^{\mu }}_{\nu }-q{{{\Theta }^{EM}}^{\mu }}_{\nu }\left(x\right)+\mathcal{O}\left(q^2\right)\right]{e^{\nu }}_a\left(x_0\right)={{\delta }^{\mu }}_a-q{{{\Theta }^{EM}}^{\mu }}_a\left(x\right)+\mathcal{O}\left(q^2\right),\end{array}$$

(18)

where we have aligned the local frame with the effective coordinates at $x_0$, i.e.${e^{\nu }}_a\left(x_0\right)={{\delta }^{\nu }}_a$.

In the free action eq.(3), the geometro-EM LLT induces an infinitesimal rotation of the boundary $T^{\mu }\stackrel{LLT}{\to }{T}^{\prime \mu }\left(x\right)$ while the Lagrangian density remains invariant and locally flat $\sqrt{-g}\simeq 1$ in the neighborhood of x (within the limited integration region of the action). The transformed action has LLT indeed recurrence cell $\mathcal{V}\left(T^{\mu }\right)\stackrel{LLT}{\to }{\mathcal{V}}_x^{\prime }\left({T^{\prime }}^{\mu }\left(x\right)\right)$,5]:

$$\begin{array}{|c|}\hline {\mathcal{S}}_{Free}\stackrel{LLT}{\to }{\mathcal{S}}_{EM}\\ \hline\end{array}={\oint }_{{\mathcal{V}}_x^{\prime }\left({T^{\prime }}^{\mu }\left(x\right)\right)}{d}^{4}y\mathcal{L}({\partial }_{\mu }{\varphi }_{p^{\prime }}^{\prime },{\varphi }_{p^{\prime }}^{\prime };m).$$

(19)

The convenient four-vector for implementing PH is however the instantaneous (tangent) spacetime periodicity, transforming as

$$\begin{array}{c}\hfill {\tau }^{\mu }=T^{\mu }\stackrel{LLT}{\to }{\tau }^{\prime a}\left(x\right)={e^a}_{\mu }\left(x\right)T^{\mu }=\left({{\delta }^a}_{\mu }+q{{{\Theta }^{EM}}^a}_{\mu }\left(x\right)\right)T^{\mu }+\mathcal{O}\left(q^2\right),\end{array}$$

(20)

rather than the transformed recurrence interval $T^{\prime \mu }\left(x\right)$. Its modulation tracks the local transport of the particle’s internal clock carried by the tetrad congruence, in the relativistic analogy of an internal gyroscope/clock precession, [29,30,55].

As in ECT, because of the covariantly transformed PBCs, LLT induces an effective internal transformation in the field mode, solution of eq.(19),

$${\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }\left(x\right)\ne {\varphi }_{\mathbf{p}}\left(x\right),$$

(21)

in contrast with the ordinary field description, where LLT of the flat metric do not induce any internal transformation of the field itself, owing to the absence of boundaries in the ordinary KG action. In fact, in our formalism each field mode is a physical standing wave whose instantaneous four-periodicity ${\tau }^{\prime \mu }\left(x\right)$ is directly determined by the local PBCs of the transformed action eq.(19). Due to the local rotation of boundary $T^{\mu }\to T^{\prime \mu }\left(x\right)$, the transformed field mode solution, allowed by the transformed PBCs, is

$${\varphi }_{\mathbf{p}}\left(x\right)\stackrel{LLT}{\to }{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }\left(x\right)={\varphi }_I{e}^{-i{\int }_{\Gamma }{p}^{\prime }\left(y\right)·dy},$$

(22)

where ${\varphi }_{\mathbf{p}}\left(x_i\right)={\varphi }_I$, up to an irrelevant phase, see eq.(6), and $\Gamma :x_i\to x$ is an arbitrary path used to measure the field evolution, or the Wilson line, as we will see. The transformed field solution is a locally modulated spacetime wave of instantaneous four-periodicity ${\tau }^{\prime a}\left(x\right)$:

$$i{\partial }_a{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }=p_a^{\prime }\left(x\right){\varphi }_{{\mathbf{p}}^{\prime }}^{\prime },$$

(23)

where the four-momentum $p_a^{\prime }\left(x\right)$, characterizing our geometro-EM interaction scheme, is locally determined by PH, eq.(1), preserving the transformed PBCs:

$$p_{\mu }{\tau }^{\mu }=p_a^{\prime }\left(x\right){\tau }^{\prime a}\left(x\right)=2\pi .$$

(24)

Thus, from eq.(18), the subclass of possible physical world-line congruences consistent with the variational principle of eq.(19), restricted to the geometro-EM generator ${\Omega }^{EM}$, is

$$\begin{array}{c}\hfill \begin{array}{|c|}\hline p_{\mu }\left(x\right)\stackrel{LLT}{\to }{p}_a^{\prime }\left(x\right)={e_a}^{\mu }\left(x\right)p_{\mu }=p_a-q{A}_a\left(x\right)\\ \hline\end{array}.\end{array}$$

(25)

This suggests introducing the effective Abelian potential as the scalarized accumulated LLT transport along the ${\Omega }^{EM}$ flow

$$\begin{array}{c}\hfill \begin{array}{|c|}\hline q{A}_a\left(x\right):=q{p}_b{{{\Theta }^{EM}}^b}_a\left(x\right)+\mathcal{O}\left(q\right)=p_{\mu }{\int }_{{\gamma }_X}(u^{\prime }·dy){{{\Omega }^{EM}}^{\mu }}_a\left(y\right)+\mathcal{O}\left(q\right)\\ \hline\end{array},\end{array}$$

(26)

so that the minimal substitution eq.(25) follows directly at first order, modulo ${\Omega }^{gauge}$ and $\Delta {\Omega }^{SR}$ contributions, as we will see. In the rest of the paper we will check whether this effective potential $A_a$ shares the properties of the ordinary EM connection. Note that, in general, the spacetime components of an EM connection, as those generated by eq.(26), need not themselves be compact. Compactness is a property of the associated gauge phase (or holonomy), not of the local potential coefficients, see effective gauge invariance in sec.(Section 4.4).

The transformation of the field mode eq.(22) can thus be written as

$${\varphi }_{\mathbf{p}}\left(x\right)\stackrel{LLT}{\to }{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }\left(x\right)=V\left(x\right){\varphi }_{\mathbf{p}}\left(x\right),$$

(27)

which now includes a path-dependent Wilson line as effective internal transformation, [5],5

$$\begin{array}{|c|}\hline V\left(x\right)=e^{iq{\int }_{\Gamma }{A}_a\left(y\right)d{y}^a}\\ \hline\end{array}.$$

(28)

It is interesting to notice that this formalism provides a kind of holographic description: interaction is (dually) encoded in the local PH-admitted transformation flow of the boundary $T^{\prime \mu }\left(x\right)$ of the theory, [5].

4. Electromagnetic Dynamics from Flat Spacetime Geometrodynamics

We investigate the dynamical properties of the effective connection $A_a\left(x\right)$ induced by the LLT transport law, eq.(26). This establishes a fundamental link between spacetime LLT geometrodynamics and the effective EM gauge interaction emerging from it. We derive the field strength $F_{ab}$ and the Lorentz force in terms of the adapted LLT geometrodynamics generated by the non-integrable contribution ${\Omega }^{EM}$, which is the only term giving a physical contribution to $F_{ab}$, as ${\Omega }^{gauge}$ results in effective compact$U\left(1\right)$ gauge orbits and $\Delta {\Omega }^{SR}$ is invisible to the gauge sector.

4.1. Generalized Field Strength from Geometrodynamics

Let us consider the non-integrable contribution ${\Omega }^{EM}$. From the eq.(16) we find

$$p_b{{\Omega }^{EM}}_a^b=m{u}_b^{\prime }(a^{\prime b}{u}_a^{\prime }-u^{\prime b}{a}_a^{\prime })+\mathcal{O}\left(q\right)=-m{a}_a^{\prime }+\mathcal{O}\left(q\right),$$

(29)

where, in general, $p_a=m{u}_a=m{u}_a^{\prime }+\mathcal{O}\left(q\right)$. Using the definition of accumulated transport and fixing the initial frame at $x_0$, we obtain

$$\begin{array}{ccc}\hfill q{A}_a{\left(x\right)|}_{{\gamma }_x}& =& p_b^{\prime }{\int }_{{\gamma }_x}ds{{\Omega }^{EM}}_a^b+\mathcal{O}\left(q\right)=p_b^{\prime }{\int }_{{\gamma }_x}ds{{\Omega }^{FW}}_a^b+\mathcal{O}\left(q\right)\hfill \\ & =& -m{\int }_{{\gamma }_x}ds{a}_a^{\prime }+\mathcal{O}\left(q\right)=-m\left[{\overline{u}}_a^{\prime }\left(x\right)-{\overline{u}}_a^{\prime }\left(x_0\right)\right]+\mathcal{O}\left(q\right).\hfill \end{array}$$

(30)

Here $A_a{\left(x\right)|}_{{\gamma }_x}$ denotes the connection evaluated along ${\gamma }_x$, while the bar denotes the pullback of the congruence fields to that worldline, e.g.${\overline{u}}_a^{\prime }\left(s\right)=u_a^{\prime }\left({\gamma }_x\left(s\right)\right)$. The endpoint form of the above equation is therefore a worldline identity and does not imply that the induced connection is integrable.

When the induced one-form $A=A_{a}d{y}^a$ is non-integrable, such as for the contribution ${\Omega }^{EM}$, the resulting geometro-EM field has generally non vanishing Lorentz curvature. This is possible because we are not describing a single worldline. In fact, the physical curvature is obtained by comparing neighboring worldlines of the congruence, namely by taking the antisymmetric gradient of the velocity field. From eq.(30) we find:

$$\begin{array}{ccc}\hfill F_{ab}& :=& {\partial }_a{A}_b-{\partial }_b{A}_a=-{\displaystyle \frac{m}{q}}({\partial }_a{u}_b^{\prime }-{\partial }_b{u}_a^{\prime })+\mathcal{O}\left(1\right)\hfill \\ & =& -{\displaystyle \frac{m}{q}}\left(u_a^{\prime }{a}_b^{\prime }-u_b^{\prime }{a}_a^{\prime }-2{\omega }_{ab}^V\right)+\mathcal{O}\left(1\right)\ne 0.\hfill \end{array}$$

(31)

By comparing with eq.(16), we finally identify the geometro-EM generator ${\Omega }^{EM}$ as the effective EM field strength resulting from PH admitted LLTs, rescaled by a mass-to-charge factor, obtaining the fundamental constitutive identity of geometro-EM

$$\begin{array}{|c|}\hline F_{ab}\equiv -{\displaystyle \frac{m}{q}}{{\Omega }^{EM}}_{ab}+\mathcal{O}\left(1\right)\ne 0\\ \hline\end{array},\mathrm{Geometro}-\mathrm{EM}\mathrm{fundamental}\mathrm{identity}.$$

(32)

up to integrable transport contributions ${\Omega }^{\mathrm{gauge}}$, which result in a gauge transformation $F_{ab}\to F_{ab}^{\prime \prime }=F_{ab}$, see eq.(40), and up to extra local rotations $\Delta {\Omega }^{\mathrm{SR}}$, whose transverse projection on $u_a^{\prime }$ is not restored by the congruence gradient in eq.(31) and therefore has no physical effect on the scalar gauge sector. Thus, the direct scalar projection defining $A_a$, eq.(26), selects the FW sector as main physical contribution, while the natural vorticity generator ${\Omega }^V$ of ${\Omega }^{EM}={\Omega }^{FW}+{\Omega }^V$ reappears only through the antisymmetric gradient of the FW congruence, as shown in eq.(31). This is the geometrical origin of the magnetic sector in the present construction and, at the same time, explains why the PH admitted extra rotational sector $\Delta {\Omega }^{\mathrm{SR}}$ may generate an anomalous magnetic misalignment between gauge and spin sectors, hence $a_e=(g-2)/2$, see sec.(Section 8). Eq.(32) is one of the main results of the paper. It is not a field equation for $F_{ab}$, but it provides the dictionary that turns the $U\left(1\right)$ curvature into a spacetime-geometrical object.

The Abelian Bianchi identity is implicit in the definition of the geometro-EM generator

$$\begin{array}{|c|}\hline {\partial }_a{{\Omega }^{EM}}_{bc}+{\partial }_b{{\Omega }^{EM}}_{ca}+{\partial }_c{{\Omega }^{EM}}_{ab}=0\\ \hline\end{array},$$

(33)

because partial derivatives commute, and it follows directly from the fundamental identity eq(32). The homogeneous Maxwell equations are automatically satisfied. This means that, locally, the geometro-EM $F_{ab}$ is the curvature of the scalarized $U\left(1\right)$ clock connection $A_a$. The inhomogeneous Maxwell equations are derived in the next section.

4.2. Generalized Lorentz force as Coriolis-like inertial effect

The Lorentz force law directly follows from the LLT transport law of the geometro-EM frame eq.(12), as geometrodynamical inertial effect, up to integrable and extra rotational contributions. Since $u^{\prime a}=e_0^a$, the velocity flow along the LLT congruence is the generalized Lorentz-force geometro-EM law

$$\begin{array}{|c|}\hline D_s{u}_a^{\prime }=-{{\Omega }^{EM}}_{ab}{u}^{\prime b}\\ \hline\end{array}.\mathrm{Geometro}-\mathrm{EM}\mathrm{Lorentz}\mathrm{force}.$$

(34)

Using the geometro-EM constitutive identity eq.(32), this geometrodynamical transport law takes the familiar Lorentz-force form

$$m{D}_s{u}_a^{\prime }\simeq q{F}_{ab}{u}^{\prime b}.$$

(35)

The Lorentz force is therefore the EM analogue of the geodesic equation in GR: gravity is encoded by spacetime curvature, while EM is encoded by local twists of the adapted Lorentz frame. It can be viewed as a Coriolis-like inertial effect associated with the local geometrodynamics of the internal clock, interpreted as gyroscope/clock precession. Worldline kinematics and Lorentz-force dynamics are not independent structures, but two equivalent readings of the same PH-admitted geometrodynamics — here the natural vorticity (magnetic) sector is restored for a non-comoving observer, similarly to ordinary Lorentz force.

4.3. Geometro-Electric and Geometro-Magnetic Fields

The resulting geometro-electric and geometro-magnetic fields measured by the comoving observer are respectively, Figure 1:

$$E_a\simeq -{\displaystyle \frac{m}{q}}{{\Omega }^{EM}}_{ab}{u}^{\prime b}={\displaystyle \frac{m}{q}}{{\Omega }^{FW}}_{ab}{u}^{\prime b},B^a\simeq -{\displaystyle \frac{1}{2}}{\displaystyle \frac{m}{q}}{ϵ}^{abcd}{u}_b^{\prime }{{\Omega }^{EM}}_{cd}={\displaystyle \frac{1}{2}}{\displaystyle \frac{m}{q}}{ϵ}^{abcd}{u}_b^{\prime }{{\Omega }^V}_{cd},$$

(36)

because ${\Omega }_{ab}^V{u}^{\prime b}=0$ and ${ϵ}^{abcd}{u}_b^{\prime }{{\Omega }^{FW}}_{cd}=0$. In fact, from the fundamental identity, eq.(32), $E_a\simeq F_{ab}{u}^{\prime b}$ and $B^a\simeq {\displaystyle \frac{1}{2}}{ϵ}^{abcd}{u}_b^{\prime }{F}_{cd}$. In the comoving frame $u^{\prime a}=(1,0,0,0)$, we have

$$\begin{array}{|c|}\hline E_i\simeq -{\displaystyle \frac{m}{q}}{{\Omega }^{FW}}_{i0},B_i\simeq -{\displaystyle \frac{1}{2}}{\displaystyle \frac{m}{q}}{ϵ}^{ijk}{u}_b^{\prime }{{\Omega }^V}_{ij}\\ \hline\end{array}.$$

(37)

In other words the ordinary electric and magnetic fields, $\mathbf{B}=(B_1,B_2,B_3)$ and $\mathbf{E}=(E_1,E_2,E_3)$, are directly generated by local Lorentz boosts and the natural local vorticity of the adapted LLT, respectively:

$$-{\Omega }^{FW}⇝\mathbf{E},-{\Omega }^V⇝\mathbf{B}.$$

(38)

This correspondence between worldlines of adapted FW frames and ordinary EM dynamics generalize gravito-EM to a more fundamental geometro-EM description, [16,17,32,33,34], and its consistency can be easily tested in basic examples such as motion in constant electric and/or magnetic fields, for a Coulomb potential or a magnetic dipole.

4.4. Gauge Invariance and Compact $U\left(1\right)$ Holonomy from Integrable Geometrodynamics

In ordinary gauge theory, EM is inferred by postulating compact $U\left(1\right)$ local invariance. Since the projection eq.(26) is taken on the scalar phase of the charged mode, a generally non-compact and non-Abelian Lorentz generator can induce a compact Abelian $U\left(1\right)$ local invariance as holonomy of the boundary $T^{\prime \mu }\left(x\right)$. Let us investigate the integrable contribution ${\Omega }^{Gauge}$ in the Abelian projection eq.(26),

$$\begin{array}{|c|}\hline p_b{\int }_{{\gamma }_x}(dy·u^{\prime }){{\Omega }^{Gauge}}_a^b=q{\partial }_a\chi \\ \hline\end{array}.$$

(39)

Such terms correspond to integrable endpoint contributions to the accumulated scalar phase along the congruence. Then, it contributes to the scalarized connection only by an exact one-form, contrarily to the general non-integrable ${\Omega }^{EM}$, and therefore generate exclusively compact$U\left(1\right)$ gauge transformations

$$A_a\stackrel{{\Omega }^{gauge}}{\to }{A}_a^{\prime \prime }=A_a+{\partial }_a\chi ,F_{ab}\stackrel{{\Omega }^{gauge}}{\to }{F}_{ab}^{\prime \prime }=F_{ab}.$$

(40)

The field mode thus acquires a local compact phase

$$\begin{array}{|c|}\hline U\left(x\right)=e^{iq\chi \left(x\right)}\in U\left(1\right)\\ \hline\end{array},{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }\left(x\right)\stackrel{{\Omega }^{gauge}}{\to }{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime \prime }\left(x\right)=U\left(x\right){\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }\left(x\right),$$

(41)

where $U\left(x\right)$ describes an effective compact$U\left(1\right)$gauge orbit of the physical gauge $A_a\left(x\right)$ induced by the LLT integrable generator ${\Omega }^{Gauge}$. Its contribution to the transformed recurrence is a boundary term of the type $T^{\prime \mu }\left(x\right)\to T^{\prime \prime \mu }\left(x\right)+{\partial }^{\mu }\mathcal{V}$, therefore leaving the PBCs unchanged, since the boundary of a boundary vanishes. Thus compactness is a property of the scalar phase, or Wilson holonomy, not of the local spacetime components of $A_a$.

Rigid Killing transformations $x^{\mu }\to x^{\prime \mu }\left(x\right)\simeq x^{\mu }+q{\xi }^{\mu }\left(x\right)$, such that $\chi =p_a{\xi }^a$, provide a simple instance of such integrable LLT contributions.

4.5. Ehresmann Connection, Spin Connection and Phase Action

The transport law induced by the full PH admitted LLT introduces a Lorentz spin-connection ${{{\omega }_{\mu }}^b}_a={e^b}_{\nu }{\partial }_{\mu }{e_a}^{\nu }$. Defining ${{\left({\omega }_{\mu }^{\Vert }\right)}^b}_a=-u_{\mu }{{{\Omega }^{PH}}^b}_a$ the pullback along the congruence, the potential in the local frame can be written as accumulated Lorentz spin-connection

$$A_a^{\prime \prime }\left(x\right)=A_a\left(x\right)+{\partial }_a\chi \left(x\right)=-p_b{\int }_{{\gamma }_X}d{y}^{\mu }{{\left({\omega }_{\mu }^{\Vert }\right)}^b}_a+\mathcal{O}\left(q\right).$$

(42)

Through projection on the scalar-phase, this singles out an effective clock fiber $S_{T_C}^1$ attached to each spacetime point, mirroring the internal fiber bundle postulated in ordinary gauge theory. By comparison, the ordinary internal fiber connection ${\varpi }_{\mu }:={\displaystyle \frac{1}{2}}{ϵ}_{ij}{{\varpi }^{ij}}_{\mu }\simeq q{A}_{\mu }$ postulated in gauge theories in terms of the spin-connection ${{\varpi }^{ij}}_{\mu }$,52,56,57], is identified with the accumulated PH admitted LLT spin-connection of ${\Omega }^{EM}$ projected onto the scalar clock phase, modulo gauge and extra rotations:

$${\varpi }_{\mu }\simeq -q{e}_{\mu }^a{p}_b{\int }_{{\gamma }_X}d{y}^{\nu }{{\left({\omega }_{\nu }^{\Vert }\right)}^b}_a.$$

(43)

In terms of this effective internal connection, the integrable contribution ${\Omega }^{gauge}$ generating $U\left(1\right)$ gauge transformations, ${\varpi }_{\mu }\stackrel{{\Omega }^{gauge}}{\to }{\varpi }_{\mu }+q{\partial }_{\mu }\chi$ and $A_{\mu }\stackrel{{\Omega }^{gauge}}{\to }{A}_{\mu }+{\partial }_{\mu }\chi$, corresponds to a local rotation of the clock fiber $S_{T_C}^1$ by an angle ${\displaystyle \frac{q}{m}}\chi \left(x\right)$. Indeed, this introduces an Ehresmann connection on the clock fiber, preserving the Compton recurrence $\eta \in S_{T_C}^1$,27,28]:

$$\begin{array}{|c|}\hline d\eta =ds-{\displaystyle \frac{q}{m}}{A}_{\mu }d{x}^{\mu },\\ \hline\end{array}$$

(44)

which is invariant under the combined gauge transformation: $A_{\mu }\stackrel{{\Omega }^{gauge}}{\to }{A}_{\mu }+{\partial }_{\mu }\chi$, and $s\stackrel{{\Omega }^{gauge}}{\to }s-{\displaystyle \frac{q}{m}}\chi$. The Ehresmann connection translates the effect of the PH-admitted Lorentz geometrodynamics, i.e.geometro-EM, directly into the underlying spacetime geometrical structure where it originates: the local spacetime-clock bundle. In other words, the LLT-induced modulation of the scalar phase is not only represented as an internal Wilson phase in the matter sector, but at a more fundamental level it is a deformation of the internal clock fiber. This observation will be a key step in sec.(Section 7), where the replacement ($ds\to d\eta$) promotes the same clock-fiber connection to a KK-like metric structure.

Finally, our formalism also suggests a useful variational reformulation. The total phase of the modulated field eq.(22) defines the particle-description counterpart of the selected field mode, namely the phase action defined in the general PH condition eq.(1):6

$${\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }\left(x\right)=e^{-i{\mathcal{S}}_{PH}\left[x\right]}{\varphi }_i,{\mathcal{S}}_{PH}\left[x\right]:={\int }_{\Gamma }\left(p_a-q{A}_a\right)d{y}^a={\int }_{\Gamma }ds\left(m-q{A}_a{\overline{u}}^{\prime a}\right).$$

(45)

4.6. Comments About Non-Abelian Extensions and General Phase Structure of Fields

The phase structure of fields and the related form of the PH condition are strictly related to the global symmetry of the theory. The projection of non-compact and non-Abelian Lorentz geometrodynamics onto the field phase is the key ingredient behind the compact $U\left(1\right)$ holonomy derived in this paper. Its extension to non-Abelian holonomies will be developed in a forthcoming work, [35].

In the present paper we consider field modes with global $U\left(1\right)$ phase symmetry. In this case the internal recurrence is therefore Abelian (scalar-valued internal clock), ${\varphi }_{\mathbf{p}}\left(x\right)\to e^{i\alpha }{\varphi }_{\mathbf{p}}\left(x\right)$. In this way, the induced internal holonomy is constrained by PH to be compact and Abelian although it originates from LLTs. The connection between PH and symmetry can also be seen from the boundary eq.(8) in terms of conserved currents, e.g.$\delta {\varphi }_{\mathbf{p}}=i\alpha {\varphi }_{\mathbf{p}}$, yields the Abelian form $i{[{\partial }^{\mu }{\varphi }_{\mathbf{p}}^{†}{\varphi }_{\mathbf{p}}-{\varphi }_{\mathbf{p}}^{†}{\partial }^{\mu }{\varphi }_{\mathbf{p}}]}_{x^{\mu }}^{x^{\mu }+T^{\mu }}=0$. More precisely, in the doublet representation ${{\varphi }_i}_{\mathbf{p}}={({{\varphi }_1}_{\mathbf{p}},{{\varphi }_2}_{\mathbf{p}})}^T$ the Abelian global symmetry is ${{\varphi }_i}_{\mathbf{p}}=exp\left(\theta {\Sigma }_{ij}\right){{\varphi }_j}_{\mathbf{p}}$ where ${\Sigma }_{ij}$ is the generator of $SO\left(2\right)\simeq U\left(1\right)$. The two-component representation of the PH is $R\left(p_{\mu }{\tau }^{\mu }\right)={\mathbb{1}}_2$, which thus reduces to the scalar PH condition eq.(1) for both components — as long as the $U\left(1\right)$ symmetry is unbroken. In sec.(Section 8) we will discuss the scalar Abelian PH of Dirac fields.

In general, however, for fields carrying several internal components, the same boundary-condition logic admits a non-Abelian (matrix-valued, or even spinorial) phase. If the free theory has a non-Abelian global symmetry group G, the multiplet ${\varphi }_i$ satisfies twisted cyclic identifications ${\varphi }_i\left(x\right)\to exp{\left(i{\alpha }^A{\Sigma }_A\right)}_i^j{\varphi }_j\left(x\right)$, where ${\Sigma }_A$ are generators of G in the representation of the multiplet. This suggests that, in this case, the preservation of PH takes a matrix form, leading naturally to a non-Abelian, matrix-valued holonomy.

The geometrodynamical mechanism investigated in this paper for charged scalar and Dirac modes, leading to the compact $U\left(1\right)$ EM holonomy, will be generalized in future work to multicomponent field theories, such as those of the Standard Model, where global non-Abelian symmetries, and the related non-abelian (matrix-valued/spinorial) phase structure suggest an extension toward a geometrodynamical description of the electroweak $SU{\left(2\right)}_L\times U{\left(1\right)}_Y$ gauge symmetry, [35] and related gauge symmetry breaking (Higgs mechanism).

5. Lagrangian Electromagnetism from Flat Spacetime Geometrodynamics

As further, partially independent confirmation of the above results, we derive the geometro-EM coupling for the single complex scalar mode, Figure 2. We assume LLT limited to the physical contribution from the generator ${\Omega }^{EM}$, eq.(16), in the definition of effective potential $A_a\left(x\right)$, eq.(26). We then introduce $A_{\mu }=e_{\mu }^a{A}_a$ and the effective covariant derivative written in terms of the accumulated LLT transport:

$$\begin{array}{|c|}\hline D_{\mu }={\partial }_{\mu }-iq{A}_{\mu }={\partial }_{\mu }-iq{p}_a{{{\Theta }^{PH}}^a}_{\mu }\\ \hline\end{array},{\partial }_{\mu }{\varphi }_{\mathbf{p}}={\partial }_{\mu }\left[V^{-1}{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }\right]=V^{-1}{D}_{\mu }{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }.$$

(46)

The combination $V^{-1}{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }$ has fixed recurrence $T^{\mu }$ as the free mode ${\varphi }_{\mathbf{p}}$, even though ${\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }$ itself carries a locally modulated recurrence $T^{\prime \mu }\left(x\right)$. Geometrically, $V^{-1}$ compensates the accumulated parallel transport of the field phase generated by the LLT, so that the modulated solution can be consistently pulled back to the fixed boundary $T^{\mu }$, removing the local phase mismatch induced by the transport of the boundary. In other words, $V^{-1}$ can locally tune the modulated mode ${\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }$ of local recurrence $T^{\prime \mu }\left(x\right)$ to the fixed recurrence $T^{\mu }$, so that it can be written as solution of an action with constant PBCs at $T^{\mu }$, preserving PH, and obtaining a more familiar and practicable formalism, [5]. The derivative term is the only relevant term to the PBCs. It is therefore sufficient to replace it with the covariant derivative term to preserve PH. The covariant derivative encodes the field modulation locally, allowing to describe the same physical mode in a formulation with static boundary.

The modulated action eq.(19) can therefore be equivalently rewritten as an action with fixed boundary:

$$\begin{array}{ccc}\hfill {\mathcal{S}}_{Free}\stackrel{LLT}{\to }{\mathcal{S}}_{EM+\left[YM\right]}& =& {\oint }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y\mathcal{L}(D_{\mu }{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime },{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime })+\left[{\mathcal{S}}_{YM}\right],\hfill \end{array}$$

(47)

and, by integrating over all field modes, it gives the ordinary gauged KG action of scalar EM. Indeed, the transformed action with modulated boundary $T^{\prime \mu }\left(x\right)$ is equivalent to the gauged version of the free action with fixed boundary $T^{\mu }$, eq.(3), while describing the same physical solution ${\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }$ consistently with PH. The integrable contribution ${\Omega }^{gauge}$ then acts as an ordinary gauge transformation and $\Delta {\Omega }^V$ is invisible to the EM sector.

For the same consistency with the fixed PBCs, the dynamics of $A_{\mu }$ must be gauge invariant, namely tunable by parallel transport and covariant derivatives to the recurrence imposed by the action boundary, besides satisfying locality and Lorentz invariance [5]. This is the standard symmetry argument leading to the Maxwell kinetic term of ordinary QFT. The unique lowest-dimension term allowed by the PBCs for the field $A_{\mu }$ is therefore of the gauge invariant form $-{\displaystyle \frac{\kappa }{4}}{F}^{\mu \nu }{F}_{\mu \nu }$, defined up to a normalization constant $\kappa$ which will be determined in sec.(Section 7). Since at this stage the Maxwell/Yang-Mills (YM) kinetic term is introduced on symmetry grounds, we denote it by $\left[YM\right]$ in square brackets. In sec.(Section 7) we will give an independent 4D geometrodynamical derivation of the same term from the Ricci tensor — removing the square brackets from $\left[YM\right]$.

Finally, the transformed action, including the kinetic term of $A_{\mu }$ written compatibly with respect to the PBCs at fixed $T^{\mu }$, becomes the standard EM action for a single scalar mode:

$$\begin{array}{|c|}\hline {\mathcal{S}}_{Free}\stackrel{LLT}{\to }{\mathcal{S}}_{EM+\left[YM\right]}\\ \hline\end{array}={\oint }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y\left[-{\displaystyle \frac{1}{4}}{\breve{F}}^{\mu \nu }{\breve{F}}_{\mu \nu }+\mathcal{L}({\breve{D}}_{\mu }{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime },{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime };m)\right],$$

(48)

where we have normalized the gauge sector: ${\breve{A}}_{\mu }=\sqrt{\kappa }{A}_{\mu }$, $\breve{q}=q/\sqrt{\kappa }$ and ${\breve{D}}_{\mu }=D_{\mu }$.

The inhomogeneous Maxwell equation follows by varying the gauge field:

$${\partial }^a{\breve{F}}_{ab}=\breve{q}{j_{{\mathbf{p}}^{\prime }}}_b:={J_{{\mathbf{p}}^{\prime }}}_b^{phys},$$

(49)

where ${J_{{\mathbf{p}}^{\prime }}}_b^{phys}$ is the physical current and the Noether current is

$${j_{{\mathbf{p}}^{\prime }}}_a=i\left[{{\varphi }^{\prime }}_{{\mathbf{p}}^{\prime }}^{†}\left(D_a{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }\right)-\left(D_a{{\varphi }^{\prime }}_{{\mathbf{p}}^{\prime }}^{†}\right){\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }\right].$$

(50)

Through the fundamental identity of geometro-EM, eq.(32), we get:

$$\begin{array}{|c|}\hline {\partial }^a{\Omega }_{ab}^{EM}=-{\displaystyle \frac{{\breve{q}}^2}{m}}{j}_b+\mathcal{O}\left(q^3\right)=-{\displaystyle \frac{\breve{q}}{m}}{\breve{J}}_b^{phys}+\mathcal{O}\left(q^2\right)\\ \hline\end{array},$$

(51)

This is the geometro-EM form of the inhomogeneous Maxwell equation. Just as Einstein’s equation relates spacetime curvature to matter stress-energy, the geometro-EM field equation relates the curvature of the induced $U\left(1\right)$ clock holonomy to the charged scalar current.

6. Geometrodynamical Origin of Electromagnetism and Gravity

Let us now consider a complex scalar mode ${\varphi }_{\mathbf{p}}$ on a curved background $g_{\mu \nu }$. We write the gravitationally coupled action in a fixed-boundary representation, with PBCs imposed at fixed $T^{\mu }$ in each local recurrence cell of base point x (as we will motivate below):

$${\mathcal{S}}_{GR}={\oint }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y\sqrt{-g}\mathcal{L}(g,{\nabla }_{\mu }{\varphi }_{\mathbf{p}},{\varphi }_{\mathbf{p}};m)+S_{EH},$$

(52)

where ${\nabla }_{\mu }$ is the Levi-Civita derivative and

$$\mathcal{L}(g,{\nabla }_{\mu }{\varphi }_{\mathbf{p}},{\varphi }_{\mathbf{p}};m)=\left[g^{\mu \nu }{\nabla }_{\mu }{\varphi }_p^{†}{\nabla }_{\nu }{\varphi }_{\mathbf{p}}-m^2{\varphi }_{\mathbf{p}}^{†}{\varphi }_p\right].$$

(53)

The Einstein–Hilbert (EH) term ${\mathcal{S}}_{EH}$ is included for completeness. The study of its boundary dynamics goes however beyond the scope of this paper, [58] — but it may be relevant for the problem of the quantization of gravity, according to ECT where quantization emerges from BCs, [3,4,5,6,7,8,9,10,11,12,13,14].

As discussed in the introduction, the action eq.(52) can be viewed as the image of the free action under local diffeomorphisms (LD), [25,26],

$${\mathcal{S}}_{Free}\stackrel{LD}{\to }{\mathcal{S}}_{GR},{\eta }_{\mu \nu }\stackrel{LD}{\to }{g}_{\mu \nu }.$$

(54)

LD generates local transformations (diffeomorphisms) of the boundary, whose modulations induced by PBCs/PH on the field mode encode the usual gravitational redshift and rulers contraction — in analogy with the the derivation of geometro-EM from the LLT-modulated boundary and related PH.

For a scalar mode, however, ${\nabla }_{\mu }{\varphi }_{\mathbf{p}}={\partial }_{\mu }{\varphi }_{\mathbf{p}}$. The gravitational modulation of ${\varphi }_{\mathbf{p}}$, i.e.of clocks/rulers, is entirely encoded in the tetrad and the metric. It is therefore convenient to work in a fixed-boundary representation, where the reference recurrence $T^{\mu }$ is kept unchanged in each base point x, while gravitational effects are carried by $g_{\mu \nu }$. This avoids introducing a separate notation for the locally transformed gravitational boundary and allows us to describe the additional geometro-EM modulation with respect to the same reference recurrence $T^{\mu }$.7

The generalization of the flat geometro-EM to this curved background is then obtained by replacing partial derivatives with Levi–Civita derivatives throughout all the demonstrations given in sec.(Section 3, Section 4 and Section 5), letting the full PH admitted LLTs to act on top of the gravitational background. Up to (gauge) integrable contributions ${\Omega }^{gauge}$ and (EM invisible) extra spatial rotations $\Delta {\Omega }^{SR}$, the geometro-EM generator ${\Omega }^{EM}$ switches on the geometro-EM interaction over the already gravitationally coupled scalar mode. The geometro-EM tetrad then satisfies

$$u^{\prime \rho }{\nabla }_{\rho }{e^{\mu }}_a=-{{{\Omega }^{EM}}_a}^b{e^{\mu }}_b,{{\Omega }^{EM}}_{ba}:=-({\nabla }_a{u}_b^{\prime }-{\nabla }_b{u}_a^{\prime }),$$

(55)

Thus gravity is carried by the Levi–Civita connection in ∇, acting on the spacetime index, whereas the EM interaction is encoded in the geometro-EM LLT generator ${\Omega }^{EM}$, acting on the local Lorentz index.

Projecting the accumulated transport onto the Abelian scalar phase gives again the effective geometro-EM connection $A_a$, formally as in eq.(26). As in the flat case, the standard EM field strength is identified by the geometro-EM LLT generator,

$$\begin{array}{|c|}\hline F_{ab}\simeq -{\displaystyle \frac{m}{q}}{{\Omega }^{EM}}_{ab}\\ \hline\end{array},$$

(56)

The corresponding geometro-EM Lorentz force law in curved spacetime is the FW-adapted velocity transport along the geometro-EM congruence,

$$\begin{array}{|c|}\hline u^{\prime b}{\nabla }_b{u}_a^{\prime }=-{{\Omega }^{EM}}_{ab}{u}^{\prime b}\\ \hline\end{array}.$$

(57)

Rewriting the LLT-modulated dynamics on curved spacetime in the fixed boundary representation, i.e.inserting parallel transport, we introduce the covariant derivative ${\breve{D}}_{\mu }={\nabla }_{\mu }-i\breve{e}{\breve{A}}_{\mu }$ to tune the modulated solution ${\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }$ and the kinematics of $A_{\mu }$ to the fixed PBCs $T^{\mu }$. This gives the common geometrodynamical description of gravitational and EM interactions for a charged scalar particle of mass m:

$$\begin{array}{|c|}\hline {\mathcal{S}}_{GR}\stackrel{LLT}{\to }{\mathcal{S}}_{GR+EM+\left[YM\right]}\\ \hline\end{array}={\oint }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y\sqrt{-g}\left[-{\displaystyle \frac{1}{4}}{\breve{F}}^{\mu \nu }{\breve{F}}_{\mu \nu }+\mathcal{L}(g,{\breve{D}}_{\mu }{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime },{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime };m)\right]+{\mathcal{S}}_{EH},$$

(58)

after canonical normalization in the gauge sector.

The resulting chain can be summarized as ${\mathcal{S}}_{free}\stackrel{LD}{\to }{\mathcal{S}}_{GR}\stackrel{LLT}{\to }{\mathcal{S}}_{GR+EM+\left[YM\right]}$. The two operations have complementary roles. LDs deform spacetime intervals and encode gravity through $g_{\mu \nu }$, while general PH admitted LLTs twist the local Lorentz frame and induce the Abelian clock holonomy identified with ordinary EM. Since LLTs are local isometries, they do not modify the gravitational metric sector; conversely, they can be applied on top of a curved background. In this sense, gravity and EM can be regarded as implemented by complementary LD and LLT sectors of the same geometrodynamical transformation, Figure 3:

$$\begin{array}{|c|}\hline {\mathcal{S}}_{free}\stackrel{LD+LLT}{\to }{\mathcal{S}}_{GR+EM+\left[YM\right]}\\ \hline\end{array}.$$

(59)

In the next section we show that the gauge kinetic term, justified so far by symmetry and PH-consistency arguments, also admits a 4D geometrodynamical derivation through the clock-fiber/KK mechanism — removing the square brackets from $\left[YM\right]$.

7. Electromagnetodynamics from the Four-Dimensional Ricci Tensor

The only distinctive assumption underlying the above derivations of geometro-EM is the intrinsic proper-time recurrence $s\in S_{T_C}^1$, such that $m{T}_C=2\pi$, implemented through PBCs in the Lagrangian formalism, whose relativistic transformations induce the modulated spacetime recurrences $T^{\prime \mu }\left(x\right)$ investigated so far. This cyclic proper-time can be represented as a local clock fiber $S_{T_C}^1$,28,46,47,59,60] attached to each spacetime point. This allows a purely four dimensional generalization of the KK miracle where EM is directly derived from the four-dimensional Ricci Tensor, in further confirmation of the above geometro-EM.

7.1. Virtual Extra Dimension and Proper-Time Formalism

The full ECT with intrinsic $s\in S_{T_C}^1$, whose fundamental scalar mode in curved spacetime is described by eq.(52), has a perfect dualism to a massless 5D theory ($d{S}^2\equiv 0$) with cyclic XD, $z\in S_{T_C}^1$, of Compton period $T_C=2\pi /m$, as proven in detail in [12,13]:

$$\begin{array}{|c|}\hline d{S}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }-d{z}^2=g_{MN}^{VXD}d{x}^{M}d{x}^N\equiv 0\left[\mathrm{VXD}\right]z\equiv sd{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }\\ \hline\end{array},$$

(60)

with $x^M=(x^{\mu },z)$. The dualism becomes a perfect equivalence as soon as the cyclic XD $z\in S_{T_C}^1$ is identified with the cyclic proper-time $s\in S_{T_C}^1$ of Compton period $T_C$ encoding the particle’s internal clock $S^1$, implicit in the PBCs of eq.(52):

$$\begin{array}{|c|}\hline z\equiv s\\ \hline\end{array},z\in S_{T_C}^1,s\in S_{T_C}^1.$$

(61)

With this identification we say that the XD is Virtual (VXD), [12,13]. Thus the extra coordinate is not a new physical spatial dimension, but the cyclic proper-time clock of the particle. From a purely 4D viewpoint, one is attaching to each spacetime point an $S^1$ fiber representing the internal clock of the particle passing through that point, Figure 4.

In what follows we retain only the fundamental mode of the corresponding virtual KK tower, which corresponds to retaining only the fundamental mode allowed by the PBCs for the scalar charged mode, as in eq.(52), or eq.(3) in the flat case. Higher virtual KK modes, interpreted in ECT as quantum excitations, have been developed in previous works and are briefly recalled in Sec. Section 9. The VXD/proper-time correspondence extended to the full harmonic expansions allowed by the PBCs is proven in [12,13].

This allows us to introduce an equivalent VXD/proper-time description of ${\mathcal{S}}_{GR}$, eq.(52), [61,62,63,64]. We start directly from curved spacetime, assuming that the LD sector has already been applied, ${\mathcal{S}}_{Free}\stackrel{LD}{\to }{\mathcal{S}}_{GR}$. Limiting the VXD/proper-time expansion to the fundamental mode ($n=1$), the cyclic VXD/proper-time formalism consists in the following equivalent ways to rewrite the action, eq.(52):

$$\begin{array}{ccc}\hfill {\mathcal{S}}_{Free}\stackrel{LD}{\to }{\mathcal{S}}_{GR}& =& {\oint }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y\sqrt{-g}\left[g^{\mu \nu }{\nabla }_{\mu }{\varphi }_{\mathbf{p}}^{†}{\nabla }_{\nu }{\varphi }_{\mathbf{p}}-m^2{\varphi }_{\mathbf{p}}^{†}{\varphi }_{\mathbf{p}}\right]+{\mathcal{S}}_{EH}\hfill \\ & =& {\oint }_{T_C}{\displaystyle \frac{ds}{T_C}}{\int }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y\sqrt{-g_{VXD}}\left[g_{VXD}^{MN}{\nabla }_M{\Phi }_{\mathbf{p}}^{†}(x,s){\nabla }_N{\Phi }_{\mathbf{p}}(x,s)\right]+{\mathcal{S}}_{EH}^{VXD}.\hfill \end{array}$$

(62)

In fact, the fundamental (classical) mode allowed by the PBCs of the VXD scalar field is

$${\Phi }_{\mathbf{p}}(x,s)={\Phi }_m\left(s\right){\varphi }_{\mathbf{p}}\left(x\right),{\Phi }_m\left(s\right)=e^{-ims},$$

(63)

where, in a chosen reference frame ($T_C\to T^{\mu }$), ${\varphi }_{\mathbf{p}}\left(x\right)$ is the gravitational coupled fundamental 4D mode solving eq.(52), while the proper-time factor is the fundamental mass eigenmode of the virtual KK tower of masses — zero-spatial-momentum mode ${\Phi }_m\left(s\right)\propto {\varphi }_{\mathbf{0}}\left(s\right)$.

After integration over the cyclic VXD/proper-time and after choosing a reference frame, labeled by $\mathbf{p}$, the fundamental VXD mode reproduces precisely the 4D scalar mode ${\varphi }_{\mathbf{p}}\left(x\right)$ of mass m, isolated through the VXD induced PBCs at $T^{\mu }$, as illustrated in eq.(62) — similarly to the KK theory, in ECT the mass has therefore a geometric origin in the Compton period of the cyclic proper-time: $m=2\pi /T_C$.

There is a crucial distinction from ordinary KK theory. In the present VXD description, the $S_{T_C}^1$ topology is not assigned to an independent physical coordinate, but to the proper-time recurrence of the particle. Consequently, the cyclicity is inherited by the 4D recurrence cell ($T_C\to T^{\mu }$), as represented by the circle symbol in integrals (denoting implicit PBCs):

$${\oint }_{T_C}{\displaystyle \frac{ds}{T_C}}{\int }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y\leftrightarrow {\oint }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y,$$

(64)

In ordinary KK theory, instead, the extra coordinate z is an additional physical cyclic coordinate, and the cyclicity of z does not by itself impose PBCs on the 4D spacetime recurrence domain: ${\oint }_{T_C}{\displaystyle \frac{dz}{T_C}}{\int }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y\leftrightarrow {\int }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y$. Thus the present construction relies essentially on the cyclic nature $s\in S_{T_C}^1$, as for all the other main results of this paper. Without this proper-time intrinsic recurrence, as in standard relativistic dynamics with $s\in \mathbb{R}$, the present purely 4D reinterpretation of the KK mechanism would not be available.

According to the geometro-EM analysis of the previous sections, switching on the 4D LLT, eq.(9), with adapted frame transport on the 4D sector, eq.(16), is equivalent to gauge the 4D terms of the original VXD theory eq.(62), as we have already seen in eq.(58). This is the VXD counterpart of the fixed-boundary tuning

$$\begin{array}{ccc}\hfill {\mathcal{S}}_{GR}& \stackrel{LLT}{\to }& {\oint }_{{\mathcal{V}}_x^{\prime }\left(T^{\prime \mu }\left(x\right)\right)}{d}^{4}y\sqrt{-g}\left[g^{\mu \nu }{\nabla }_{\mu }{{\varphi }^{\prime }}_{{\mathbf{p}}^{\prime }}^{†}{\nabla }_{\nu }{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }-m^2{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime †}{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }\right]+{\mathcal{S}}_{EH}+\left[{\mathcal{S}}_{YM}\right]\hfill \\ & =& {\oint }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y\sqrt{-g}\left[g^{\mu \nu }{D}_{\mu }{{\varphi }^{\prime }}_{{\mathbf{p}}^{\prime }}^{†}{D}_{\nu }{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }-m^2{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime †}{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }\right]+{\mathcal{S}}_{EH}+\left[{\mathcal{S}}_{YM}\right]\hfill \\ & =& {\oint }_{T_C}{\displaystyle \frac{ds}{T_C}}{\int }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y\sqrt{-g^{VXD}}\left[g_{VXD}^{MN}{D}_M{\Phi }_{\mathbf{p}}^{\prime †}{D}_N{\Phi }_{\mathbf{p}}^{\prime }\right]+{\mathcal{S}}_{EH+\left[YM\right]}^{VXD}\hfill \\ & =& {\mathcal{S}}_{GR+EM+\left[YM\right]}.\hfill \end{array}$$

(65)

The effect of the LLT on the VXD EH sector, ${\mathcal{S}}_{EH}^{VXD}\stackrel{LLT}{\to }{\mathcal{S}}_{EH+\left[YM\right]}^{VXD}$, will be described below. As discussed in sec.(Section 5), the tuning to the fixed boundary $T^{\mu }$ is obtained by replacing the derivative with the effective covariant derivative in the 4D terms, according to eq.(46):

$${\partial }_M\stackrel{LLT}{\to }{D}_M=({\nabla }_{\mu }-iq{A}_{\mu },{\partial }_s),$$

(66)

while the proper-time direction encodes the mass mode. In particular, the fundamental identity of geometro-EM, eq.(32), between geometro-EM (observer-adapted) LLT generator ${\Omega }_{ab}^{EM}$ and the field strength $F_{ab}$ still holds in this description.

7.2. The Clock-Fiber Geometry of Electromagnetism

We now transfer geometro-EM from the Wilson phase back to the underlying spacetime geometry where it originates. Geometro-EM introduces the Wilson line $V\left(x\right)$, eq.(28), into the original field mode, as fundamental parallel transport of the induced interaction:

$$\begin{array}{ccc}\hfill {\Phi }_{\mathbf{p}}(x,s)\stackrel{LLT}{\to }{\Phi }_{\mathbf{p}}^{\prime }(x,s)& =& {\Phi }_m\left(s\right){\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }\left(x\right)={\Phi }_m\left(s\right)V\left(x\right){\varphi }_{\mathbf{p}}\left(x\right)={\Phi }_m\left(\eta \right){\varphi }_{\mathbf{p}}\left(x\right),\hfill \end{array}$$

(67)

where the ${\Phi }_m\left(\eta \right)$ component now encodes the parallel transport $V\left(x\right)$ coming from ${\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }\left(x\right)$,

$${\Phi }_m\left(s\right)\stackrel{LLT}{\to }{\Phi }_m\left(\eta \right)={\Phi }_m\left(s\right)V\left(x\right)=e^{-im\eta },s\stackrel{LLT}{\to }\eta =s-{\displaystyle \frac{q}{m}}{\int }_{\Gamma }{A}_{\mu }\left(y\right)d{y}^{\mu },$$

(68)

being expressed in terms of the coordinate $\eta$. This transfers the effect of the PH-admitted LLT geometrodynamics from the effective internal transformational $\Phi \stackrel{LLT}{\to }{\Phi }^{\prime }$ to the reparametrization $s\stackrel{LLT}{\to }\eta$:

$$\begin{array}{|c|}\hline {\Phi }_{\mathbf{p}}(x,s)\stackrel{LLT}{\to }{\Phi }_{\mathbf{p},\eta }(x,\eta )={\Phi }_m\left(\eta \right){\varphi }_{\mathbf{p}}\left(x\right)\\ \hline\end{array}.$$

(69)

At differential level, the clock coordinate $\eta$ defines the effective Ehresmann connection obtained in eq.(44), [27,28]. It describes the same cyclic proper-time clock in the locally twisted, EM-interacting frame, $\eta \in S_{T_C}^1$.8 Thus the Compton period $T_C$, and hence the mass $m=2\pi /T_C$, are not changed by the EM connection. The construction does not introduce a XD massive trajectory: the VXD dynamics remains massless. The LLT geometrodynamics ($s\stackrel{LLT}{\to }\eta$) thus result in an effective KK-like metric $G_{MN}^{VXD}$,48,49,65,66]:

$$\begin{array}{ccc}\hfill d{S}^2\stackrel{LLT}{\to }d{S}_{\eta }^2& =& g_{MN}^{VXD}d{\eta }^{M}d{\eta }^N=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }-d{\eta }^2\hfill \\ & =& g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }-{\left(ds-{\displaystyle \frac{q}{m}}{A}_{\mu }d{x}^{\mu }\right)}^2=G_{MN}^{VXD}d{x}^{M}d{x}^N\equiv 0,\hfill \end{array}$$

(70)

where ${\eta }^M=(x^{\mu },\eta )$, s is the cyclic VXD/proper-time and the proper-time KK metric is9

$$\begin{array}{|c|}\hline g_{MN}^{VXD}\stackrel{LLT}{\to }{G}_{MN}^{VXD}=\left(\begin{array}{cc}{g}_{\mu \nu }-{\displaystyle \frac{q^2}{m^2}}{A}_{\mu }{A}_{\nu }& {\displaystyle \frac{q}{m}}{A}_{\mu }\\ {\displaystyle \frac{q}{m}}{A}_{\nu }& -1\end{array}\right)\\ \hline\end{array}.$$

(71)

From the VXD viewpoint, this Ehresmann connection modifies the original metric eq.(60), encoding the LLT in clock fiber $S_{T_C}^1$. In fact, the matter sector of eq.(65) can equivalently be written either in terms of the shifted clock coordinate $\eta$, or in terms of the original cyclic proper-time s and the proper-time KK metric $G_{MN}^{VXD}$, preserving PH:

$$\begin{array}{ccc}\hfill {\mathcal{S}}_{GR}& \stackrel{LLT}{\to }& {\oint }_{T_C}{\displaystyle \frac{d\eta }{T_C}}{\int }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y\sqrt{-g^{VXD}}\left[g_{VXD}^{MN}{\nabla }_M^{\left(\eta \right)}{\Phi }_{\mathbf{p},\eta }^{†}{\nabla }_N^{\left(\eta \right)}{\Phi }_{\mathbf{p},\eta }\right]+{\mathcal{S}}_{EH+\left[YM\right]}^{VXD}\hfill \\ & =& {\oint }_{T_C}{\displaystyle \frac{ds}{T_C}}{\int }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y\sqrt{-G^{VXD}}\left[G_{VXD}^{MN}{\nabla }_M^{\left(G\right)}{\Phi }_{\mathbf{p}}^{†}{\nabla }_N^{\left(G\right)}{\Phi }_{\mathbf{p}}\right]+{\mathcal{S}}_{EH+\left[YM\right]}^{VXD}\hfill \\ & =& {\mathcal{S}}_{GR+EM+\left[YM\right]},\hfill \end{array}$$

where ${\nabla }_M^{\left(\eta \right)}=({\nabla }_{\mu },{\partial }_{\eta })$, similar to ${\nabla }_M=({\nabla }_{\mu },{\partial }_s)$, ${\nabla }_M^{\left(G\right)}=({\nabla }_{\mu },{\nabla }_s)$, and ${\oint }_{T_C}d\eta ={\oint }_{T_C}ds$ on each fiber at fixed base point x. Geometro-EM, as gravity, is thus encoded by the proper-time KK metric $G_{VXD}^{MN}$, having a geometric origin in the general LLT geometrodynamics of the underlying spacetime — in fact, ${\nabla }_M^{\left(G\right)}{\Phi }_{\mathbf{p}}={\partial }_M{\Phi }_{\mathbf{p}}$ for the scalar.10

7.3. Kaluza Klein Mechanism in 4D Spacetime

The relevant remaining sector to investigate is the EH sector. A comment is in order. The matter sector above shows two equivalent representations of the same geometro-EM coupling: either the Wilson phase is absorbed into the shifted clock coordinate $\eta$, or the same information is encoded in the proper-time KK metric $G_{MN}^{VXD}$. If this were only a rewriting of the matter phase, the replacement $s\to \eta$ would amount to a field reparametrization. In the VXD formulation, however, the proper-time variable s is not an external auxiliary parameter: it is the cyclic coordinate of the local clock fiber $S_{T_C}^1$. Therefore the replacement is a change of the clock-fiber coframe itself. [28,46,47,59,60]. The corresponding VXD line element is consequently transformed as $d{S}^2⟶d{S}_{\eta }^2$. Thus the same clock-fiber geometry that reproduces the covariant derivative in the matter sector must also be used in the geometrical sector. The EH action is therefore evaluated with the same clock-fiber lift associated with PH/PBCs and the same metric $G_{MN}^{VXD}$.

We can now apply the KK mechanism, [48,49,65,66], to the cyclic VXD action, where $s\in S_{T_C}^1$ is the cyclic proper-time rather than a physical XD. In the EH sector, in parallel with the reduction of the matter term yielding the Abelian gauging of ${\mathcal{S}}_{GR}$, see eq.(52) and eq.(65), the proper-time KK metric eq.(70) gives the standard KK reduction if applied to the corresponding ${\mathcal{S}}_{EH+YM}^{VXD}$, as schematically described by

$$\begin{array}{ccc}\hfill \begin{array}{|c|}\hline {\mathcal{S}}_{EH}^{VXD}\stackrel{LLT}{\to }{\mathcal{S}}_{EH+YM}^{VXD}\\ \hline\end{array}& =& {\displaystyle \frac{1}{16\pi G}}{\oint }_0^{T_C}{\displaystyle \frac{ds}{T_C}}{\int }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y\sqrt{|G^{VXD}|}{R}^{VXD}\hfill \\ & =& {\oint }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y{\displaystyle \frac{\sqrt{-g}}{16\pi G}}\left[R-{\displaystyle \frac{1}{4}}{\displaystyle \frac{q^2}{m^2}}{F}_{\mu \nu }{F}^{\mu \nu }\right]=\begin{array}{|c|}\hline {\mathcal{S}}_{EH}+{\mathcal{S}}_{YM}\\ \hline\end{array},\hfill \end{array}$$

(72)

where G is Newton’s constant, $R^{VXD}$ and R are the cyclic VXD/proper-time lifted and the purely 4D Ricci scalars, respectively. The Maxwell kinetic term follows from the Ricci scalar, since the LLT-induced clock connection is encoded in the geometry of the VXD bundle.

Therefore, by retaining only the fundamental virtual KK mode (i.e.the fundamental scalar 4D mode allowed by the PBCs) and canonically normalizing the gauge sector with

$$\begin{array}{|c|}\hline \kappa ={\displaystyle \frac{q^2}{m^2}}{\displaystyle \frac{1}{16\pi G}}\\ \hline\end{array},$$

(73)

we recover directly the action eq.(58). The Maxwell kinetic term has thus been obtained from the Ricci tensor, rather than introduced only by consistency arguments as in eq.(58).

This is a common geometrodynamical description of gravitational and EM interactions for a charged scalar particle of mass m, where now also the Maxwell kinetic term of the geometro-EM field has been explicitly obtained from effectively purely 4D geometrodynamical arguments, i.e.from the Ricci tensor, rather than from consistency arguments.

From eq.(58), the resulting geometro-EM Maxwell equation (i.e.the geometro-EM analogue of the Einstein equation for gravity) in curved spacetime is given by the generalization of eq.(51) to curved spacetime. Expressed in terms of the non-normalized charge q we get

$$\begin{array}{|c|}\hline {\nabla }^a{\Omega }_{ab}^{EM}\simeq -{\displaystyle \frac{q^2}{m\kappa }}{j}_b=-16\pi Gm{j}_b.\\ \hline\end{array}$$

(74)

This relation is interesting because the Newton gravitational constant G relates geometrodynamical quantities such as ${\Omega }^{EM}$ with field quantities such as the Noether current $j_a$.

7.4. Comments About the Einstein-Maxwell Equations

The 4D interpretation of the KK mechanism developed above is closely related to recent cotangent-bundle/Hamilton-geometry approaches in which gravity and EM are encoded in a phase-space scalar Hamiltonian and the Einstein-Maxwell equations are derived from a scalar field equation on phase space, [41]. The present construction provides, however, a complementary physical interpretation of this Hamiltonian structure: the gauge-covariant momentum $p_{\mu }-e{A}_{\mu }$ arises here from the accumulated LLT/Fermi–Walker holonomy of the particle’s internal $S^1$ clock, implemented by means of PBCs in the field action, rather than being introduced as an independent ansatz [41].

Indeed, the phase action ${\mathcal{S}}_{PH}$, eq.(45), can be written as ${\mathcal{S}}_{PH}={\int }_{\Gamma }d{\theta }_{PH}=m{\int }_{\Gamma }d\eta ={\int }_{\Gamma }{p}_{\mu }^{\prime }\left(y\right)d{y}^{\mu }$, so that ${\partial }_{\mu }{\mathcal{S}}_{PH}=p_{\mu }^{\prime }\left(x\right)=p_{\mu }-q{A}_{\mu }$. Hence the VXD/Ehresmann one-form $d\eta$ is the local periodicity one-form of the interacting mode, describing the clock fiber. The corresponding mass-shell condition becomes the HJ equation $H(x,{\partial }_{\mu }{\mathcal{S}}_{PH})=g^{\mu \nu }{\partial }_{\mu }{\mathcal{S}}_{PH}{\partial }_{\nu }{\mathcal{S}}_{PH}=m^2$. Equivalently, in phase-space variables, this gives

$$H(x,p)=g^{\mu \nu }\left[p_{\mu }-e{A}_{\mu }\left(x\right)\right]\left[p_{\nu }-e{A}_{\nu }\left(x\right)\right].$$

(75)

Thus the Hamiltonian structure considered in [41,64] is recovered here as the quadratic Hamiltonian constraint associated with the internal-clock one-form $md\eta =\left(p_{\mu }-q{A}_{\mu }\left(x\right)\right)d{x}^{\mu }$. In this sense, the phase-space Hamiltonian is not merely a formal assumption: it is the HJ expression of the interacting particle’s intrinsic clock recurrence.

A related geometric interpretation of classical electromagnetism in terms of $U\left(1\right)\simeq SO\left(2\right)$ internal rotations, gauge connections and matter-field angular velocity has been proposed in [42,43]. In that framework, the gauge fiber and its connection provide the geometrical arena for matter-field electromagnetism. In the present construction, instead, the effective $U\left(1\right)$ fiber is not introduced as an independent internal structure: it is identified with the particle’s $S^1$ Compton clock fiber selected by the intrinsic proper-time recurrence, while the electromagnetic connection arises from the scalar projection of PH-admitted local Lorentz/FW transport.

Finally, the additional metric structure induced by $d\eta =ds-(q/m)A_{\mu }d{x}^{\mu }$ may also suggest links with other geometric approaches to EM and gravity, including double-copy motivated constructions [44,45] and broader geometrodynamical unification scenarios, see [40] for a review.

8. Geometro-Electromagnetism for Dirac Fermions, Spin Transport and Related Phenomenology

In the Dirac sector, our framework necessarily implies that the very same PH-admitted geometrodynamics that generates EM interaction also governs spin transport. This common origin of gauge dynamics and spin kinematics constitutes a further phenomenological diagnostic of the framework. It yields a generalized geometrodynamical description of Thomas precession and the BMT equation in which the gauge-spin link, previously only suggested in the gravito-EM context, is derived from the same PH-admitted LLT underlying both the physical EM gauge sector and the spin-transport sector. The correspondence is therefore not accidental, but structural, being rooted in spacetime geometrodynamics. In particular, the additional spatial rotations $\Delta {\Omega }^{SR}$ admitted by PH, although invisible to the scalar (U(1)) gauge sector, remain active in the spinorial sector. This magnetic gauge-spin misalignment provides a natural geometrodynamical carrier for the anomalous BMT structure $(g-2)$.

8.1. Dirac Geometro-EM Dynamics

We extend the geometro-EM description of the charged scalar to a Dirac fermion. For the sake of simplicity, we assume flat spacetime. Again, we consider the fundamental single free mode ${\psi }_{\mathbf{p}}$ of momentum $\mathbf{p}$, such that $p_{\mu }{\tau }^{\mu }=2\pi$, solution of the free Dirac action with global boundary $T^{\mu }={\tau }^{\mu }$ and implicit PBCs encoding the particle’s internal clock $s\in S_{T_C}^1$:

$${\mathcal{S}}_{free}^D={\oint }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y{\overline{\psi }}_{\mathbf{p}}\left(i{\gamma }_{\mu }{\partial }^{\mu }-m\right){\psi }_{\mathbf{p}},\mathrm{PBCs}\text{:}{\psi }_{\mathbf{p}}\left(x^{\mu }\right)={\psi }_{\mathbf{p}}(x^{\mu }+T^{\mu }).$$

(76)

Also in the fermionic case, the PBCs make the boundary variation vanish. Moreover, because the free Dirac theory has a global $U\left(1\right)$ symmetry, the scalar phase structure is the same as for the charged scalar. The Dirac mode satisfies the same Abelian PH relation, eq.(1). Thus the full LLT generator ${\Omega }^{PH}$ of the scalar sector, eq.(15), is equally PBCs/PH admitted.

Let us examine the full PH admitted contribution ${\Omega }^{PH}$, eq.(15), to the free Dirac theory with PBCs eq.(76). As in the charged scalar case, the LLT induces a local transformation of the spacetime boundary, $T^{\mu }\to {T^{\prime }}^a\left(x\right)$, and a corresponding modulation of the instantaneous spacetime recurrence, ${\tau }^{\mu }\to {{\tau }^{\prime }}^a\left(x\right)$, in the phase of the field solution. In addition, however, the Dirac field carries a spinorial orientation, which is transported by the Lorentz generators. The locally modulated solution selected by the transformed boundary is therefore

$$\begin{array}{ccc}\hfill {\psi }_{\mathbf{p}}\left(x\right)=e^{-i{p}_{\mu }{x}^{\mu }}{\psi }_{\mathbf{p}}\left(0\right)\stackrel{LLT}{\to }{\psi }_{{\mathbf{p}}^{\prime }}^{\prime }\left(x\right)& =& e^{-i{\int }_{\Gamma }{p}_a^{\prime }d{y}^a}D\left({\Omega }^{PH}\right){\psi }_{\mathbf{p}}\left(0\right)\hfill \\ & =& V\left(x\right)D\left({\Omega }^{PH}\right){\psi }_{\mathbf{p}}\left(x\right),\hfill \end{array}$$

(77)

where the spinorial transport is — see also the spin connection eq.(42):

$$D\left({\Omega }^{PH}\right)=\mathcal{P}\left[exp\left(-{\displaystyle \frac{i}{2}}{\int }_{\Gamma }{\Omega }_{ab}^{PH}{\Sigma }^{ab}ds\right)\right],{\Sigma }^{ab}={\displaystyle \frac{i}{4}}[{\gamma }^a,{\gamma }^b].$$

(78)

This is in fact the modulated solution of the local Lorentz transformed Dirac action

$$\begin{array}{ccc}\hfill {\mathcal{S}}_{free}^D\stackrel{LLT}{\to }{\mathcal{S}}_{EM}^D& =& {\oint }_{{\mathcal{V}}_x^{\prime }\left(T^{\prime \mu }\left(x\right)\right)}{d}^{4}y{\overline{\psi }}_{{\mathbf{p}}^{\prime }}^{\prime }\left[i{\gamma }_{\mu }{\partial }^{\mu }+{\displaystyle \frac{1}{4}}{\Omega }_{\mu ab}^{PH}{\Sigma }^{ab}-m\right]{\psi }_{{\mathbf{p}}^{\prime }}^{\prime }\hfill \\ & =& {\oint }_{{\mathcal{V}}_x\left(T^{\mu }\right)}{d}^{4}y{\overline{\psi }}_{{\mathbf{p}}^{\prime }}^{\prime }\left[i{\gamma }_{\mu }{D}^{\mu }-m\right]{\psi }_{{\mathbf{p}}^{\prime }}^{\prime }.\hfill \end{array}$$

(79)

In flat spacetime, the Lorentz spin-connection term, ${\displaystyle \frac{1}{4}}{\Omega }_{\mu ab}^{PH}{\Sigma }^{ab}$, is a local frame contribution associated with the LLT. It can be removed from the Dirac operator by working in the corresponding transported spin frame. Its physical content is then retained in the spinorial Wilson factor $D\left({\Omega }^{PH}\right)$ of the field, which determines the spin transport.

The scalar phase sector allows us to proceed in perfect parallelism with the bosonic derivation. Thus $p_a\stackrel{LLT}{\to }{p}_a^{\prime }\left(x\right)$ represents the geometrodynamically induced transformation of the four-momentum, constrained to preserve PH $p_a^{\prime }\left(x\right){{\tau }^{\prime }}^a\left(x\right)=2\pi$. This transformation can be written in the standard minimal-substitution form $p_a^{\prime }\left(x\right)=p_a-q{A}_a\left(x\right)$, where the effective geometro-EM potential is again the scalarized accumulated transport projected on the four-momentum, $A_a\left(x\right)=p_b{{{\Theta }^{PH}}^b}_a\left(x\right)$, see eq.(26). In turns, it yields to the fundamental identification of the geometro-EM LLT generator ${\Omega }^{EM}$ with the effective EM field strength, up to gauge contributions ${\Omega }^{gauge}$ and gauge invisible $\Delta {\Omega }^{SR}$. The local phase modulation of the Dirac mode is therefore given by the same path-dependent Wilson line $V\left(x\right)$, eq.(28), and the Lagrangian formulation is obtained by gauging the free Dirac action, eq.(79), in terms of the covariant derivative eq.(46), as in the scalar case.

The gradient of the congruence of geometro-EM frame then identifies the EM field tensor, the Lorentz force and the Maxwell equation, consistently with ordinary EM, in terms of the underlying LLT geometrodynamics, modulo ${\Omega }_{ab}^{Gauge}$ and $\Delta {\Omega }_{ab}^{SR}$ contributions:

$$\begin{array}{ccc}\hfill F_{ab}& \simeq & -{\displaystyle \frac{m}{q}}{\Omega }_{ab}^{EM},\mathrm{Geometro}-\mathrm{EM}\mathrm{fundamental}\mathrm{identity},\hfill \\ \hfill D_s{\overline{u}}_a^{\prime }& =& -{\Omega }_{ab}^{EM}{{\overline{u}}^{\prime }}^b\simeq {\displaystyle \frac{q}{m}}{F}_{ab}{{\overline{u}}^{\prime }}^b,\mathrm{Generalized}\mathrm{Lorentz}\mathrm{force},\hfill \\ \hfill {\partial }^a{\Omega }_{ab}^{EM}& \simeq & -{\displaystyle \frac{{\breve{q}}^2}{m}}{j}_b^f=-{\displaystyle \frac{\breve{q}}{m}}{\breve{J}}_b^{f,phys},\mathrm{Generalized}\mathrm{Maxwell}\mathrm{equations}.\hfill \end{array}$$

(80)

Here $j_b^f$ is the charge-free fermionic Noether current and ${\breve{J}}_b^{f,phys}:=\breve{q}{j}_b^f$ is the physical EM current. These relations summarize geometro-EM in the Dirac sector: the EM interaction is the inertial effect associated with the PH admitted LLT transport. As in the bosonic case, ${\Omega }_{ab}^{Gauge}$ generates compact $U\left(1\right)$ gauge orbits without changing the physical field strength, while $\Delta {\Omega }_{ab}^{SR}$ is invisible to the gauge sector because its projection on the clock phase vanishes: ${\Delta {\Omega }^{SR}}^{ab}{u}_b^{\prime }=0$. However, contrarily to the bosonic case, $\Delta {\Omega }^{SR}$ can nevertheless contribute to the Dirac spin transport, as discussed below.

8.2. Unified Thomas Spin Transport and BMT Dynamics

The spinorial polarization $D\left({\Omega }^{PH}\right)$ in general transforms under the full PH-admitted LLT ${\Omega }^{PH}={\Omega }^{EM}+{\Omega }^{gauge}+\Delta {\Omega }^{SR}$, eq.(15), where ${\Omega }^{EM}={\Omega }^{FW}+{\Omega }^V$, and $\Delta {\Omega }^{SR}=a_e{\Omega }^V$. The integrable contribution, ${\Omega }^{Gauge}$, doesn’t affect the spin sector: it is in fact a pure endpoint Lorentz-frame transformation, ${\Omega }^{Gauge}\sim \dot{R}{R}^{-1}$, with $R\in SO(1,3)$. It can be absorbed into the choice of spin frame and therefore changes only the representation of the spin components. The physical spin may be in fact represented by the Lorentz-generator bilinear $S^{ab}={\overline{\psi }}_{\mathbf{p}}\left(0\right){\Sigma }^{ab}{\psi }_{\mathbf{p}}\left(0\right)$, or equivalently by the Pauli–Lubanski projection $S^a={\displaystyle \frac{1}{2m}}{ϵ}^{abcd}{p}_b{S}_{cd}$,67,68,69]. Its full PH-admitted relativistic transport is

$$\begin{array}{|c|}\hline D_s{S}^a=-{{{\Omega }^{PH}}^a}_b{S}^b=-{{\left({\Omega }^{FW}+{\Omega }^V+{\Omega }^{gauge}+a_e{\Omega }^V\right)}^a}_b{S}^b\\ \hline\end{array}.$$

(81)

This is the kinematical core of spin transport.

If ${\Omega }^{PH}$ is restricted to its pure FW part, ${\Omega }^{FW}$, one obtains exactly the standard Thomas precession, which is a consequence of the noncommutativity of successive Lorentz boosts:

$${\left.D_s{S}^a\right|}_{FW}=-{{{\Omega }^{FW}}^a}_b{S}^b,\mathrm{Ordinary}\mathrm{Thomas}\mathrm{Precession}.$$

(82)

In the standard interpretation, Thomas precession is primarily a kinematical effect: it follows from accelerated relativistic motion, independently of the particular force responsible for the acceleration [53,55,70]. Excluding gravitational dynamics, which can be easily included in this description consistently, and nuclear forces in general neglected in this spin studies, the ultimate responsible of the Thomas kinematical effects is the EM dynamics. The interplay between Thomas precession describing kinematics and EM dynamics is so fundamental that it is at the base of gravito-EM, i.e.a precession associated to gravitational fields that take the form of an EM interaction, [16,32,33,34]. The theory of geometro-EM generalizes this interplay to a fundamental level. Gauge dynamics and spin kinematics in fact are derived from the same underlying geometrodynamics in our framework.

From the fundamental identity of geometro-EM, eq.(32), restricted to the FW part, $F_{ab}^{FW}\simeq -{\displaystyle \frac{m}{q}}{\Omega }_{ab}^{FW}$, our framework implies the following generalized formulation of Thomas precession directly in terms of EM kinematics

$$\begin{array}{|c|}\hline {\left.D_s{S}^a\right|}_{FW}=-{{{\Omega }_{FW}}^a}_b{S}^b\simeq {\displaystyle \frac{q}{m}}{{F_{FW}}^a}_b{S}^b\\ \hline\end{array},\mathrm{Generalized}\mathrm{Thomas}\mathrm{Precession}.$$

(83)

In fact, this equation shows that, according to our geometro-EM framework, the Thomas precession is essential to EM interaction (or gravitational interaction) and vice versa.

The BMT equation is usually introduced from a different viewpoint, [70]. In standard treatments one starts from a charged particle with magnetic moment in a prescribed EM field, i.e.from dynamics, and writes the covariant spin dynamics directly in terms of $F_{ab}$. The Lorentz force fixes the acceleration by $a^a={\displaystyle \frac{q}{m}}{F^a}_b{u}^b$ while the magnetic part of $F_{ab}$ determines the rest-frame spin precession:

$${\left.D_s{S}^a\right|}_{EM}\simeq {\displaystyle \frac{q}{m}}{F^a}_b{S}^b,\mathrm{Ordinary}g=2\mathrm{BMT}\mathrm{equation}.$$

(84)

For the minimally coupled Dirac value $g=2$, this is exactly the spin transport law with the full geometro-EM LLT generator ${\Omega }^{EM}={\Omega }^{FW}+{\Omega }^V$ expressed through the fundamental identity of geometro-EM eq.(32):

$$\begin{array}{|c|}\hline {\left.D_s{S}^a\right|}_{EM}=-{{{\Omega }^{\mathrm{EM}}}^a}_b{S}^b\simeq {\displaystyle \frac{q}{m}}{F^a}_b{S}^b\\ \hline\end{array},\mathrm{Generalized}g=2\mathrm{BMT}\mathrm{equation}.$$

(85)

BMT equation with $g=2$ is therefore inferred from geometrodynamics, while, in ordinary presentations, the passage from the kinematical spin transport to the EM tensor $F_{ab}$ is justified dynamically. Our framework gives a more unified interpretation of this deep relationship. The $g=2$ dynamics are not merely postulated as EM spin-precession law, but follow as the realization of the same geometro-EM LLT that governs the frame transport and generates geometro-EM. The boost sector ${\Omega }^{FW}$ of ${\Omega }^{EM}={\Omega }^{FW}+{\Omega }^V$ determines the orbital acceleration, while its vorticity sector ${\Omega }^V$ determines the magnetic spin precession. Thomas kinematics and BMT dynamics are encoded in the same geometrodynamics of the EM interaction.

For instance, when the acceleration is generated by the Coulomb field, the FW/Thomas part of the transport gives the usual spin–orbit corrections entering the fine structure of hydrogen-like atoms. In the present description, these corrections are not a separate kinematical addendum to the EM dynamics, but the boost-sector manifestation of the same PH admitted Lorentz holonomy.

8.3. Geometrodynamical Contribution to the Anomalous Magnetic Moment

Let us therefore consider all the relevant contributions to the spin transport allowed by PH, ${\Omega }^{EM}+\Delta {\Omega }^{SR}={\Omega }^{FW}+(1+a_e){\Omega }^V$, with arbitrary coefficient $a_e$, modulo gauge orbits. The additional spatial rotations, $\Delta {\Omega }^{SR}=a_e{\Omega }^V$, extra contribution to the natural vorticity of the congruence contained in ${\Omega }^{EM}$, although invisible to the scalar $U\left(1\right)$ gauge sector, remain active in the spinorial sector, giving the full BMT equation for $g\ne 2$, where therefore also the anomalous $(g-2)$ term has a precise geometrodynamical origin.

Thus the spin full PH admitted transportation law eq.(81), modulo gauge orbits, is

$$\begin{array}{ccc}\hfill \begin{array}{|c|}\hline D_s{S}^a\\ \hline\end{array}& =& -{{\Omega }^{PH}}_b^a{S}^b=-{{\Omega }^{EM}}_b^a{S}^b-a_e{{\Omega }^V}_b^a{S}^b\hfill \\ & =& -(1+a_e){{\Omega }^{EM}}_b^a{S}^b-a_e{u}^{\prime a}\left(S_c{{\Omega }^{EM}}^{cd}{u}_d^{\prime }\right)\hfill \\ & =& \begin{array}{|c|}\hline -{\displaystyle \frac{g}{2}}{{\Omega }^{EM}}_b^a{S}^b-\left({\displaystyle \frac{g}{2}}-1\right)u^{\prime a}\left(S_c{{\Omega }^{EM}}^{cd}{u}_d^{\prime }\right)\\ \hline\end{array}.\hfill \end{array}$$

(86)

where we have used the relations ${{{\Omega }^V}^a}_b{S}^b={{{\Omega }^{\mathrm{EM}}}^a}_b{S}^b+u^{\prime a}\left(S_c{{\Omega }^{\mathrm{EM}}}^{cd}{u}_d^{\prime }\right)$ and the identification $1+a_e={\displaystyle \frac{g}{2}}$ coming from the first term. This is a geometrodynamical equation. Finally, we apply the fundamental identity of geometro-EM, ${{\Omega }^{EM}}_{ab}\simeq -{\displaystyle \frac{q}{m}}{F}_{ab}$, eq.(32), obtaining the ordinary dynamical BMT equation,

$$D_s{S}^a={\displaystyle \frac{q}{m}}\left[{\displaystyle \frac{g}{2}}{F}_b^a{S}^b+\left({\displaystyle \frac{g}{2}}-1\right)u^{\prime a}\left(S_c{F}^{cd}{u}_d^{\prime }\right)\right].$$

(87)

EM dynamics and spin kinematics necessarily originate from the very same underlying geometrodynamics and are therefore strictly correlated in our generalized formulation. This unified description of the gauge and spin sectors is a main phenomenological diagnostics of our framework: the Lorentz force, Maxwell dynamics, Thomas precession and BMT spin transport are not introduced as independent structures, but consistently arise from different projections of the same PH-admitted local Lorentz geometrodynamics. The anomalous term $a_e={\displaystyle \frac{(g-2)}{2}}$ further sharpens this diagnostic role. We have in fact obtained the ordinary BMT equation for fermions with $g\ne 2$, with an important difference with respect to the standard derivation, where the anomalous term is added phenomenologically to preserve $S·u^{\prime }=0$. Here this tensorial structure originates as a geometrodynamical contribution associated with the particle’s internal clock. While the numerical derivation of the Schwinger value $a_e=\alpha /\left(2\pi \right)$ goes beyond the scope of this paper, which is limited to a leading-order classical-relativistic treatment, the anomalous BMT structure is identified with the extra spin-frame geometric contribution $\Delta {\Omega }^{\mathrm{SR}}$, naturally admitted by the variational consistency of our framework, i.e.by PH. Unlike the minimal $g=2$ geometro-EM transport, this additional contribution is invisible by construction to the scalar $U\left(1\right)$ gauge projection defining geometro-EM, and therefore appears precisely as a magnetic misalignment between the gauge and spin sectors11.

9. Comments and Outlooks

The main original results of the present paper are the geometro-EM identity $F_{ab}\simeq -{\displaystyle \frac{m}{q}}{\Omega }_{ab}^{EM}$, eq.(32), the derivation of the Lorentz force and Maxwell equations from the LLT/PBC construction, and the 4D clock-fiber interpretation of the KK mechanism. In the Dirac sector this directly implies a common description of EM-dynamics and BMT spin-kinematics in terms of PH admitted geometrodynamics, including the $(g-2)$ term. The lagrangian mechanism of sec.(Section 5) was in fact originally introduced in [5]. However, the specific PH admitted geometrodynamics identified here as at the origin of EM was not developed. Nevertheless, the full extension of the tuning mechanism to QED, proven [5], can then be applied to the present geometrodynamical formulation, thus extending our result to scalar QED; see also [3] and the discussion below.

In the present paper we deliberately retain only the fundamental mode allowed by the covariant PBCs; nevertheless, this restriction admits two complementary quantum generalizations. Apart from the intrinsic Compton recurrence, this single-mode restriction is the only additional working assumption adopted in the present construction. In the conservative QFT reading, the fundamental on-shell mode selected by the PBCs used to derive the geometro-EM Abelian gauge structure is nothing but the ordinary KG or Dirac field mode entering the construction of standard fields, e.g.see eq.(7), which can be therefore second-quantized in the standard way, so that the present construction embeds directly into scalar QED. In particular the normal ordered energy spectrum resulting from the second quantization of the field mode of momentum $\mathbf{p}$ is $\widehat{H}\left(\mathbf{p}\right)|{{\varphi }_{\mathbf{p}}}_n\rangle ={\omega }_n\left(\mathbf{p}\right)|{{\varphi }_{\mathbf{p}}}_n\rangle =n\omega \left(\mathbf{p}\right)|{{\varphi }_{\mathbf{p}}}_n\rangle$, with $\omega \left(\mathbf{p}\right)=\sqrt{{\mathbf{p}}^2+m^2}$.

In the full ECT reading, instead, one retains the entire harmonic expansion ${{\varphi }_{\mathbf{p}}}_n\left(x\right)$ allowed by the PBCs, without any explicit second-quantization ($n=1$ is the fundamental mode investigated so far ${{\varphi }_{\mathbf{p}}}_1={\varphi }_{\mathbf{p}}$). Indeed, in previous works it was shown that this full PBC harmonic sector reproduces the same ladder algebra structure of ordinary second quantization [3]. This correspondence to quantum mechanics is further reinforced by the general result that PH/PBCs on symplectic manifolds lead to canonical quantization [4] and the Feynman path-integral formulation [5,6,7,8,11,12,14] .

The same point has a natural interpretation in the VXD/proper-time formalism as illustrated in detail in [12,13]. Since the intrinsic recurrence $s\sim s+T_C$ of the proper time induces the corresponding spacetime recurrence $T^{\mu }$, with a corresponding “generalized semiclassical” quantization of the four-momentum given by the single-valued generalization of the PH relation (generalized Bohr-Sommerfeld quantization) ${\oint }_{T^{\prime \mu }\left(z\right)}{p_n^{\prime }}_{\mu }\left(y\right)d{y}^{\mu }=2\pi n$, which for the free field modes yields an harmonic energy spectrum ${\omega }_n\left(\mathbf{p}\right)=\left|n\right|\omega \left(\mathbf{p}\right)$, with $n\in \mathbb{Z}$.12 In this sense, since the cyclic proper-time/VXD recurrence induces the corresponding spacetime recurrence, eq.(64), the associated KK-like tower is virtual: the KK levels $m_n=\left|n\right|m$, with $n\in \mathbb{Z}$, are not independent classical particles propagating in a physical extra dimension, as in ordinary KK theory. Rather, they are naturally identified with the harmonic quantum excitations of the same fundamental field mode ${\varphi }_{\mathbf{p}}$, equivalently described by the normal-ordered spectrum obtained after standard second quantization, after the usual separation of positive- and negative-frequency sectors.13 In the rest frame, this virtual KK tower is precisely the normal-ordered rest-energy spectrum of the zero-momentum mode, $\widehat{H}\left(\mathbf{0}\right)|{{\varphi }_{\mathbf{0}}}_n\rangle ={\omega }_n\left(\mathbf{0}\right)|{{\varphi }_{\mathbf{p}}}_n\rangle =n\omega \left(\mathbf{0}\right)|{{\varphi }_{\mathbf{0}}}_n\rangle$ with $\omega \left(\mathbf{0}\right)=m$ and $n\in \mathbb{N}$. Thus, the fundamental-mode approximation adopted in the main text is the leading classical sector of a broader PBC/QFT correspondence, rather than an ad hoc truncation. This “generalized semiclassical” correspondence between the virtual KK tower $m_n=nm$ and the normal-ordered second-quantized rest-energy spectrum $\widehat{H}\left(\mathbf{0}\right)|{{\varphi }_{\mathbf{0}}}_n\rangle =nm|{{\varphi }_{\mathbf{0}}}_n\rangle$, with $n\in \mathbb{N}$, realizes, within the cyclic proper-time formalism, a correspondence between XD classical-geometry and 4D quantum behavior of the general type the classical-geometry/quantum-behavior relation underlying AdS/CFT and gauge-gravity duality, as shown in [12,13]. These quantum extensions are not required for the geometro-EM derivation, but they show that the single-mode construction used in this paper sits naturally inside a broader QFT-compatible framework.

Although this paper is restricted to the Abelian $U\left(1\right)$ case, it also identifies the key ingredients for a geometrodynamical formulation of non-Abelian gauge symmetries, sec.(Section 4.6). The crucial point is that the compact Abelian gauge orbit obtained here arises from the projection of generally non-compact and non-Abelian LLT geometrodynamics onto the scalar phase structure of charged scalar or Dirac modes, both carrying a global $U\left(1\right)$ symmetry. A non-Abelian generalization therefore requires field multiplets with a richer internal phase structure induced by an extended global non-Abelian symmetry group G. Thus, the PH condition is expected to take a matrix-valued, or spinorial, form, and the LLT holonomy can be projected onto the corresponding compact internal non-Abelian gauge sector.

This observation provides the starting point for a forthcoming extension of the present construction from compact $U\left(1\right)$ geometro-EM to an electroweak geometrodynamical framework based on $SU{\left(2\right)}_L\times U{\left(1\right)}_Y$,35]. In such a setting, the phase structure of Standard Model fields becomes essential, and the Higgs mechanism may admit a complementary interpretation in terms of geometrodynamical recurrence locking and internal clock alignment. In particular, in the present framework, we have already noticed that particle masses have a geometrical meaning as inverse proper-time recurrences, namely as the periods of the internal clocks. This suggests natural connections with XD extensions of the Standard Model, [5,12,48,49,50,51].

In sec.(Section 7) we discussed the connection with the Hamiltonian cotangent-bundle construction of [41]. The framework is also conceptually related to the Rainich–Misner–Wheeler program [36,37], where EM is encoded in spacetime curvature, and to gauge-theoretic formulations of gravity, such as those of Utiyama and Sciama–Kibble [38,39], where gravity is formulated as a gauge theory of local Lorentz/Poincaré symmetry; see [40] for a review. The difference is that EM is not reconstructed here from Einstein–Maxwell solutions, nor is the gauge connection postulated as an independent internal field. Instead, it is induced by the scalar projection of the PH-admitted Lorentz-frame holonomy. EM and gravity arise from complementary spacetime-geometrical transformations: PH-admitted LLT for the former, LD for the latter. Our framework also shares analogies with double-copy ideas relating gauge and gravitational descriptions [44,45], although the mechanism proposed here is formulated at the level of PH, boundary conditions, and clock-fiber holonomy rather than scattering amplitudes.

10. Conclusions

We have identified a spacetime-geometrodynamical structure underlying $U\left(1\right)$ EM, by requiring compatibility with the variational principle of a field description implemented with covariant PBCs, namely by requiring PH in the particle’s internal-clock phase dynamics. The variational admitted geometrical structure for EM is the congruence of observer-adapted Fermi-Walker frames, generated by Local Lorentz Transport and described by an antisymmetric Lorentz generator ${\Omega }_{\mu \nu }^{EM}$. At leading order in the electric charge q, the fundamental geometro-EM identity $F_{\mu \nu }\simeq -{\displaystyle \frac{m}{q}}{\Omega }_{\mu \nu }^{EM}$ defines the field strength of EM, $F_{\mu \nu }$, as the generator of admitted local Lorentz isometries. This identification holds up to integrable contributions ${\Omega }_{\mu \nu }^{gauge}$, which generate compact$U\left(1\right)$ gauge orbits without changing the field strength, and up to additional local spatial frame rotations $\Delta {\Omega }_{\mu \nu }^{SR}$, which are invisible to the scalar gauge sector but relevant in the Dirac spin sector.

The main novelty is that the $U\left(1\right)$ gauge connection is not postulated as an internal symmetry: PH projects the generally non-compact and non-Abelian local Lorentz transport onto the scalar phase of fields with global $U\left(1\right)$ symmetry, thereby producing the compact Abelian $U\left(1\right)$ local invariance of EM alongside the complementary standard gravitational geometrodynamics based on curved spacetime. The Lorentz-force law then emerges as a Coriolis-like inertial effect associated with the adapted Fermi-Walker transport, in analogy with the way gravitational motion is encoded in General Relativity. Furthermore, the PBCs implementing the proper-time recurrence $s\sim s+T_C$ associated to the particle’s mass, $T_C=2\pi /m$, naturally define an internal clock fiber $S_{T_C}^1$. The Ricci geometry of this clock fiber reproduces the Maxwell kinetic term through a purely four-dimensional reinterpretation of the Kaluza-Klein mechanism. This reinforces the geometro-EM obtained independently in both the dynamical and Lagrangian formulations.

The extension to Dirac fermions shows that EM and spin transport are governed by the same underlying Lorentz geometrodynamics. This gives a geometrodynamical interpretation of classical-relativistic EM phenomena such as Thomas/Larmor precession and the BMT equation, where the relationship between EM dynamics and spin kinematics is not accidental, but reflects two dual aspects of the same relativistic frame structure. In particular, extra contributions to the frame vorticity generate the tensorial structure of the anomalous $(g-2)$ term. Thus the present framework provides a geometrodynamical carrier for the anomalous magnetic-moment structure itself. More generally, this common origin of gauge dynamics and spin transport, although suggested in a limited form by gravito-EM analogies, appears not to have been previously formulated as a single geometrodynamical mechanism. It therefore constitutes a further nontrivial phenomenological consistency check successfully passed by the framework, while the anomalous magnetic moment probes the PH admitted magnetic misalignment between the scalar gauge holonomy and the spinorial transport sector. This also connects the present geometro-EM construction with one of the most sensitive phenomenological probes of QED, while the numerical Schwinger value $a_e=\alpha /\left(2\pi \right)$ remains a higher-order QED result beyond the leading classical-relativistic construction developed here. The present analysis is a leading-order (tree-level), purely four-dimensional geometrodynamical derivation of Abelian gauge invariance, one of the pillars of Quantum Field Theory. It also introduces an original mathematical method for tracking field interactions through covariant transformations of PBCs. Although the analysis has been restricted to $U\left(1\right)$ gauge invariance, it sets the basis for a generalization to non-Abelian gauge groups, such as the electroweak gauge group and related symmetry breaking, to be discussed in a forthcoming work [35].

The present framework is based on ECT which provides a natural extension of its validity to scalar QED, [5], and with second quantization prescription implicit in the PBCs, [3]. Further quantum aspects and a variety of quantum phenomena in condensed matter and high-energy physics, are developed in [2,3,4,5,6,7,8,9,10,11,12,13,14,15].

The combination of Local Diffeomorphisms and PH admitted Local Lorentz Transformations shows that gravity and EM can therefore be derived as complementary sectors of a single spacetime-geometrodynamical structure: the former deforms the metric and generates gravitational dynamics, whereas the latter induces the compact $U\left(1\right)$ clock holonomy identified here with EM. In this sense, electromagnetic interaction (light), much like gravity, can be understood as a geometrodynamical phase connection governing the synchronization of elementary internal clocks.

Appendix A.1. Second Quantization — [3] —

The most general solution of the free action, eq.(3), is not limited to the fundamental mode ${\varphi }_{\mathbf{p}}\left(x\right)$, but contains the full harmonic expansion allowed by the PBCs. We denote this full PBC-allowed solution by the tilde symbol, ${\varphi }_{\mathbf{p}}\left(x\right)\stackrel{\mathrm{PBCs}}{\to }{\tilde{\varphi }}_{\mathbf{p}}\left(x\right)$. The corresponding harmonic modes form the natural basis of a Hilbert-space description and, in the ordinary QFT formulation, are associated with ladder operators ${\widehat{a}}_{\mathbf{p}}$ and ${\widehat{a}}_{\mathbf{p}}^{†}$. In [3], is a kind of quantum-relativistic generalization of semiclassical quantization methods, it was shown that the PBC constraint imposed on the field mode plays a role formally equivalent to the canonical ladder algebra of second quantization:

$$\left\{\begin{array}{ccc}{\tilde{\varphi }}_{\mathbf{p}}(x^{\mu }+T^{\mu })\hfill & \stackrel{\mathrm{PBCs}}{\equiv }& {\tilde{\varphi }}_{\mathbf{p}}\left(x^{\mu }\right),\hfill \\ T^{\mu }{p}_{\mu }\hfill & =& 2\pi \hfill \end{array}\right.⟺[{\widehat{a}}_{\mathbf{p}},{\widehat{a}}_{\mathbf{p}}^{†}]=1.$$

(A1)

This observation suggests two consistent ways of connecting the geometro-EM construction of the main text with QED.

The first route is conservative and independent of the full ECT interpretation. One retains only the fundamental mode ${\varphi }_{\mathbf{p}}\left(x\right)$, as done throughout the main text, and then quantizes the corresponding KG field in the standard QFT sense by imposing the canonical commutator $[{\widehat{a}}_{\mathbf{p}},{\widehat{a}}_{\mathbf{p}}^{†}]=1$. Since the set of modes ${\varphi }_{\mathbf{p}}\left(x\right)$, for all $\mathbf{p}$, reconstructs the ordinary KG field in eq.(7), this route embeds geometro-EM directly into the standard scalar-QED framework. In this interpretation the VXD/KK correspondence and the derivation of the Maxwell kinetic term from the Ricci scalar are to be understood at the fundamental-mode level. Including both the full PBC harmonic tower and the standard second-quantization prescription would otherwise double-count the same quantum excitations (ladder algebra).

The second route is to take seriously the equivalence summarized in eq.(A1) and to investigate quantum effects directly through the full harmonic expansion allowed by the PBCs, without imposing an additional second-quantization prescription. In this case PH is imposed in its single-valued form, ${\oint }_{T^{\prime \mu }\left(z\right)}{p_n^{\prime }}_{\mu }\left(y\right)d{y}^{\mu }=2\pi n$, namely as a generalized Bohr–Sommerfeld condition. The higher harmonics are then interpreted as quantum excitations, of the same fundamental mode ${\varphi }_{\mathbf{p}}\left(x\right)$ which constitutes the virtual KK tower in the proper-time formalism, rather than as independent classical fields.

Appendix A.2. Internal Clock S 1 Fiber in the Virtual Extra Dimension Formalism — [12,13] —

The PBCs reproduce, in the harmonic sector, the normal-ordered excitation of the spectrum second-quantized description of the classical mode. In particular, the time component of the PBCs fixes the allowed frequencies, so that the positive-frequency sector gives $E_n\left(\mathbf{p}\right)=n\omega \left(\mathbf{p}\right)$ associated with the ordinary second-quantized description of the classical mode ${\varphi }_{\mathbf{p}}\left(x\right)$, consistently with ordinary second quantization. For the zero-momentum mode ${\varphi }_{\mathbf{0}}$, this gives the rest-energy spectrum $m_n=n\omega \left(0\right)=nm$, where $m={\displaystyle \frac{2\pi }{T_C}}$ and $n\in \mathbb{N}$, which is precisely the spectrum generated by the Compton recurrence of the internal clock $S_{T_C}^1$ when the full positive-frequency harmonic sector is retained. This observation provides the heuristic bridge to the VXD description: ECT admits a dual representation in terms of a massless XD theory whose extra coordinate is virtual, namely identified with the cyclic proper time. We illustrate the interpretation in which the higher KK-like masses $m_n$ are not independent classical particles, but the normal-ordered excitation levels of the same field mode of mass m. Thus the virtual KK tower corresponds to the second-quantized rest-energy spectrum of the zero-momentum mode, as discussed in sec.(Section 7). The full demonstration of this correspondence is given in [12,13].

In particular here we shortly illustrate this point by investigating the VXD correspondence with respect to the full harmonic expansion of the free 5D massless KK-theory in flat XD spacetime. The fundamental hypothesis (postulate) of ECT, at the base of the main results given in the main text, is that elementary particles are characterized by an intrinsic proper-time periodicity $s\in S_{T_C}^1$ of Compton time $T_C=2\pi /m$. This, in the rest frame, implies that the full harmonic expansion allowed by the PBCs for zero momentum field mode ${\varphi }_{\mathbf{0}}\left(s\right)$ is

$${\tilde{\varphi }}_{\mathbf{0}}\left(s\right)\equiv {\tilde{\varphi }}_{\mathbf{0}}(s+T_C)\Rightarrow {\tilde{\varphi }}_{\mathbf{0}}\left(s\right)=\sum _{n\in \mathbb{Z}}{\alpha }_n\left(0\right)e^{-inms},m_n=\left|n\right|{\displaystyle \frac{2\pi }{T_C}}=\left|n\right|m,$$

(A2)

where ${\alpha }_n$ are Fourier coefficients.

Through LLT transformation ${\Omega }^{EM}$ into a PH admitted frame, the general solution of the modulated action eq.(19), with all the expansion modes allowed by the PBCs at $T^{\prime \mu }\left(x\right)$ is

$${\varphi }_{\mathbf{0}}\left(s\right)\stackrel{LLT}{\to }{\tilde{\varphi }}_{{\mathbf{p}}^{\prime }}^{\prime }\left(x\right)=\sum _{n\in \mathbb{Z}}{{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }}_n\left(x\right)=\sum _{n\in \mathbb{Z}}{\alpha }_n\left({\mathbf{p}}^{\prime }\right)e^{-i{\int }_{\Gamma }{p_n^{\prime }}_{\mu }\left(y\right)d{y}^{\mu }}.$$

(A3)

The fundamental mode solution of eq.(19) or the gauged eq.(48) is thus ${\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }\left(x\right)\equiv {{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }}_{n=1}\left(x\right)$.

In the free case, eq.(3), we write explicitly the action of all these energy eigenmodes. Using the proper-time formalism, we observe the direct dualism with a 5D massless KK theory with cyclic XD of Compton compactification length. We say that the proper-time is a Virtual XD: $z\equiv s$ implies the exact equivalence with the full ECT expansion

$$\begin{array}{ccc}\hfill {\tilde{\mathcal{S}}}_{free}& =& {\oint }_{T^{\mu }}{d}^{4}x\sum _{n\in \mathbb{Z}}\left[{\partial }_{\mu }{{\varphi }_{\mathbf{p}}}_n^{†}\left(x\right){\partial }^{\mu }{{\varphi }_{\mathbf{p}}}_n\left(x\right)-m_n^2{{\varphi }_{\mathbf{p}}}_n^{†}\left(x\right){{\varphi }_{\mathbf{p}}}_n\left(x\right)\right]\hfill \\ & =& {\oint }_{T_C}{\displaystyle \frac{ds}{T_C}}{\int }_{T^{\mu }}{d}^{4}x\sum _{n\in \mathbb{Z}}\left[{\partial }_M{{\tilde{\Phi }}_{\mathbf{p}}}^{†}(x,s){\partial }^M{\tilde{\Phi }}_{\mathbf{p}}(x,s)\right].\hfill \end{array}$$

(A4)

In fact, the PBCs on the proper-time s implies the following decomposition

$${\tilde{\Phi }}_{\mathbf{p}}(x,s)=\sum _{n\in \mathbb{Z}}{N}_n{\Phi }_{n\in \mathbb{Z}}\left(s\right){{\varphi }_{\mathbf{p}}}_n\left(x\right)=\sum _{n\in \mathbb{Z}}{e}^{-inms}{{\varphi }_{\mathbf{p}}}_n\left(x\right).$$

(A5)

Notice that such a dualism with XD is only possible if the proper-time has an intrinsically cyclic nature, as in ECT.

We shortly illustrate that the higher modes can be regarded as quantum excitations without pretending to be exhaustive. Full details are given in [12,13], see also [2,3,4,5,6,7,8,9,10,11,14,15]. Being a harmonic expansion, the modes associated with the cyclic proper-time recurrence define a natural orthonormal basis on the clock fiber. With the normalization

$$u_n\left(s\right):={\displaystyle \frac{1}{\sqrt{T_C}}}{e}^{-inms},m_n={\displaystyle \frac{2\pi \left|n\right|}{T_C}}=\left|n\right|m,$$

(A6)

one has

$${\oint }_0^{T_C}ds{u}_{n^{\prime }}^{*}\left(s\right)u_n\left(s\right)={\delta }_{n^{\prime }n},\sum _{n\in \mathbb{Z}}{u}_n\left(s\right)u_n^{*}\left(s^{\prime }\right)={\delta }_{S_{T_C}^1}(s-s^{\prime }).$$

(A7)

For the same reason, at fixed time and separating positive and negative frequencies as in ordinary QFT, the corresponding field modes define a Hilbert space, with base eigenvectors $|{\varphi }_{{\mathbf{p}}^{\prime },n}^{\prime }\rangle$ and relative orthogonality relation normalized as

$$\begin{array}{ccc}\hfill |{{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }}_n\rangle ,s.t.{{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }}_n\left(x\right):=\langle x|{{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }}_n\rangle ,〈{\varphi }_{{\mathbf{p}}^{\prime \prime },n^{\prime }}^{\prime }\left(t\right)|{\varphi }_{{\mathbf{p}}^{\prime },n}^{\prime }\left(t\right)〉& :=& \int {\displaystyle \frac{d^{3}x}{{\left(2\pi \right)}^3}}{{\varphi }_{{\mathbf{p}}^{\prime \prime }}^{\prime †}}_{n^{\prime }}\left(\mathbf{x}\right){{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }}_n\left(\mathbf{x}\right)\hfill \\ & =& {\delta }^{\left(3\right)}({\mathbf{p}}^{\prime \prime }-{\mathbf{p}}^{\prime }){\delta }_{n^{\prime }n},\hfill \end{array}$$

up to the conventional relativistic normalization factor. Thus the full PBC-allowed solution can be written as a Hilbert state with related completeness relation

$$|{\tilde{\varphi }}_{{\mathbf{p}}^{\prime }}^{\prime }\left(t\right)\rangle =\sum _{n\in \mathbb{Z}}{\alpha }_n\left({\mathbf{p}}^{\prime }\right)|{\varphi }_{{\mathbf{p}}^{\prime },n}^{\prime }\left(t\right)\rangle ,{\delta }^{\left(3\right)}({\mathbf{x}}^{\prime }-\mathbf{x})=\sum _{n\in \mathbb{Z}}\int {\displaystyle \frac{d^3\mathbf{p}}{{\left(2\pi \right)}^3}}\langle {\mathbf{x}}^{\prime }|{{\varphi }_{\mathbf{p}}^{\prime }}_n\left(t\right)\rangle \langle {{\varphi }_{\mathbf{p}}^{\prime }}_n\left(t\right)|\mathbf{x}\rangle .$$

(A8)

Similarly the energy and momentum spectrum define related Hamiltonian and momentum Hilbert operators, respectively:

$${\widehat{H}}^{\prime }\left(x\right)|{{\varphi }_{{\mathbf{p}}^{\prime }}}_n\rangle ={\omega }_n^{\prime }\left(x\right)|{{\varphi }_{{\mathbf{p}}^{\prime }}}_n\rangle ,{\widehat{\mathbf{P}}}^{\prime }\left(x\right)|{{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }}_n\rangle ={\mathbf{p}}_n^{\prime }\left(x\right)|{{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }}_n\rangle ,$$

(A9)

The time component of bulk equations of motion coming from the KG equation is $({\partial }_0^2+{\omega }_n^2\left(t\right)){{\varphi }_{{\mathbf{p}}^{\prime }}}_n\left(x\right)=0$, which can be written in the Hilbert space as a Shrödinger equation $i{\partial }_0|{\tilde{\varphi }}_{{\mathbf{p}}^{\prime }}^{\prime }\rangle ={\widehat{H}}^{\prime }|{\tilde{\varphi }}_{{\mathbf{p}}^{\prime }}\rangle$. Thus, states at different times are not related by a Dirac delta in time, but by the time-evolution operator,

$$|{{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }}_n\left(t^{\prime }\right)\rangle =\widehat{U}(t^{\prime },t)|{{\varphi }_{{\mathbf{p}}^{\prime }}^{\prime }}_n\left(t\right)\rangle ,\widehat{U}(t+dt;t)=e^{-i\widehat{H}dt}.$$

(A10)

According to the correspondence with QM recalled above, the KK mass eigenmodes of the VXD expansion should not be interpreted as independent classical KK particles. Rather, they mimic the normal-ordered quantum excitation spectrum of the same fundamental cyclic field mode. It is convenient to distinguish the positive and negative energy sectors, $n>0$ and $n<0$, respectively, in analogy with standard QFT (the zero mode is purely translational). In the free case described in eq.(A4), this is indeed analogous to the second quantization of the zero-momentum mode. According to the correspondence eq.(A1) we have ($n\in \mathbb{N}$):

$$\widehat{H}|{{\varphi }_{\mathbf{0}}}_n\rangle =\omega \left(0\right){\widehat{a}}_{\mathbf{0}}^{†}{\widehat{a}}_{\mathbf{0}}|{{\varphi }_{\mathbf{0}}}_n\rangle =n\omega \left(0\right)|{{\varphi }_{\mathbf{0}}}_n\rangle =m_n|{{\varphi }_{\mathbf{0}}}_n\rangle ,$$

(A11)

where $\omega \left(0\right)=m$ and therefore $m_n=n\omega \left(0\right)=nm$.

Thus the integer n labels the harmonic excitation level of the same cyclic mode, not a tower of independent massive particles. Since the XD is virtual, the same integer n labels both the proper-time harmonic and the corresponding four-momentum eigenvalue, as shown in [12,13]. Indeed, the Lorentz transformation of the rest-frame mass spectrum gives $p_{na}={e_a}^0{m}_n=n{e_a}^{0}m=n{p}_a$, so that, in an inertial frame, ${\omega }_n\left(\mathbf{p}\right)=\sqrt{{\mathbf{p}}_n^2+m_n^2}=n\sqrt{{\mathbf{p}}^2+m^2}$, ${\mathbf{p}}_n=n\mathbf{p}$. This agrees with the normal-ordered spectrum obtained from standard second quantization, while preserving the VXD interpretation of the higher harmonics as quantum excitations of one elementary cyclic mode. This also shows that the proper-time periodicity is induced on the spacetime dimensions, eq.(64). The field dynamics associated with the particle’s internal clock is thus equivalent to the dynamics of a massless 5D theory with virtual KK-tower, once the extra coordinate is identified with the proper time $z\equiv s$, including the full harmonic expansion allowed by the PBCs.

Appendix A.3. Feynman Path Integral and Scalar QED — [5] —

In agreement with the previous correspondences, ECT also admits a classical worldline correspondence with the ordinary Feynman path integral [5,6,7,8,11,12,13,14]. This illustrates the extension of the PBCs tuning mechanism to scalar QED developed in [5].

Here we recall only the minimal ingredients relevant to the geometro-EM construction of the main text. The evolution operator $\widehat{U}(t_f,t_i)$, eq.(A10), satisfies the usual composition property. Hence the evolution from an initial time $t_i$ to a final time $t_f$ can be decomposed into N infinitesimal intervals $ϵ=(t_f-t_i)/N$, with $t_m=t_i+mϵ$:

$$\widehat{U}(t_f,t_i)={\displaystyle \prod _{m=0}^{N-1}}\widehat{U}(t_{m+1},t_m),t_N=t_f.$$

(A12)

By inserting at each intermediate time the momentum/eigenmode completeness relation eq.(A8), one obtains the standard time-sliced kernel

$$\mathcal{K}(x_f,t_f;x_i,t_i)=\underset{N\to \infty }{lim}\int {\displaystyle \prod _{m=1}^{N-1}}{d}^3{x}_m{\displaystyle \prod _{m=0}^{N-1}}〈{\mathbf{x}}_{m+1}\left|\widehat{U}(t_{m+1},t_m)\right|{\mathbf{x}}_m〉,$$

(A13)

where ${\mathbf{x}}_0={\mathbf{x}}_i$ and ${\mathbf{x}}_N={\mathbf{x}}_f$ and the infinitesimal kernel can be written schematically as

$$〈{\mathbf{x}}_{m+1}\left|\widehat{U}(t_{m+1},t_m)\right|{\mathbf{x}}_m〉\simeq \int {\displaystyle \frac{d^3{p}_m}{{\left(2\pi \right)}^3}}exp\left\{i\left[{\mathbf{p}}_m^{\prime }·\Delta {\mathbf{x}}_m-H_m^{\prime }\Delta t_m\right]\right\},$$

(A14)

where $H_m$ denotes the classical Hamiltonian symbol associated with the locally modulated geometro-EM phase transport in the dual particle description. In fact, in the continuum limit this gives the single-mode worldline path integral

$${\mathcal{K}}_{EM}(x_f,t_f;x_i,t_i)=\int \mathcal{D}\mathbf{x}{e}^{i{\mathcal{S}}_{PH}(t_f,t_i)},$$

(A15)

with the phase action ${\mathcal{S}}_{PH}$ given in eq.(45). In a $3+1$ splitting this can be written, up to the chosen sign convention for $p_{\mu }d{x}^{\mu }$, as

$${\mathcal{S}}_{PH}(t_f,t_i)={\int }_{t_i}^{t_f}dt\left[\mathbf{P}·\dot{\mathbf{x}}-H-q{A}_{\mu }{J}^{\mu }\right],J^{\mu }={\displaystyle \frac{d{x}^{\mu }}{dt}}.$$

(A16)

Thus the same phase action introduced in the main text is the EM interacting classical-relativistic particle action entering the Feynman kernel of a charged scalar mode, according to standard QED.

The modulation of the full PBC harmonic set induced by the adapted LLT generator ${\Omega }^{EM}$ can equivalently be written in terms of a scattering operator $S[t_f,t_1]$ acting on the corresponding Hilbert state. Introducing the operator ${\widehat{A}}_{\mu }$ associated with the geometro-EM connection, one has schematically the

$$|{\tilde{\varphi }}_{{\mathbf{p}}^{\prime }}^{\prime }\rangle =S[t_f,t_1]|{\tilde{\varphi }}_{\mathbf{p}}\rangle ,S[t_f,t_1]=e^{iq{\int }_{t_i}^{t_f}dt{\widehat{A}}_{\mu }{J}^{\mu }}.$$

(A17)

This is the single-mode counterpart of the scalar-QED coupling discussed in [5].

The path-integral formulation has a simple geometrical interpretation in ECT. Because of the PBCs, for fixed initial and final spacetime points the cyclic proper-time support admits infinitely many equivalent classical representatives (degenerate solutions), distinguished by the winding number $n^{\prime }\in \mathbb{Z}$ of the internal clock. The cyclic part of the proper-time kernel can therefore be represented by the Dirac comb $K_{PH}(s_f,s_i)=T_C{\sum }_{n^{\prime }\in \mathbb{Z}}\delta (s_f-s_i+n^{\prime }{T}_C)$, i.e.a sum over all cyclic paths between $s_f$ and $s_i$ allowed by the cyclic geometry of the proper-time. In previous works this winding-number representation was shown to reproduce the standard Feynman path-integral structure [6,7,8,14].

Appendix A.4. Canonical Quantization — [4] —

We finally mention that, more generally, as proven in [4], for a symplectic manifold equipped with a non-degenerate closed two-form, imposing PH through PBCs constrains the classical dynamics in a way that is formally equivalent to canonical quantization. In particular, the PBC constraint plays the role of the canonical commutation structure of ordinary QM. In the present notation, this correspondence is schematically summarized as

$${\tilde{\varphi }}_{{\mathbf{p}}^{\prime }}^{\prime }\left(\mathbf{x}\right)\stackrel{\mathrm{PBCs}}{\equiv }{\tilde{\varphi }}_{{\mathbf{p}}^{\prime }}^{\prime }({\mathbf{x}}^{\mu }+T^{\prime \mu }\left(\mathbf{x}\right))⟺[{\widehat{x}}^i,{\widehat{P}}_j^{\prime }]=i\{x^i,p_j^{\prime }\}=i{{\delta }^i}_j,$$

(A18)

where ${\widehat{P}}_j^{\prime }=-i\partial /\partial x^j$, $\{,\}$ denotes the Poisson bracket, and we use units $\hslash =1$.

The generality of this result is important for the present discussion because it applies to arbitrary symplectic dynamics, including the phase-space structures underlying general-relativistic (Riemannian) and field-theoretic (infinite degrees of freedom) systems. It therefore suggests a possible perspective on quantum aspects of gravity when the relevant geometrical dynamics are formulated in terms of boundary conditions and intrinsic recurrences.

Figure A1. Schematic illustration of the correspondence between intrinsic time periodicity, vibrating string, and the Hilbert-space formulation of QM. A free relativistic particle of energy E is represented as an elementary clock with internal cyclic time $t\in S_T^1$ of period $T=h/E$ (left), covariant description of the cyclic proper-time $s\in S_{T_C}^1$, implying periodic dynamics in the field description that admits (Fourier) a discrete harmonic expansion with frequencies ${\omega }_n=n\omega =2\pi n/T$ and thus a quantized energy spectrum $E_n=\hslash {\omega }_n$ (center). The same complete set of harmonics can be therefore mapped to a Hilbert-space basis $|n\rangle$, with state $|\Phi \rangle ={\sum }_n{c}_n|n\rangle$ and Schrödinger evolution generated by $\widehat{H}$ (right), a dictionary illustrating how cyclic internal time leads naturally to discretization and standard QM notation, [2,3,4,5,6,7,8,9,10,11,12,13,14,15]).

Figure A1. Schematic illustration of the correspondence between intrinsic time periodicity, vibrating string, and the Hilbert-space formulation of QM. A free relativistic particle of energy E is represented as an elementary clock with internal cyclic time $t\in S_T^1$ of period $T=h/E$ (left), covariant description of the cyclic proper-time $s\in S_{T_C}^1$, implying periodic dynamics in the field description that admits (Fourier) a discrete harmonic expansion with frequencies ${\omega }_n=n\omega =2\pi n/T$ and thus a quantized energy spectrum $E_n=\hslash {\omega }_n$ (center). The same complete set of harmonics can be therefore mapped to a Hilbert-space basis $|n\rangle$, with state $|\Phi \rangle ={\sum }_n{c}_n|n\rangle$ and Schrödinger evolution generated by $\widehat{H}$ (right), a dictionary illustrating how cyclic internal time leads naturally to discretization and standard QM notation, [2,3,4,5,6,7,8,9,10,11,12,13,14,15]).

A key role in the derivation of eq.(A18) is played by gauge invariance, and is therefore worth recalling in this context. Owing to the $S^1$ clock fiber, general holonomies arise from canonical transformations, or symplectic flows, in close analogy with the geometrodynamical holonomy mechanism discussed in the main text. As shown in [15], this gauge structure plays a central role in defining the correct prequantum operators of Geometric Quantization, leading to eq.(A18) and revealing a deep relationship between gauge invariance and QM.

Finally, the particle’s internal clock can be viewed as a covariant realization of a continuous cellular automaton with Compton period $T_C$, rather than a discrete dynamics at the Planck scale. This suggests that the correspondence between deterministic cyclic dynamics and ordinary QM suggested by ’t Hooft may be implemented in an exact relativistic form within the ECT framework [71,72].

Appendix A.5. Stokes, Bohr-Sommerfeld and WKB Approximation — [11,14]—

The PBCs of eq.(19) applied to the generic interacting solution ${\varphi }_{p^{\prime }}^{\prime }\left(x\right)$, eq.(22), directly implies a generalization of the Bohr-Sommerfeld (WKB) condition — single valued version of Stokes’ lemma:

$${\oint }_{T^{\prime \mu }\left(x\right)}{p_n^{\prime }}_{\mu }\left(x\right)d{x}^{\mu }=2\pi n.$$

(A19)

A large class of quantum phenomenology follows, such as the Bohr energy levels of the atomic orbitals — closed orbits. The fine structure of the atomic orbitals then follows by considering the Larmon and Thomas precession, discussed above, where the gauge field necessary to guarantee the PH in each point is the emitted EM atomic spectrum.

Appendix A.6. Dirac Monopoles, Superconductivity — [5,10] —

If we consider the minimal substitution in the above Bohr-Sommerfeld condition we obtain the correct framework to interpret EM phenomena such as Aharonov–Bohm effect or Landau levels.

In the pure gauge we immediately find the quantization of the Dirac string

$$e\oint d{x}^{\mu }{A}_{\mu }\left(x\right)=2\pi n,$$

(A20)

and, from the spatial part, the Dirac quantization of magnetic monopoles

$$e\oint d\mathbf{x}·{\mathbf{A}}_i\left(x\right)=e{\int }_S\mathbf{B}·d\mathbf{S}=e{\Phi }_B=2\pi n$$

(A21)

where ${\Phi }_B=4\pi g$ and g is the gyromagnetic factor. In [5,10] we have proven that all the fundamental aspects of superconductivity and Josephson effect directly follow from the first physical principle of single-valued PH, without invoking any empirical model such BCS.

Appendix A.7. Graphene Physics as General Relativity Laboratory — [9] —

Graphene provides a clean arena for ECT and our geometrodynamical view of EM field. Rolling a graphene sheet into a carbon nanotube compactifies one dimension: massless Dirac electrons acquire an effective mass fixed by the Compton relation with the tube circumference (internal clock $S^1$). Carbon nanotubes are thus interesting, from the point of view of ECT, because they allow to rescale the ultrafast internal clock of electrons from the Compton time, $\sim {10}^{-21}$ sec, to a time scale accessible to modern timekeepers — offering the possibility of an indirect experimental access to “possible new physics beyond QM”. Motion along the axis is given by induced spatial periodicities; with ECT-on-lattice this correctly reproduces graphene bands and electrical properties.

Lattice deformations act as an emergent connection whose holonomy enters into the equations as a pseudo-EM field. In ECT terms this is the local twist of the intrinsic clock, i.e.twisted PBCs, producing pseudo-Landau levels and Aharonov–Bohm–like band shifts. Thus graphene pseudomagnetism realizes, in condensed matter, the boundary-condition/holonomy origin of EM investigated in this paper.

Appendix A.8. Holography and AdS/CFT Correspondence — [12,13] —

We have already mentioned that ECT provides a sort of holographic description in which the dynamics are dually encoded on the modulated boundary $T^{\prime \mu }\left(x\right)$ of the theory. We add that, as proven in [12,13], the dualism between classical geometry XD and 4D quantum behavior, in particular through the VXD formalism, effectively reproduces basic aspects of the the AdS/CFT correspondence and gauge-gravity duality, [12,13]. This correspondence is realized in terms of the VXD formalism, applied in the main text to infer the YM kinetic term from the Ricci tensor, in a purely 4D reinterpretation of the KK “miracle”.

Finally, ECT can be read as a minimal string theory, where the non-compact world-sheet parameter is dispensed, leaving only the compact world-sheet parameter $S^1$, characterizing string theory. In fact, this closed proper parameter alone, if identified with the particle’s internal clock $S^1$, is sufficient to describe particles causal and local dynamics in a consistent purely 4D spacetime — avoiding the problematic XD of standard string theory — as seen in this paper.