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Relativistic Beating of Compton Clocks, Action Discretization, and the Emergence of Planck's Constant

Jonatan P. camargo

Published Mar 10, 2026• 0 citations
Publisher

Abstract

We investigate the relationship between phase interference associated with action, the relativistic dynamics of proper time, and the structural origin of action quantization. We start from the general principle of interferential compatibility of action introduced in previous works, in which the dynamic phase of a system is given by $\phi=S/\hbar$ and phase coincidences between distinct evolutions occur only when $\Delta S = 2\pi n\hbar$. We show that when this formalism is specialized to the relativistic action of a free particle $S=-mc^{2}\tau$, the phase evolution naturally implies an internal frequency $\omega_C = mc^{2}/\hbar$, corresponding to the Compton clock. When two systems evolve with distinct proper times, their Compton phases diverge, producing a relativistic phase beating. We demonstrate that the interferential coherence condition $\Delta\phi = 2\pi n$ defines a discrete sequence of synchronization instants and directly implies the relation $mc^{2}\Delta\tau = nh$, showing that discrete action increments emerge as a geometric consequence of the relativistic periodicity of the phase. We also show that the used phase coincides exactly with the rest phase of the Dirac equation solutions, establishing the connection with the relativistic dynamics of fields. The beating between Compton clocks defines an operational discretization of the action that can be interpreted as a discrete sum, whose continuous form is recovered in the limit of almost perfect synchronization between reference frames. We also discuss the relationship of this formalism with the notion of the internal clock introduced by de Broglie and with the interferential structure of relativistic matter interferometry. In this framework, Planck's constant emerges as the geometric invariant associated with the first closure of the relativistic beating between Compton clocks. Quantization can then be interpreted as a coherence condition between relativistic phases, establishing a direct bridge between quantum interference, proper time relativistic dynamics, and the variational principle of stationary action.

DOI
10.5281/zenodo.18931690

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We investigate the relationship between phase interference associated with action, the relativistic dynamics of proper time, and the structural origin of action quantization. We start from the general principle of interferential compatibility of action introduced in previous works, in which the dynamic phase of a system is given by ϕ=S/ℏ\phi=S/\hbarϕ=S/ℏ and phase coincidences between distinct evolutions occur only when ΔS=2πnℏ\Delta S = 2\pi n\hbarΔS=2πnℏ. We show that when this formalism is specialized to the relativistic action of a free particle S=−mc2τS=-mc^{2}\tauS=−mc2τ, the phase evolution naturally implies an internal frequency ωC=mc2/ℏ\omega_C = mc^{2}/\hbarωC​=mc2/ℏ, corresponding to the Compton clock. When two systems evolve with distinct proper times, their Compton phases diverge, producing a relativistic phase beating. We demonstrate that the interferential coherence condition Δϕ=2πn\Delta\phi = 2\pi nΔϕ=2πn defines a discrete sequence of synchronization instants and directly implies the relation mc2Δτ=nhmc^{2}\Delta\tau = nhmc2Δτ=nh, showing that discrete action increments emerge as a geometric consequence of the relativistic periodicity of the phase. We also show that the used phase coincides exactly with the rest phase of the Dirac equation solutions, establishing the connection with the relativistic dynamics of fields. The beating between Compton clocks defines an operational discretization of the action that can be interpreted as a discrete sum, whose continuous form is recovered in the limit of almost perfect synchronization between reference frames. We also discuss the relationship of this formalism with the notion of the internal clock introduced by de Broglie and with the interferential structure of relativistic matter interferometry. In this framework, Planck's constant emerges as the geometric invariant associated with the first closure of the relativistic beating between Compton clocks. Quantization can then be interpreted as a coherence condition between relativistic phases, establishing a direct bridge between quantum interference, proper time relativistic dynamics, and the variational principle of stationary action.
The shape of the classical Lagrangian L=T−V L = T - V L=T−V is universally used in analytical mechanics and in the variational formulation of physics. Despite its conceptual centrality, this structure is historically introduced as a working hypothesis, not usually being derived from more fundamental principles. In this work, we present a systematic derivation of this form from a minimal interferential structure based on the action phase. We start from three physical hypotheses widely observed in interferential phenomena: (i) the existence of an accumulative quantity associated with trajectories, identified as action SSS; (ii) the dependence of physical amplitudes on phase factors of the type eiS/ℏe^{iS/\hbar}eiS/ℏ; and (iii) the possibility of parametrizing evolutions by monotonic proper parameters. We initially show that action differences between systems generate isolated episodic coincidences, implying an operational discretization of interferential compatibility events. These events naturally organize into a discrete network of interactions. Under minimal variational hypotheses—global phase invariance, adjacency locality, and flux conservation—we demonstrate that the structural operator associated with the network coincides with the combinatorial Laplacian K=D−A K = D - A K=D−A, where AAA is the adjacency matrix and DDD the degree matrix. The linear dynamics of complex amplitudes on this network, imposing linearity, unitarity, and locality, leads to a unitary generator of the form H=αK+diag(V) H = \alpha K + \mathrm{diag}(V) H=αK+diag(V) \cite{vrev}. In the continuous limit, the network Laplacian converges to the differential operator −∇2-\nabla^2−∇2, recovering the effective Schrödinger equation. The semiclassical decomposition ψ=a eiS/ℏ \psi = a\,e^{iS/\hbar} ψ=aeiS/ℏ leads, to leading order in ℏ\hbarℏ, to the Hamilton-Jacobi equation ∂tS+(∇S)22m+V=0, \partial_t S + \frac{(\nabla S)^2}{2m} + V = 0, ∂t​S+2m(∇S)2​+V=0, from which we identify the effective Hamiltonian H=p22m+V H = \frac{p^2}{2m} + V H=2mp2​+V. Applying the Legendre transform, we then obtain the Lagrangian L=12mx˙2−V, L = \frac{1}{2}m\dot{x}^2 - V, L=21​mx˙2−V, showing that the T−VT-VT−V decomposition emerges as a direct consequence of the phase's interferential structure and the spectral dynamics of the network operator. As a physical test and validation of the formalism, we extend the result to the relativistic regime, identifying the action phase with proper time. We show that the local term of the network generator acts as a modifier of the effective spacetime metric, naturally leading to the isotropic form g00≈1+2Φ/c2g_{00} \approx 1 + 2\Phi/c^2g00​≈1+2Φ/c2. Under the stationary phase condition for null geodesics, we derive the exact gravitational refractive index. This produces the Einstein light deflection angle (4GM/c2b4GM/c^2b4GM/c2b) purely as a consequence of phase curvature, without postulating the General Relativity field equations. The central result of the work is that classical mechanics (L=T−VL=T-VL=T−V) and the geometric structure of gravitation do not need to be assumed as independent postulates, but emerge simultaneously as effective limits of a variational architecture based on phase interference, episodic discretization, and spectral dynamics in networks.
The divergence of proper times between distinct physical systems, predicted by Special Relativity, naturally leads to a relative phase evolution between the systems. When this evolution is described by the fundamental relationship between action, phase, and proper time, ϕ=S/ℏ=ω0τ\phi = S/\hbar = \omega_0 \tauϕ=S/ℏ=ω0​τ, the phase difference grows with a beat frequency that determines the instants of interferential coincidence between the systems. The resolution of this condition leads to a discrete sequence of synchronization events tn=nπ/Ωt_n = n\pi/\Omegatn​=nπ/Ω, which define the instants at which phase coherence occurs between the reference frames. In this work, we investigate the effect of introducing a linear time-invariant operator—specifically an RC circuit—into the observation process of a system whose phase dynamics are described by this relativistic structure. Using the spectral formalism based on the Laplace Transform, we demonstrate that the circuit is characterized by the transfer function HC(s)=sRC1+sRCH_C(s)=\frac{sRC}{1+sRC}HC​(s)=1+sRCsRC​, whose frequency response introduces an additional phase ϕC(Ω)=arctan⁡(1/ΩRC)\phi_C(\Omega)=\arctan(1/\Omega RC)ϕC​(Ω)=arctan(1/ΩRC) into the processed signal. It is shown that this additional phase modifies the interferential coincidence condition, leading to the relationship Ωtn′+ϕC=nπ, \Omega t_n' + \phi_C = n\pi ,Ωtn′​+ϕC​=nπ, whose solution defines the new synchronization instants tn′=nπ−ϕCΩ. t_n'=\frac{n\pi-\phi_C}{\Omega}. tn′​=Ωnπ−ϕC​​. This result demonstrates that the spectral operator introduced into the observation process does not alter the beat frequency that determines the synchronization periodicity, but it produces a constant translation of the sequence of phase coincidence events. We also analyze the consequences of this shift on the accumulated dynamic evolution of a system characterized by an effective frequency ωB(τ(t))\omega_B(\tau(t))ωB​(τ(t)). It is shown that the observed value of the accumulated evolution at the synchronization instant becomes dependent on the phase introduced by the spectral operator, while the periodicity of the sequence of events remains unchanged, ensuring the preservation of the causal structure. The obtained results establish a formal connection between three mathematical structures traditionally treated separately: the phase evolution associated with relativistic proper time, the theory of linear time-invariant systems, and the spectral description of electronic circuits. The developed formalism shows that the temporal position of interferential synchronization events can depend on the parameters of the spectral operator introduced into the observation system, while preserving the causality conditions and the relativistic structure of the temporal evolution.
This short note proposes a structural interpretation of relativistic time dilation in terms of the dynamical phase associated with the action. The starting point is the general relation ϕ=Sℏ, \phi=\frac{S}{\hbar}, ϕ=ℏS​, and, in the case of a free massive particle, the minimal relativistic action S=−mc2τ. S=-mc^2\tau. S=−mc2τ. It follows immediately that the dynamical phase is proportional to the proper time, ϕ=−mc2ℏτ=−ωCτ,ωC=mc2ℏ. \phi=-\frac{mc^2}{\hbar}\tau=-\omega_C\tau, \qquad \omega_C=\frac{mc^2}{\hbar}. ϕ=−ℏmc2​τ=−ωC​τ,ωC​=ℏmc2​. Thus, the proper time can be interpreted as a geometric parameterization of the phase of the relativistic action. It is then shown that the time dilation observed in relative reference frames can be read as a modification of the accumulation rate of the dynamical phase when the geometry of the worldline is altered. This work does not intend to fully derive special relativity nor substitute its geometric formulation. Its objective is more restricted: to formally register that the relativistic structure of proper time admits a natural interpretation as the phase dynamics of the action. Finally, it is indicated that the truly new element of the broader program is not only in this reinterpretation, but in the condition ΔS=2πnℏ \Delta S=2\pi n\hbar ΔS=2πnℏ as a structural criterion of interferential compatibility between dynamical systems.
We present a variational formulation in which the phase is identified as the operational manifestation of the accumulated action, establishing a structural unification between different physical regimes. It is shown that the fundamental phase of any admissible system is given by ϕ=Sℏ, \phi = \frac{S}{\hbar}, ϕ=ℏS​, from which it follows that the frequency — and, therefore, the operational content of time — emerges directly from the rate of action accumulation, ω=S˙ℏ. \omega = \frac{\dot S}{\hbar}. ω=ℏS˙​. The analysis of the relative phase evolution between independent systems leads to the demonstration that interferential coherence is, in general, episodic, being characterized by a discrete set of instants of compatibility. This results in a precise formulation of the physical interaction as Eint=Eesp∩Eϕ, E_{\mathrm{int}} = E_{\mathrm{esp}} \cap E_\phi, Eint​=Eesp​∩Eϕ​, in which the occurrence of observable events is filtered by phase compatibility. It is further shown that massive and massless regimes correspond to different parameterizations of the same phase structure, eliminating the need for distinct ontologies for these cases. The introduction of linear classical operators, such as macroscopic electromagnetic systems, allows us to demonstrate that such operators act exclusively on the observed phase, ϕobs=ϕfund+ϕC, \phi_{\mathrm{obs}} = \phi_{\mathrm{fund}} + \phi_C, ϕobs​=ϕfund​+ϕC​, shifting the synchronization instants without altering the underlying variational dynamics. As a result, a rigorous distinction is established between the structural level and the observational level: the action governs the fundamental dynamics, while the observed phase corresponds to its operational projection under reading transformations. Finally, the possibility of an extended variational framework in which the experimental apparatus is incorporated into a global functional StotS_{\mathrm{tot}}Stot​ is discussed, leaving this construction as a program for future work.
This work develops a phase-based formulation of relativistic wave dynamics, demonstrating that the Klein--Gordon equation emerges naturally from the foundational assumption of intrinsic phase periodicity in material fields. Mapping the phase directly onto the classical action, we postulate that localized excitations possess an invariant rest-frame oscillation governed by a proper frequency ω0ω_0ω0​. This physical condition establishes an operational mass-frequency relation, m=ℏω0/c2m = \hbar ω_0 / c^2m=ℏω0​/c2, without requiring rest energy as an independent, axiomatic input. We show that the Klein--Gordon equation arises as the minimal local, linear, Lorentz-invariant field equation compatible with this internal phase structure. Within this framework, mass acts as an intrinsic frequency scale governing wave propagation, and relativistic kinematics is fully recovered as a structural consequence of phase coherence. This approach provides a unified wave-mechanical interpretation where particle dynamics maps onto the group velocity of dispersive wave packets, offering an intuitive account of free propagation, dispersion, and tunneling across potential barriers.
Starting from only three ingredients: (i) variational formulation with cumulative action, (ii) interferential dependence on SSS, and (iii) PFu's episodic compatibility theorem, this paper assumes a single additional hypothesis of the universality of the action constant (hhh) and a minimal hypothesis of macroscopic stability, and derives mathematical consequences in an axiomatic and constructive manner, without introducing new fields, potentials, or \textit{ad hoc} phenomenological terms. Three main results are obtained. First, it is demonstrated that the existence of a universal action constant hhh is a structural requirement for interferential consistency among multiple systems: the phase ϕ\phiϕ must be treated as a global structure, defined modulo hhh, common to all admissible processes. Second, it is shown that dynamic compatibility between systems can only occur at discrete events of phase coincidence, characterized by δϕ=0\delta\phi = 0δϕ=0; the observable dynamics are, therefore, reconstructible as sampling over a discrete sequence of compatibility events, whose density is proportional to the rate of variation of SSS (abstract relativistic sampling theorem). Third, under hypotheses of invariance, extensivity, and coercivity, a global functional I\mathcal{I}I is constructed that measures the compatibility stability of a configuration of systems, and it is proven that, in the infrared regime, it is equivalent to a quadratic spectral functional Q=⟨ψ,Aψ⟩Q = \langle \psi, \mathcal{A} \psi \rangleQ=⟨ψ,Aψ⟩, where A\mathcal{A}A is a positive self-adjoint operator that encodes the compatibility structure. In the continuous limit, when A\mathcal{A}A admits a realization as an effective elliptic operator and its spectrum is described by a spectral density ρ(λ)\rho(\lambda)ρ(λ), the abstract functional I\mathcal{I}I is rewritten as a geometric functional of the PESG type, but now interpreted as a consequence of universal action and episodic compatibility, rather than as an independent geometric principle.