In the Action--Phase program, physical dynamics are formulated from the action $S$ as the primary structure, with the phase $\phi = S/\hbar$ playing the role of the universal operational variable. Previous works have established the emergence of quantum dynamics, geometric structure, and gauge interactions as different modes of action upon the phase, as well as the closure of the relationship between spectrum, energy, and momentum. However, it remains to formulate intrinsically the notion of an observable and the measurement process, without the introduction of independent postulates. In this work, we show that this gap can be naturally filled through the spectral theory of self-adjoint operators. In particular, we demonstrate that physical observables are functionals of the spectral measure associated with the state and the relevant structural operator, while measurement corresponds to the effective restriction of this measure to dominant spectral regions. In this framework, the Born rule is recovered as a particular case in the discrete spectrum regime, while in more general situations, physical results are described by spectral regions with finite width. The proposed formulation eliminates the need for additional axioms for observables and measurement, unifying the dynamical and operational structure of quantum mechanics within the same fundamental principle based on action.
In the Action--Phase program, physical dynamics are formulated from the action S as the primary structure, with the phase ϕ=S/ℏ playing the role of the universal operational variable. Previous works have established the emergence of quantum dynamics, geometric structure, and gauge interactions as different modes of action upon the phase, as well as the closure of the relationship between spectrum, energy, and momentum. However, it remains to formulate intrinsically the notion of an observable and the measurement process, without the introduction of independent postulates. In this work, we show that this gap can be naturally filled through the spectral theory of self-adjoint operators. In particular, we demonstrate that physical observables are functionals of the spectral measure associated with the state and the relevant structural operator, while measurement corresponds to the effective restriction of this measure to dominant spectral regions. In this framework, the Born rule is recovered as a particular case in the discrete spectrum regime, while in more general situations, physical results are described by spectral regions with finite width. The proposed formulation eliminates the need for additional axioms for observables and measurement, unifying the dynamical and operational structure of quantum mechanics within the same fundamental principle based on action.
We present a structural formulation of physical dynamics based on discrete unitary propagation, in which the time evolution is directly described by a unitary operator W^=f(H^) associated with a self-adjoint structural operator H^. Starting from the relationship between action and phase, ϕ=S/ℏ, and the variational construction of structural operators, we show that the propagation can be written exactly as the action of a bilocal kernel W(x,y) on the state, in continuity with recent developments of the Action-Phase program. We show that the spectral decomposition of H^ induces a diagonalization of the unitary propagation, in which the state evolution becomes multiplicative in the eigenvector basis of the structural operator. Consequently, the spectral structure organizes the propagation modes of the system, providing a description in which the dynamics is expressed in a decoupled manner between modes. It is evidenced that the distinction between nodal representation and spectral representation emerges naturally from the operator structure, corresponding to different decompositions of the same state in distinct bases of the same Hilbert space. In the case of translational invariance, the decomposition into Fourier modes is recovered as a particular situation. The results presented do not constitute a complete reconstruction of quantum mechanics nor a general gravitational theory. In particular, aspects of measurement, multiparticle structures, or complete covariant formulations are not treated. The objective of this work is more restricted and precise: to establish a formulation in which the dynamics is determined by the propagation operator and organized by its spectral decomposition, providing a structural layer from which more complete physical descriptions can be developed.
Spectral Mechanics of Phase is presented as a variational formulation in which energy, dynamics, and observables emerge from a spectral structure associated with the phase ϕ=S/ℏ. Starting from the variational principle and spectral functional calculus, it is demonstrated that unitary evolution can be constructed via the Cayley transform of a self-adjoint operator, leading to a nonlinear relationship between energy and spectrum: \begin{equation} E(\lambda) = \hbar \frac{2}{\Delta t}\arctan(\tau \lambda). \end{equation} It is shown that standard quantum mechanics is recovered as a low-energy limit, in which the spectral relationship becomes approximately linear. In contrast, in the high-energy regime, the theory predicts energy saturation and spectral compression, arising from the limitation of the spectral phase. Applications to canonical systems, including the harmonic oscillator, infinite well, and hydrogen atom, show the presence of systematic corrections in energy levels and transition frequencies. In particular, modifications in atomic spectral series are derived that preserve the known structure in asymptotic regimes but introduce controlled deviations at higher orders. Order-of-magnitude analysis and comparison with spectroscopic data indicate that the theory is compatible with current experimental results, while providing explicit criteria for falsification. An experimental protocol based on low-energy transitions is proposed to distinguish the predictions of Spectral Mechanics of Phase from those of quantum electrodynamics. The presented formulation reproduces the established results of quantum mechanics in its domain of validity, while introducing a more general spectral structure, derived directly from the action, without the need to postulate spectral linearity or operators as primitive entities.
This work addresses the structural problem of defining physical observables in regimes where the spectral decomposition of self-adjoint operators is not purely discrete. In particular, it is demonstrated that, outside the ultraviolet regime, the identification between mass and eigenvalues ceases to be consistent when the associated spectral measure has a continuous component. Starting exclusively from minimal spectral hypotheses -- self - adjoint operator, resolvent, spectral measure, and its density -- it is established, without resorting to ad hoc hypotheses, that the observable physical structure is determined by the spectral measure associated with the state. As a consequence, it is proven that the physical dimension of a system is not given by the dimension of the Hilbert space nor by the cardinality of the spectrum, but by the structure of the dominant spectral regions of the density, defining the notion of effective spectral dimension. In this framework, physical observables emerge as functionals of the spectral measure. In particular, mass is characterized as the first-order spectral moment, while the spectral width quantifies the dispersion associated with the energy distribution. It is further shown that dominant spectral regions act as effective degrees of freedom, providing a distributed description of the dynamics. The developed formalism recovers the discrete limit of quantum mechanics, is compatible with the spectral representation of quantum field theory, and admits a natural interpretation in terms of scale flow. Explicit falsifiability criteria are also established, and the structural limitations of the proposed framework are delineated. The central result is the formulation of a principle: physical dimension and observables emerge from the effective spectral structure, implying the necessary substitution of pointlike descriptions with distributed descriptions in non-discrete regimes.
The objective of this work is to investigate whether the position--momentum duality can be reformulated, without ad hoc hypotheses and without circular reasoning, as a direct consequence of the spectral structure of bilocal unitary propagation. We start from the architecture already established in the Action--Phase program: the physical phase is given by ϕ=S/ℏ; self-adjoint structural operators organize the dynamics on a discrete lattice; such operators admit interpretation as unitary filters via Cayley realization; and the discrete unitary evolution can be written exactly as propagation by a bilocal kernel W(x,y). The central question is to determine under what hypotheses this bilocal kernel can be represented as a function of a self-adjoint structural operator, W^=f(H^),H^=H^†, and how this structure implies the existence of an intrinsic spectral basis in which the propagation becomes diagonal. It is shown, step by step, that the spectral decomposition of H^ induces a corresponding decomposition of the kernel, W(x,y)=∑nf(λn)ϕn(x)ϕn∗(y), so that the unitary evolution begins to act multiplicatively on the spectral modes, Ψn(t+Δt)=f(λn)Ψn(t), without the introduction of additional structures. In the homogeneous case, it is demonstrated that the eigenfunctions of the structural operator are discrete Fourier modes and that the continuous spectral parameter coincides with the usual wavenumber. In this regime, the spectral representation exactly reproduces the momentum sector of standard quantum mechanics. Outside this case, ``momentum'' is reinterpreted more generally as the spectral coordinate of the operator generating the propagation. The result of this stage is, therefore, conceptually precise: the position--momentum duality does not need to be postulated as an ontological duplication, but emerges as a duality between the nodal basis and the spectral basis of the same structural operator that governs the unitary propagation. The work does not intend, at this stage, to completely reconstruct the theory of canonical observables, nor to definitively resolve multiparticle systems or formulate a general covariant theory. Its role is to establish the minimum layer in which the spectral structure of propagation provides the mathematical origin of position--momentum duality, opening a controlled continuation program for these problems.
We develop a spectral formulation of physical dynamics based on the action phase ϕ=S/ℏ, in which the evolution is governed by Cayley-type unitary operators and organized through the resolvent structure of the underlying structural operator. We show that the propagation can be described by a bilocal kernel associated with the resolvent, whose spectral decomposition defines independent modes of evolution. In this formalism, physical connectivity emerges as an intrinsically nonlocal property of the resolvent kernel, and physical states can be interpreted as stabilized spectral modes. We demonstrate that the existence of massive states is conditioned on the presence of a global phase structure, encoded by holonomy, which prevents the variational collapse of the structural coupling. We also show that phase factors associated with gauge connections admit an interpretation as a manifestation of the accumulated phase of the resolvent under discrete reduction. From this structure, we derive finite-dimensional effective operators as infrared spectral projections of the global operator, establishing the emergence of tridiagonal operators as natural approximations of low-energy dynamics. Finally, we demonstrate that the limitations observed in truncated effective models stem from the suppression of the tail of the bilocal kernel, which dominantly affects the description of highly delocalized states. In particular, the discrepancy associated with the ground state is interpreted as a structural consequence of the loss of global connectivity induced by spectral truncation. These results allow formulating the dynamics as a process of spectral filtering of the action phase, in which structural operators act as unitary filters and physical states emerge as stabilized spectral modes of the resolvent. This perspective leads to a framework we call Spectral Mechanics of Phase.
We present a structural formulation of physical dynamics in which time evolution is directly described by a unitary propagation operator, rather than being primarily introduced by continuous differential equations. Starting from the relationship between action and phase, ϕ=S/ℏ, and the variational construction of self-adjoint structural operators, it is shown that propagation can be exactly expressed through a bilocal kernel W(x,y) associated with an operator W^=f(H^), with H^=H^†. We demonstrate that the structural homogeneity of the kernel, characterized by the condition W(x,y)=W(x−y), implies the spectral diagonalization of the propagation operator. In this regime, the evolution becomes decoupled between modes, and the absence of spectral mixing provides a minimal structural expression of conservation. It is further shown that the off-diagonal structure of the bilocal kernel completely determines the connectivity of propagation between states, allowing the effective locality of the dynamics to be characterized in terms of the support of W(x,y). The analysis establishes, in a controlled manner, the relationship between the bilocal description and the Lagrangian formulation, indicating that the operatorial structure replaces, at the considered level, the need for a continuous sum over trajectories, without excluding its recovery as an effective limit in appropriate regimes. The obtained results show that dynamics can be entirely organized in terms of the propagation operator and its spectral decomposition, in which the nodal representation and the spectral representation emerge as complementary descriptions of the same structure. The scope of this work is strictly structural: it does not propose a complete reconstruction of quantum mechanics, nor a general gravitational theory, nor a formulation for multiparticle systems or measurement processes. The objective is to establish a minimal layer in which physical propagation is encoded by the unitary operator and its spectral structure.