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Spectral Structure of Physical Propagation: Modes, structural selection, and organization of dynamics

Jonatan P. camargo

Published Apr 13, 2026• 0 citations
Publisher

Abstract

We present a structural formulation of physical propagation in which dynamics are described by a unitary operator $\hat W=f(\hat H)$, associated with a self-adjoint structural operator $\hat H$, and by its bilocal representation $W(x,y)$. Time evolution is formulated as the iterative application of this operator, without assuming differential equations as a fundamental starting point. It is shown that the spectral decomposition of $\hat H$ induces a complete organization of the evolution into independent eigenmodes, each governed by a spectral factor $f(\lambda_n)$. From this structure, it is demonstrated that the bilocal propagation kernel is entirely reconstructed from the spectrum and the eigenvector basis of the structural operator. In the regime of translational homogeneity, it is proven that the propagation operator is diagonalizable in a basis compatible with translations, resulting in the absence of mixing between spectral modes. Outside this regime, it is shown that the support of the kernel $W(x,y)$ completely determines the connectivity of propagation, while its off-diagonal structure encodes relations between distinct nodal states. Additionally, it is established that quantities defined as functions of the structural operator can be evaluated directly in the spectral basis, providing an intrinsic characterization of the dynamics in terms of the spectrum of $\hat H$. The results obtained do not constitute a complete physical theory nor do they establish a direct identification between the operator's spectrum and specific empirical observables. The objective of this work is to make explicit the structural level at which propagation is organized by unitary operators and by their spectral decomposition, providing a coherent basis for further developments.

DOI
10.5281/zenodo.19555929

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We present a structural formulation of physical propagation in which dynamics are described by a unitary operator W^=f(H^)\hat W=f(\hat H)W^=f(H^), associated with a self-adjoint structural operator H^\hat HH^, and by its bilocal representation W(x,y)W(x,y)W(x,y). Time evolution is formulated as the iterative application of this operator, without assuming differential equations as a fundamental starting point. It is shown that the spectral decomposition of H^\hat HH^ induces a complete organization of the evolution into independent eigenmodes, each governed by a spectral factor f(λn)f(\lambda_n)f(λn​). From this structure, it is demonstrated that the bilocal propagation kernel is entirely reconstructed from the spectrum and the eigenvector basis of the structural operator. In the regime of translational homogeneity, it is proven that the propagation operator is diagonalizable in a basis compatible with translations, resulting in the absence of mixing between spectral modes. Outside this regime, it is shown that the support of the kernel W(x,y)W(x,y)W(x,y) completely determines the connectivity of propagation, while its off-diagonal structure encodes relations between distinct nodal states. Additionally, it is established that quantities defined as functions of the structural operator can be evaluated directly in the spectral basis, providing an intrinsic characterization of the dynamics in terms of the spectrum of H^\hat HH^. The results obtained do not constitute a complete physical theory nor do they establish a direct identification between the operator's spectrum and specific empirical observables. The objective of this work is to make explicit the structural level at which propagation is organized by unitary operators and by their spectral decomposition, providing a coherent basis for further developments.
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\phi=S/\hbarϕ=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)W(x,y)W(x,y) associated with an operator W^=f(H^)\hat W=f(\hat H)W^=f(H^), with H^=H^†\hat H=\hat H^\daggerH^=H^†. We demonstrate that the structural homogeneity of the kernel, characterized by the condition W(x,y)=W(x−y)W(x,y)=W(x-y)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)W(x,y)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.
We present a structural formulation of physical dynamics in which temporal evolution is described by a unitary propagation operator W^=f(H^)\hat W=f(\hat H)W^=f(H^), defined from a self-adjoint structural operator H^\hat HH^. In this context, dynamics is not primarily introduced by differential equations or continuous sums over trajectories but is characterized by the spectral structure of the propagation operator. It is shown that the spectral decomposition of H^\hat HH^ induces an organization of the evolution into independent modes, each governed by a spectral factor f(λn)f(\lambda_n)f(λn​). The unitarity of W^\hat WW^ implies that these factors admit a phase representation, leading to a modal description in terms of spectral phases. It is further demonstrated that frequency emerges as a well-defined spectral function associated with the temporal variation of the phase of each mode. Additionally, it is established that dynamical quantities can be formulated as spectral functions of the eigenvalues of H^\hat HH^, providing a minimal spectral dictionary in which phase, frequency, and dynamical scales are defined intrinsically to the formalism. The results obtained are strictly structural and do not involve the introduction of external phenomenological hypotheses. In particular, this work does not establish a direct identification between eigenvalues and specific physical quantities, nor does it construct a complete theory of interactions or measurement. It is concluded that the spectral structure of the propagation operator provides the minimum level at which dynamics can be organized, establishing a formal basis for the development of future extensions of the formalism toward the description of specific physical systems.
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^)\hat W=f(\hat H)W^=f(H^) associated with a self-adjoint structural operator H^\hat HH^. Starting from the relationship between action and phase, ϕ=S/ℏ\phi=S/\hbarϕ=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)W(x,y)W(x,y) on the state, in continuity with recent developments of the Action-Phase program. We show that the spectral decomposition of H^\hat HH^ 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.
We present a spectral formulation of unitary propagation in which the dynamics is characterized directly by the resolvent structure of a self-adjoint operator. A propagation operator defined by the Cayley transform is considered, whose form allows expressing the evolution in terms of the resolvent of the structural operator, without direct reference to continuous differential equations. It is shown that the bilocal propagation kernel is completely determined by the resolvent, implying that the dynamic connectivity does not reduce, in general, to the local support of the structural operator. In the perturbative regime, a controlled expansion is obtained that characterizes an effective notion of locality, while in the homogeneous regime the evolution is organized by a nonlinear spectral phase associated with the eigenvalues of the operator. From this structure, dynamical quantities such as frequency and group velocity are defined, which emerge as well-defined spectral functions, modulated by nonlinear factors arising from the unitarity of the evolution. Additionally, it is demonstrated that the spectrum responds in a controlled manner to internal perturbations associated with phase parameters, establishing a general mechanism of spectral response. The obtained results are strictly structural and independent of external phenomenological hypotheses. It is concluded that unitary propagation admits a consistent formulation in terms of the resolvent of a self-adjoint operator and its spectral decomposition, providing a minimal level at which connectivity, modal evolution, and response to perturbations are determined internally by the structure of the operator.
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\phi=S/\hbarϕ=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)W(x,y)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^†, \hat W=f(\hat H), \qquad \hat H=\hat H^\dagger, 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^\hat HH^ induces a corresponding decomposition of the kernel, W(x,y)=∑nf(λn) ϕn(x)ϕn∗(y), W(x,y)=\sum_n f(\lambda_n)\,\phi_n(x)\phi_n^*(y), W(x,y)=∑n​f(λn​)ϕn​(x)ϕn∗​(y), so that the unitary evolution begins to act multiplicatively on the spectral modes, Ψ~n(t+Δt)=f(λn)Ψ~n(t), \widetilde\Psi_n(t+\Delta t)=f(\lambda_n)\widetilde\Psi_n(t), 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.
The spectral formulation of dynamics, based on self-adjoint structural operators, allows organizing the evolution of physical systems in terms of independent eigenmodes. However, the spectral decomposition, although mathematically complete, does not provide, by itself, a criterion of physical admissibility. In this work, a variational principle is introduced in the space of structural operators, in the form of a spectral functional whose stationarity condition imposes restrictions on the spectrum. It is shown that physical admissibility must be formulated as a spectral selection problem, in which only specific subsets of eigenvalues are compatible with conditions of structural stability. It is demonstrated that spectral decomposition does not determine the admissibility of the structure, and that the introduction of an independent variational criterion leads to non-trivial relations between the eigenvalues of the operator. As a consequence, it is obtained that admissibility implies spectral restrictions, establishing a direct connection between the operatorial organization of dynamics and underlying variational principles. The scope of the work is strictly structural. A direct identification between eigenvalues and specific physical quantities is not established, nor is a complete phenomenological theory proposed. The objective is to determine the minimum level at which physical selection can be formulated as a consequence of spectral properties and stability criteria.