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Physical Propagation as Spectral Dynamics of Unitary Operators Foundations of a spectral description of propagation

Jonatan P. camargo

Published Apr 9, 2026• 0 citations
Publisher

Abstract

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 $\hat W=f(\hat H)$ associated with a self-adjoint structural operator $\hat H$. Starting from the relationship between action and phase, $\phi=S/\hbar$, 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 $\hat H$ 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.

DOI
10.5281/zenodo.19484075

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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 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 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.
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.
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 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.
In the Action--Phase program, physical dynamics are formulated from the action SSS as the primary structure, with the phase ϕ=S/ℏ\phi = S/\hbarϕ=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.