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Position--Momentum Duality as Spectral Diagonalization of Bilocal Propagation A Continuation of the Action--Phase Program

Jonatan P. camargo

Published Apr 6, 2026• 0 citations
Publisher

Abstract

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 \(\phi=S/\hbar\); 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, \[ \hat W=f(\hat H), \qquad \hat H=\hat H^\dagger, \] 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 \(\hat H\) induces a corresponding decomposition of the kernel, \[ W(x,y)=\sum_n f(\lambda_n)\,\phi_n(x)\phi_n^*(y), \] so that the unitary evolution begins to act multiplicatively on the spectral modes, \[ \widetilde\Psi_n(t+\Delta t)=f(\lambda_n)\widetilde\Psi_n(t), \] 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.

DOI
10.5281/zenodo.19433966

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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 develop a spectral formulation of physical dynamics based on the action phase ϕ=S/ℏ\phi = S/\hbarϕ=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.
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.
This work develops the direct continuation of the Action--Phase program following the reinterpretation of structural operators as unitary filters and the derivation of the scalar delay equation. The objective is to establish, without ad hoc introductions, circular reasoning, or undue duplication of degrees of freedom, the minimal bilocal formulation in which discrete unitary propagation is treated as the action of a spatial convolution kernel. Starting from the chain action  ⟹  phase  ⟹  structural operator  ⟹  unitary filter, \text{action} \;\Longrightarrow\; \text{phase} \;\Longrightarrow\; \text{structural operator} \;\Longrightarrow\; \text{unitary filter}, action⟹phase⟹structural operator⟹unitary filter, it is initially shown that the Cayley realization of a self-adjoint structural operator admits an exact rewriting as propagation by a strictly unitary bilocal kernel W(x,y)W(x,y)W(x,y) on a discrete spatial basis. Next, this kernel is identified as the minimal structural variable of the new formulation, strictly distinguishing it from the propagated state Ψ\PsiΨ, the local scalar field Θ(x)\Theta(x)Θ(x), and the quasi-energy Ω(x,k)\Omega(x,k)Ω(x,k), which are now understood at different levels of the theory's physical hierarchy. Under hypotheses of minimal bilocal locality, structural hermiticity, coercivity, minimal differential order, and minimal coupling to the phase state, the variational functional of the kernel is constructed and the corresponding Euler--Lagrange equation is derived. Finally, it is shown that the previous scalar delay theory does not constitute an independent construction but arises as the local/diagonal sector of the bilocal structure obtained here. The result of this stage is, therefore, precise and limited: to establish the minimal bilocal layer of adaptive unitary propagation over a structural network. The work does not intend, at this stage, to provide a complete theory of multipartite entanglement, turbulence, strong gravitational regimes, or covariant extensions. Its role is more fundamental: to provide the explicit variational basis from which such problems can be reformulated and investigated within a unified framework of discrete unitary propagation.