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Resolving the Representation Crisis in Relativistic Quantum Mechanics_ The Relational Event-Coordination Framework

Muhammad yahya

Published Jan 1, 2026• 0 citations
Publisher

Abstract

The Dirac and Foldy-Wouthuysen (FW) representations of relativistic quantum mechanics produce conflicting predictions for fundamental observables like location, spin, and probability density, creating a fundamental stalemate. This representation crisis undermines quantitative results across the field, yet researchers continue to work within restrictive methodologies—low-dimensional models, paraxial approximations, or classical circuit simulations—that mask rather than resolve the underlying mathematical tension. We propose the Relational Event-Coordination (REC) framework, which redefines position as a relational property emerging from a Hilbert space of spacetime events rather than an intrinsic particle operator. Within this framework, Lorentz symmetry acts as a geometric unitary, and observables arise from covariant clock POVMs conditioned against spacetime foliation. The REC framework recovers the Dirac density where covariance and continuity requirements dominate, while mapping to FW variables when classical alignment becomes necessary. Neither representation holds privileged status; both describe the same underlying events under different conditioning regimes. We identify testable predictions involving twisted electron beams in non-inertial frames and high-energy particles in intense laser fields, offering pathways toward empirical resolution of the representation paradox. Keywords: Dirac representation, Foldy–Wouthuysen transformation, relational quantum mechanics, relativistic observables, Event-Hilbert Space, Newton–Wigner localization, POVM, representation crisis

DOI
10.17605/osf.io/qmu3v

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The Dirac and Foldy-Wouthuysen (FW) representations of relativistic quantum mechanics produce conflicting predictions for fundamental observables like location, spin, and probability density, creating a fundamental stalemate. This representation crisis undermines quantitative results across the field, yet researchers continue to work within restrictive methodologies—low-dimensional models, paraxial approximations, or classical circuit simulations—that mask rather than resolve the underlying mathematical tension. We propose the Relational Event-Coordination (REC) framework, which redefines position as a relational property emerging from a Hilbert space of spacetime events rather than an intrinsic particle operator. Within this framework, Lorentz symmetry acts as a geometric unitary, and observables arise from covariant clock POVMs conditioned against spacetime foliation. The REC framework recovers the Dirac density where covariance and continuity requirements dominate, while mapping to FW variables when classical alignment becomes necessary. Neither representation holds privileged status; both describe the same underlying events under different conditioning regimes. We identify testable predictions involving twisted electron beams in non-inertial frames and high-energy particles in intense laser fields, offering pathways toward empirical resolution of the representation paradox. Keywords: Dirac representation, Foldy–Wouthuysen transformation, relational quantum mechanics, relativistic observables, Event-Hilbert Space, Newton–Wigner localization, POVM, representation crisis
Relativistic quantum theory poses a number of conceptual problems over and above the non-relativistic mechanics. From a purely mathematical point of view it is also much more sophisticated. Even the kinematic theory poses considerable difficulties: locality, antimatter, negative energy, and charge are already systematically linked in ways that cannot simply be paraphrased. A further complication is that in the midst of this structure we have what is usually called the Newton-Wigner representation (or, in the case of the Dirac theory, the Foldy-Wouthuysen representation), in which fields and states are no longer covariantly described and where the “local” self-adjoint quantities (NW-local observables) do not obey microcausality, but only satisfy equal-time commutators with respect to a particular inertial frame. If we pass to this represenation, we obtain at a stroke the basic structure of non-relativistic quantum field theory; it is not too hard to descend from that to a many-particle mechanics, and to recover the usual definitions of localization.
This paper is the thirty-first in a sequence developing an interpretive framework that redescribes physical reality as a causally consistent history of information updates rather than as a collection of independently existing objects. It is the third panel — the capstone — of the eighth grouping, a foundational-physics trilogy. Its subject is spacetime localization: where and when a record or system is — its position, orientation, relative phase, and time. The claim is that localization is real but not authoritatively global. There is no Global Frame: no physically privileged reference system that fixes absolute location and orientation, a privileged global simultaneity, a classical frame standing outside quantum theory, and one physically privileged, globally valid perspectival gauge-fixing. Localization is relational: positions and times are defined relative to a declared quantum reference frame (QRF), itself a quantum system that may be superposed and entangled with what it localizes (Giacomini, Castro-Ruiz, & Brukner, 2019; Loveridge, Miyadera, & Busch, 2018). What keeps this objective rather than subjective is a strict typing of cross-frame comparison: a single gauge-invariant relational (Dirac) observable has one invariant prediction, its perspectival representations intertwined by a frame-change map (case A); distinct relational observables form a relationally consistent web (case B); and where two frames share no common refinement their claims are declared incomparable (case C). "Objective" means invariance of the physical predictions under the declared symmetry group and the frame-change groupoid over the admissible frame family, to a declared tolerance — never frame-independence in an absolute sense. The flagship R-A is a dynamics-independent kinematic existence witness: a finite-dimensional discrete-ring model exhibits one physical state and two admissible ideal QRFs under which the localization profile and the entanglement structure differ — a sharp product relative to one particle; an entangled state with a delocalized, mixed marginal relative to another — while the distribution of the same relational observable is preserved. R-B splits into a domain covariance on shared refinements (ideal QRFs compose and agree on shared Dirac predictions) and a model-dependent global-perspective obstruction: where a globally valid relational gauge-fixing fails, as in known Gribov-type models (Vanrietvelde, Höhn, & Giacomini, 2023), no single perspective covers the whole physical state space — which is neither the same as, nor a consequence of, the absence of a privileged frame, and does not preclude a perspective-neutral description. R-C places Paper 8's relational time as the time-translation special case, with its finite-resolution-clock, conditional-covariance, and record–clock qualifications, separating the retired claim (no global clock) from the deferred one (relativistic foliation with dynamical geometry). R-D identifies and transports the frame-representable localization sector of certain Paper-30 grains, partially discharging that paper's grain-coherence deferral while leaving interaction, fragmentation, redundancy, and history-family structure grain-relative. This completes the trilogy's dismantling of the Global Bookkeeper while keeping geometry fixed: the quantum-mechanics–gravity interface is prepared, not entered. Localization is frame-relative but objective: no absolute frame, no globally gauge-fixed perspective claimed beyond what models establish, and no anarchy. ALAC Paper 31. Third panel (capstone) of the eighth grouping — a foundational-physics trilogy (29 Public Quantum Records -> 30 Classicality Without a Global Classicalizer -> 31 Spacetime Localization Without a Global Frame). Part of the Consistency-Constrained Information History (Akashic Ledger Architecture Cosmology) series. Presents no new quantum mechanics; results are interpretive (L2) on imported physics (L1).
We have previously shown that three approaches to relational quantum dynamics—relational Dirac observables, the Page-Wootters formalism and quantum deparametrizations—are equivalent. Here we show that this “trinity” of relational quantum dynamics holds in relativistic settings per frequency superselection sector. Time according to a clock subsystem is defined via a positive operator-valued measure (POVM) that is covariant with respect to the group generated by its (quadratic) Hamiltonian. This differs from the usual choice of a self-adjoint clock observable conjugate to the clock momentum. It also resolves Kuchař's criticism that the Page-Wootters formalism yields incorrect localization probabilities for the relativistic particle when conditioning on a Minkowski time operator. We show that conditioning instead on the covariant clock POVM results in a Newton-Wigner type localization probability commonly used in relativistic quantum mechanics. By establishing the equivalence mentioned above, we also assign a consistent conditional-probability interpretation to relational observables and deparametrizations. Finally, we expand a recent method of changing temporal reference frames, and show how to transform states and observables frequency-sector-wise. We use this method to discuss an indirect clock self-reference effect and explore the state and temporal frame-dependence of the task of comparing and synchronizing different quantum clocks.
We extend the framework of Unified Quantum Mechanics (UQM) to the relativistic domain by formulating the Dirac equation entirely in terms of real-valued fields. In this approach, the complex imaginary unit iii is systematically replaced by the temporal Hilbert transform operator Ht\mathcal{H}_{t}Ht​, which satisfies the anti-involution property Ht2=−I\mathcal{H}_{t}^{2}=-\mathcal{I}Ht2​=−I and acts as a dimensionless isometric operator. The resulting theory is expressed as a coupled system of real-valued equations for a primary field ψ\psiψ and its Hilbert-transformed partner ϕ=Ht[ψ]\phi=\mathcal{H}_{t}[\psi]ϕ=Ht​[ψ], whose evolution is governed exclusively by local differential operators. We demonstrate that all essential algebraic and symmetry structures of relativistic quantum mechanics are preserved within this real-valued formulation. In particular, the full Clifford algebra of the Dirac matrices, Lorentz covariance, and the relativistic energy-momentum relation are recovered exactly. A rigorous isomorphism with the conventional complex Dirac equation is established via the analytic signal representation ψa=ψ−iϕ\psi_{a}=\psi-i\phiψa​=ψ−iϕ, under which the standard U(1)U(1)U(1) gauge symmetry emerges naturally as a global SO(2)SO(2)SO(2) rotation in the real field space. The resulting formulation is therefore operationally and predictively equivalent to the standard complex theory. The conserved scalar density ρ=ψ†ψ+ϕ†ϕ=∣ψa∣2\rho=\psi^{\dagger}\psi+\phi^{\dagger}\phi=|\psi_{a}|^{2}ρ=ψ†ψ+ϕ†ϕ=∣ψa​∣2 is shown to arise as the squared envelope (intensity) of a real physical field and admits a direct interpretation as a local field intensity with well-defined physical dimensions. Probability conservation is recovered through the continuity equation, while normalization in standard quantum mechanics is reinterpreted as a rescaling by the conserved total field. Stability of matter, expressed as energy positivity, is shown to restrict the physical solution space to analytic signals, enforcing a Hilbert-pair structure and eliminating independent complex degrees of freedom. This work demonstrates that the complex structure of relativistic quantum mechanics is not fundamental but emerges from the analytically constrained dynamics of real-valued fields, providing a unified and physically transparent foundation for the Dirac theory.
The problem of the position and spin in relativistic quantum mechanics is analyzed in detail. It is definitively shown that the position and spin operators in the Foldy-Wouthuysen representation (but not in the Dirac one) are quantum-mechanical counterparts of the classical position and spin variables. The probabilistic interpretation is valid only for Foldy-Wouthuysen wave functions. The relativistic spin operators are discussed. The spin-orbit interaction does not exist for a free particle if the conventional operators of the orbital angular momentum and the rest-frame spin are used. Alternative definitions of the orbital angular momentum and the spin are based on noncommutative geometry, do not satisfy standard commutation relations, and can allow the spin-orbit interaction.