Momentum Primacy - A Deeper Foundation from which Physical Limits Emerge

This paper shows that taking Momentum (four-momentum) as the primitive concept of physics resolves several weaknesses in the foundations of Relativity, extends its explanatory power to Quantum Mechanics, and reveals two new physical limits.

Part I rebuilds Relativity on two primitives: Momentum and a minimal spacetime arena carrying succession and extension. Proper time and proper distance are defined from Momentum through E = hν and p = h/λ; Minkowski spacetime and the Momentum expression follow from the symmetries of the universe; mass becomes the Minkowskian norm of Momentum; events are constructed by integrating Momentum over proper time. The resulting framework, Momentum Primacy, removes the circular definitions of mass and proper time and turns the clock postulate into a law.

Momentum Primacy excludes any return to one’s own past, together with the causality-violating geometries themselves. It thereby secures the Causality Principle, a fundamental principle whose possible violation is a major weakness of the standard formulation, protected there only by an external conjecture (Hawking’s chronology protection).

In Quantum Mechanics, Momentum Primacy gives a common origin to both Heisenberg relations, position–momentum and energy–time. Momentum and events form conjugate pairs whose invariant contraction P·M/ℏ is the relativistic phase. The time spread  thus acquires a direct meaning as the temporal localization of an event, and the remaining asymmetry between the two relations, expressed by Pauli’s theorem, traces back to the future orientation of Momentum, P⁰ > 0.

Part II proposes that Momentum dynamics possesses an intrinsic discrete structure at the microscopic level, giving rise to two fundamental limits.

Conjecture 1 sets a proper-time quantum Δτ₀ = h/(mc²), the minimum interval between two distinguishable changes of the Momentum norm. This conjecture is supported by the Margolus–Levitin theorem and corroborated by quantum jumps in single trapped ions and by strong-interaction timescales. De Broglie’s internal clock then acquires a physical meaning as the rhythm of Momentum changes, each quantum being one cycle of the relativistic proper phase and one quantum of action h.

Conjecture 1 also brings an axiomatic gain: the Momentum expression follows directly for inertial and accelerated entities alike, without the hypothesis of locality. Its limit of validity follows as well, an acceleration length c²/a large compared with the Compton wavelength, a condition amply satisfied in the muon storage-ring experiment, in agreement with the measured lifetime dilation.

Conjecture 2 sets a maximum four-force ‖ ‖ = m²c³/h and expresses a condition of continuity. Below this threshold, the succession of discrete Momentum states approximates a differentiable trajectory, and each state can be inferred from the previous one. When the four-force reaches it over several consecutive proper-time quanta, no continuous trajectory exists, predictability is lost, and the state undergoes a structural transition. Sustained predictability is precisely what classical objectivity requires in Zurek’s analysis of decoherence, whose progressive character has been observed directly (Brune et al. 1996).

Conjecture 2 is corroborated quantitatively: the four-force on a neutron at the Tolman–Oppenheimer–Volkoff limit reaches the order of ‖ ‖ , as does the force exerted on an electron by the Schwinger critical field. For matter containing quarks, the formation of a singularity in gravitational collapse loses its foundation, since the differentiable trajectories underlying the classical extrapolation break down at a finite scale.

Both conjectures are testable with present or near-term experiments. One test has already been confronted with Particle Data Group data: a lifetime threshold for composite states, which holds as a graded indicator of particle status across nearly five orders of magnitude.

Both limits open new perspectives for quantum gravity and cosmology, developed in the companion volume Momentum Primacy in Cosmology — A Common Origin for Dark Matter and Dark Energy.