Wave-particle duality at eleven parameters: the half-sum of the two readings, and the experiments
Abstract
The relativity at eleven parameters describes a body by one element of u(2, 2) read in two ways, the ordinary reading with the position leading and the reverse one with the momentum leading, exchanged by the Weyl reflection η of the algebra through the Planck scale. This paper asks what it means to treat the two readings as equiprobable and to take their half-sum. The reflection is the Cartan involution of the algebra, and the half-sum of an element and its reflection is its even part, the projection on the maximal compact subalgebra, an element whose position and momentum coincide in Planck units and whose mass moment and dilatation charge vanish, and which is not in general the element of a body; a formula that preserves the orbit is the quarter turn, conjugation by the square root of the reflection, and the linear, quadratic and exponential formulas between the readings are sorted by the symmetric pair, the weighting of the two readings being a choice of scale; the half-sum of a state is the reading-symmetric mixture; the half-sum of the energy and of its reflection is the conformal Hamiltonian 1 2 (P 0 +K 0 ), the compact generator of the energytime sl(2, R) of the algebra, with the discrete spectrum E 0 + n, which on the canonical pair of ordinary quantum mechanics is half the harmonic oscillator and generates the fractional Fourier family between the position reading and the momentum reading. Equiprobability is a place, not a principle: the reading-symmetric states are Planck-scale states, a micron wave packet being squeezed away from them by some sixty-seven e-folds. In the two readings a body carries a wave of wavevector p/ℏ and one of wavevector x/L 2 P , with angular velocities E/ℏ and t/t 2 P , coinciding on the free worldline at the gravitational time Gm/c 3 , the reverse wavelength at the gravitational radius being the Compton wavelength. On renormalisation the half-sum is the reciprocal method of the companion paper, the loop integral on the momentum sphere whose two hemispheres are images of each other: the logarithms become finite, the powers Planckian and doubled, and no half-sum removes the light-cone singularity or the mass-shell pole, which are physical. Wave-particle duality is the choice of reading, made continuous by the fractional family and realised in Young's experiment by the Fresnel region between the image of the slits and the fringes; the distinguishability of the slits and the overlap of their patterns satisfy D 2 + V 2 ≤ 1 along the family. The reduction of the wave packet is reading-covariant and not derived; the amplitude is what the readings share, the entropies of its two squared moduli bounded by the entropic uncertainty relation. Feynman's formalism in Hamiltonian form, the phase-space path integral, is the readingindependent form of which the two Lagrangian integrals are the two polarisations, and the Hamiltonian of the unified theory is the one Dirac's method gives, a sum of constraints with the compensator gauge fixing the Planck mass. Bell's inequalities are untouched by every reading: the angular momentum and the helicity are what the reflection keeps, the CHSH value of the singlet, 2 √ 2, and Aspect's measured 2.697 are the same in every reading, and the quadrature measurements that the readings are cannot violate a Bell inequality on a Gaussian state.
Related
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BibTeX
@misc{reynaud2026waveparticle,
title = {{Wave-particle duality at eleven parameters: the half-sum of the two readings, and the experiments}},
author = {Patrick .B Reynaud},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05750046v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Wave-particle duality at eleven parameters: the half-sum of the two readings, and the experiments
AU - Patrick .B Reynaud
PY - 2026
DA - 2026/09/15
PB - HAL
UR - https://hal.science/hal-05750046v2
AB - The relativity at eleven parameters describes a body by one element of u(2, 2) read in two ways, the ordinary reading with the position leading and the reverse one with the momentum leading, exchanged by the Weyl reflection η of the algebra through the Planck scale. This paper asks what it means to treat the two readings as equiprobable and to take their half-sum. The reflection is the Cartan involution of the algebra, and the half-sum of an element and its reflection is its even part, the projection on the maximal compact subalgebra, an element whose position and momentum coincide in Planck units and whose mass moment and dilatation charge vanish, and which is not in general the element of a body; a formula that preserves the orbit is the quarter turn, conjugation by the square root of the reflection, and the linear, quadratic and exponential formulas between the readings are sorted by the symmetric pair, the weighting of the two readings being a choice of scale; the half-sum of a state is the reading-symmetric mixture; the half-sum of the energy and of its reflection is the conformal Hamiltonian 1 2 (P 0 +K 0 ), the compact generator of the energytime sl(2, R) of the algebra, with the discrete spectrum E 0 + n, which on the canonical pair of ordinary quantum mechanics is half the harmonic oscillator and generates the fractional Fourier family between the position reading and the momentum reading. Equiprobability is a place, not a principle: the reading-symmetric states are Planck-scale states, a micron wave packet being squeezed away from them by some sixty-seven e-folds. In the two readings a body carries a wave of wavevector p/ℏ and one of wavevector x/L 2 P , with angular velocities E/ℏ and t/t 2 P , coinciding on the free worldline at the gravitational time Gm/c 3 , the reverse wavelength at the gravitational radius being the Compton wavelength. On renormalisation the half-sum is the reciprocal method of the companion paper, the loop integral on the momentum sphere whose two hemispheres are images of each other: the logarithms become finite, the powers Planckian and doubled, and no half-sum removes the light-cone singularity or the mass-shell pole, which are physical. Wave-particle duality is the choice of reading, made continuous by the fractional family and realised in Young's experiment by the Fresnel region between the image of the slits and the fringes; the distinguishability of the slits and the overlap of their patterns satisfy D 2 + V 2 ≤ 1 along the family. The reduction of the wave packet is reading-covariant and not derived; the amplitude is what the readings share, the entropies of its two squared moduli bounded by the entropic uncertainty relation. Feynman's formalism in Hamiltonian form, the phase-space path integral, is the readingindependent form of which the two Lagrangian integrals are the two polarisations, and the Hamiltonian of the unified theory is the one Dirac's method gives, a sum of constraints with the compensator gauge fixing the Planck mass. Bell's inequalities are untouched by every reading: the angular momentum and the helicity are what the reflection keeps, the CHSH value of the singlet, 2 √ 2, and Aspect's measured 2.697 are the same in every reading, and the quadrature measurements that the readings are cannot violate a Bell inequality on a Gaussian state.
ER -