We work in a fully discrete framework where physical motions are realised aspaths on an integer spatial display Z3, sampled at integer tick time. Observers arenot primitive frames but realised paths, and comparison is defined operationally asthe direct reading of one path relative to another through the tick–sampled differenceRB|A(t) = XB(t) − XA(t).Relative kinematic information is extracted from finite comparison windows byexact integer arithmetic, without introducing a prior velocity space or a global Lorentzstructure. We formulate multi–observer (three–leg) comparison as an explicit chainingof such path readings on canonical lcm windows and show that, at the discrete tick–path level, the output consists solely of windowed integer displacements and associatedscalar invariants.Within this discrete comparison layer there is no intrinsic notion of rotation. Anonzero Wigner or Thomas rotation angle appears only after imposing a continuousSpecial Relativity reporting map (a “shadow”) and performing Lorentz–group compositionin that shadow space. We finally state a grain–limited resolution boundfor reconstructing any rotation angle from transverse readouts over a lever arm D,clarifying the operational status of such effects in a discrete observer–centred setting.We work in a fully discrete framework where physical motions are realised aspaths on an integer spatial display Z3, sampled at integer tick time. Observers arenot primitive frames but realised paths, and comparison is defined operationally asthe direct reading of one path relative to another through the tick–sampled differenceRB|A(t) = XB(t) − XA(t).Relative kinematic information is extracted from finite comparison windows byexact integer arithmetic, without introducing a prior velocity space or a global Lorentzstructure. We formulate multi–observer (three–leg) comparison as an explicit chainingof such path readings on canonical lcm windows and show that, at the discrete tick–path level, the output consists solely of windowed integer displacements and associatedscalar invariants.Within this discrete comparison layer there is no intrinsic notion of rotation. Anonzero Wigner or Thomas rotation angle appears only after imposing a continuousSpecial Relativity reporting map (a “shadow”) and performing Lorentz–group compositionin that shadow space. We finally state a grain–limited resolution boundfor reconstructing any rotation angle from transverse readouts over a lever arm D,clarifying the operational status of such effects in a discrete observer–centred setting.

Feb 2, 20260 citations

Quantum Mechanical Operators

George H. Duffey

Springer eBooksJan 1, 19840 citations

Quantum Mechanical Operators

WORLD SCIENTIFIC eBooksApr 1, 19990 citations

Wave–Particle Duality as Evolution–Vantage Duality on the Thales Constraint Surface

Elias De Jesus

The Englert–Greenberger wave–particle duality relation V 2 + D2 ≤ 1, where V is fringevisibility and D is which-path distinguishability, is algebraically identical to the Thales con-straint L2 + h2 = R2 under the identification V = 2h (visibility equals twice the Thalesaltitude) and D = 2L (distinguishability equals twice the lateral displacement). We showthat this identification unifies three previously separate frameworks: (i) the hypotenuse rota-tion diagnostic for binary black hole inspiral, where the depth angle θh = arctan(h/L) tracksprocess progress; (ii) the torus vantage formula, where the observer angle α = arccos √aclassifies the measurement; and (iii) the Englert–Greenberger complementarity, where thewhich-path angle θ = arctan(D/V ) = arctan(L/h) quantifies wave–particle balance.The three frameworks are related by the exact identity 2α−θ = 90, which states that theobserver angle α always exceeds half the which-path angle by 45. This encodes a fixed 2:1angular relation: the two angles are linked so that each degree of which-path angle increasecorresponds to two degrees of observer-angle decrease toward equipartition at 45.Applied to laboratory data from entangled nuclear spins in isotopically engineered sili-con carbide (Bourassa et al., 2020), all identities hold to machine precision: the Englert–Greenberger constraint V 2 + D2 = 1 is conserved to 10−16, and 2α − θ = 90 holds to10−14 degrees. Through the structural mapping to effective Englert coordinates, the densitymatrix gives V = 0.990 (near-perfect fringe contrast in the interferometric analogue) andD = 0.139 (minimal which-path information), placing the system near the wave pole atθ = 8.0—consistent with the near-maximal entanglement (S = 0.986 bits) found indepen-dently from the ER = EPR diagnostic. Decoherence maps to which-path information leakingto the spin bath, with CPMG dynamical decoupling extending fringe persistence by 12×.The Born rule P = cos2(θBorn/2), the observer equation a = cos2 α, and the fringe-minimum intensity a = (1 − D)/2 are shown to be the same number: the partition fractionat the observer’s angle on the constraint circle.The Englert–Greenberger wave–particle duality relation V 2 + D2 ≤ 1, where V is fringevisibility and D is which-path distinguishability, is algebraically identical to the Thales con-straint L2 + h2 = R2 under the identification V = 2h (visibility equals twice the Thalesaltitude) and D = 2L (distinguishability equals twice the lateral displacement). We showthat this identification unifies three previously separate frameworks: (i) the hypotenuse rota-tion diagnostic for binary black hole inspiral, where the depth angle θh = arctan(h/L) tracksprocess progress; (ii) the torus vantage formula, where the observer angle α = arccos √aclassifies the measurement; and (iii) the Englert–Greenberger complementarity, where thewhich-path angle θ = arctan(D/V ) = arctan(L/h) quantifies wave–particle balance.The three frameworks are related by the exact identity 2α−θ = 90, which states that theobserver angle α always exceeds half the which-path angle by 45. This encodes a fixed 2:1angular relation: the two angles are linked so that each degree of which-path angle increasecorresponds to two degrees of observer-angle decrease toward equipartition at 45.Applied to laboratory data from entangled nuclear spins in isotopically engineered sili-con carbide (Bourassa et al., 2020), all identities hold to machine precision: the Englert–Greenberger constraint V 2 + D2 = 1 is conserved to 10−16, and 2α − θ = 90 holds to10−14 degrees. Through the structural mapping to effective Englert coordinates, the densitymatrix gives V = 0.990 (near-perfect fringe contrast in the interferometric analogue) andD = 0.139 (minimal which-path information), placing the system near the wave pole atθ = 8.0—consistent with the near-maximal entanglement (S = 0.986 bits) found indepen-dently from the ER = EPR diagnostic. Decoherence maps to which-path information leakingto the spin bath, with CPMG dynamical decoupling extending fringe persistence by 12×.The Born rule P = cos2(θBorn/2), the observer equation a = cos2 α, and the fringe-minimum intensity a = (1 − D)/2 are shown to be the same number: the partition fractionat the observer’s angle on the constraint circle.

Apr 6, 20260 citations

Ideal intersections for radio-frequency trap networks

Janus H. Wesenberg

We investigate the possible form of ideal intersections for two-dimensional rf trap networks suitable for quantum information processing with trapped ions. We show that the lowest order multipole component of the rf field that can contribute to an ideal intersection is a hexapole term uniquely determined by the tangents of the intersecting paths. The corresponding ponderomotive potential does not provide any confinement perpendicular to the paths if these intersect at right angles, indicating that ideal right-angle $\mathsf{X}$ intersections are impossible to achieve with hexapole fields. Based on this result, we propose an implementation of an ideal oblique- $\mathsf{X}$ intersection using a three-dimensional electrode structure.We investigate the possible form of ideal intersections for two-dimensional rf trap networks suitable for quantum information processing with trapped ions. We show that the lowest order multipole component of the rf field that can contribute to an ideal intersection is a hexapole term uniquely determined by the tangents of the intersecting paths. The corresponding ponderomotive potential does not provide any confinement perpendicular to the paths if these intersect at right angles, indicating that ideal right-angle $\mathsf{X}$ intersections are impossible to achieve with hexapole fields. Based on this result, we propose an implementation of an ideal oblique-$\mathsf{X}$ intersection using a three-dimensional electrode structure.

arXiv:0802.3162Physical Review AJan 16, 200917 citations

Quantum-Mechanical Operators

WORLD SCIENTIFIC eBooksJun 1, 20010 citations

\mathsf{X}

intersections are impossible to achieve with hexapole fields. Based on this result, we propose an implementation of an ideal oblique-

\mathsf{X}

intersection using a three-dimensional electrode structure.