Probability amplitude: from Feynman rules to the reciprocal relationship and the Born rule

Yoel Tikochinsky

F.J. Duarte

The Dirac–Feynman probability amplitude is introduced from first principles. This introduction starts from the basic two intra-interferometric spaces and continues to multiple intra-interferometric spaces. The probability amplitudes are then multiplied by their complex conjugate, via Born’s rule, to yield practical usable generalized interferometric quantum probabilities. A series of mathematically explicit quantum probabilities for N = 2, 3, 4, 5 are given. Various measurements and practical examples for N = 2 up to N = 100 are discussed in detail. Generalized N -slit quantum probability equations for two dimensions and three dimensions are also given.

Leonard Eisenbud

Eric Needham

The Complete Geometrical Derivation of Born’s Rule develops a geometrical derivation of the finite-dimensional Born weight law, The term complete geometrical derivation is used in a defined sense: the construction accounts for conjugation, orthogonal amplitudes, the half-angle structure, quadratic weighting, normalization, and the finite-dimensional projector weight law without independently inserting a probability exponent or binary probability parameter. It does not claim to derive measurement-basis selection, individual event occurrence, event-realization dynamics, or the equality of empirical terminal frequencies with the derived weights without an additional realization mechanism. A companion Python validation ledger checks 21 algebraic and numerical consequences of the construction, including reciprocal closure, conjugation, chord geometry, the half-angle law, the reciprocal-ratio weight map, golden-ratio calibration, Bloch-meridian consistency, and random finite-dimensional projector tests. For further information about the ENSO Framework, please contact Eric Needham:ensotheory1@gmail.com

Прохоров Лев Васильевич

Charles R. Giardina