A polynomial probability functional on complex amplitudes that is phase invariant, phase-averaged additive, and normalized must equal the squared modulus, the Born rule.
Consistency, Amplitudes and Probabilities in Quantum Theory
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abstract
Quantum theory is formulated as the only consistent way to manipulate probability amplitudes. The crucial ingredient is a consistency constraint: if there are two different ways to compute an amplitude the two answers must agree. This constraint is expressed in the form of functional equations the solution of which leads to the usual sum and product rules for amplitudes. A consequence is that the Schrodinger equation must be linear: non-linear variants of quantum mechanics are inconsistent. The physical interpretation of the theory is given in terms of a single natural rule. This rule, which does not itself involve probabilities, is used to obtain a proof of Born's statistical postulate. Thus, consistency leads to indeterminism. PACS: 03.65.Bz, 03.65.Ca.
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Local Uniqueness of the Born Rule on Categories with Complex-Weighted Morphisms
A polynomial probability functional on complex amplitudes that is phase invariant, phase-averaged additive, and normalized must equal the squared modulus, the Born rule.