Five physical assumptions are proposed that together entail the general qualitative results, including the Born rule, of non-relativistic quantum mechanics by physical and information-theoretic reasoning alone. Two of these assumptions concern fundamental symmetries of physical interactions. The third concerns the Hilbert-space dimensions and the fourth and fifth the self-interaction Hamiltonians of the systems that function as "observers" within the theory. These assumptions are shown to provide a sufficient motivation for the usual Hilbert-space formalism, and to obviate the observation-related axioms and most interpretative assumptions with which minimal quantum mechanics is typically supplemented.
Five physical assumptions are proposed that together entail the general qualitative results, including the Born rule, of non-relativistic quantum mechanics by physical and information-theoretic reasoning alone. Two of these assumptions concern fundamental symmetries of physical interactions. The third concerns the Hilbert-space dimensions and the fourth and fifth the self-interaction Hamiltonians of the systems that function as "observers" within the theory. These assumptions are shown to provide a sufficient motivation for the usual Hilbert-space formalism, and to obviate the observation-related axioms and most interpretative assumptions with which minimal quantum mechanics is typically supplemented.
Conventional quantum mechanics with a complex Hilbert space and the Born Rule is derived from five axioms describing properties of probability distributions for the outcome of measurements. Axioms I,II,III are common to quantum mechanics and hidden variable theories. Axiom IV recognizes a phenomenon, first noted by Turing and von Neumann, in which the increase in entropy resulting from a measurement is reduced by a suitable intermediate measurement. This is shown to be impossible for local hidden variable theories. Axiom IV, together with the first three, almost suffice to deduce the conventional rules but allow some exotic, alternatives such as real or quaternionic quantum mechanics. Axiom V recognizes a property of the distribution of outcomes of random measurements on qubits which holds only in the complex Hilbert space model. It is then shown that the five axioms also imply the conventional rules for all dimensions. 1
We present a heuristic derivation of Born's rule and unitary transforms in Quantum Mechanics, from a simple set of axioms built upon a physical phenomenology of quantization. This approach naturally leads to the usual quantum formalism, within a new realistic conceptual framework that is discussed in details. Physically, the structure of Quantum Mechanics appears as a result of the interplay between the quantized number of "modalities" accessible to a quantum system, and the continuum of "contexts" that are required to define these modalities. Mathematically, the Hilbert space structure appears as a consequence of a specific "extra-contextuality" of modalities, closely related to the hypothesis of Gleason's theorem, and consistent with its conclusions.
We present a heuristic derivation of Born's rule and unitary transforms in Quantum Mechanics, from a simple set of axioms built upon a physical phenomenology of quantization. This approach naturally leads to the usual quantum formalism, within a new realistic conceptual framework that is discussed in details. Physically, the structure of Quantum Mechanics appears as a result of the interplay between the quantized number of "modalities" accessible to a quantum system, and the continuum of "contexts" that are required to define these modalities. Mathematically, the Hilbert space structure appears as a consequence of a specific "extra-contextuality" of modalities, closely related to the hypothesis of Gleason's theorem, and consistent with its conclusions.
The paper proposes a new approach in the foundations of Quantum Mechanics.It does not make any assumption about the physical world, but looks at the consequences of the formalism used in models.Whenever a system is represented by variables which meet precise, but common, mathematical properties, one can prove theorems which are very close to the axioms of Quantum Mechanics (Hilbert spaces, observables, eigen values,...).It is then possible to explore the conditions of the validity of these axioms and to give a firm ground to the usual computations.Moreover this approach sheds a new ligth on the issues of determism, and interacting systems.In the third edition of this paper developments have been added about the statistical procedures used to detect anomalies in Physical Laws.
The paper proposes a new approach in the foundations of Quantum Mechanics.It does not make any assumption about the physical world, but looks at the consequences of the formalism used in models.Whenever a system is represented by variables which meet precise, but common, mathematical properties, one can prove theorems which are very close to the axioms of Quantum Mechanics (Hilbert spaces, observables, eigen values,...).It is then possible to explore the conditions of the validity of these axioms and to give a firm ground to the usual computations.Moreover this approach sheds a new ligth on the issues of determism, and interacting systems.In the third edition of this paper developments have been added about the statistical procedures used to detect anomalies in Physical Laws.
The Born rule stands as the fundamental probability postulate of quantum mechanics, yet its first-principle derivation remains a longstanding open problem. This paper presents a rigorous, collapse-free, and observer-independent derivation of the Born rule. Starting from four physically natural and minimal axioms—unitary dynamics, uniqueness of classical records, unitary invariance, and consistency with the classical limit and ergodicity—we uniquely deduce the Born rule from first principles. We prove that the Born rule is not an independent postulate, but a necessary mathematical consequence of the core axiomatic structure. Our approach is self-contained, assumption-free, and purely unitary, establishing that the Born rule is canonically determined by the fundamental mathematical and physical principles of quantum mechanics.