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The Mathematical Foundations of Quantum Indeterminacy

Steve Faulkner

Published Apr 1, 2016• 0 citations
OpenAlex

Abstract

Abstract: In 2008, Tomasz Paterek et al published ingenious research, proving that quantum randomness is the output of measurement experiments, whose input commands a logically independent response. Following up on that work, this paper develops a full mathematical theory of quantum indeterminacy. I explain how, the Paterek experiments imply, that the measurement of pure eigenstates, and the measurement of mixed states, cannot both be isomorphically and faithfully represented by the same single operator. Specifically, unitary representation of pure states is contradicted by the Paterek experiments. Profoundly, this denies the axiomatic status of Quantum Postulates, that state, symmetries are unitary, and observables Hermitian. Here, I show how indeterminacy is the information of transition, from pure states to mixed. I show that the machinery of that transition is unpreventable, logically circular, unitary-generating self-reference: all logically independent. Profoundly, this indeterminate system becomes apparent, as a visible feature of the mathematics, when unitarity --- imposed by Postulate --- is given up and abandoned. Keywords: foundations of quantum theory, quantum mechanics, quantum randomness, quantum indeterminacy, quantum information, prepared state, measured state, pure states, mixed states, unitary, redundant unitarity, orthogonal, scalar product, inner product, mathematical logic, logical independence, self-reference, logical circularity, mathematical undecidability.

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Abstract:
In 2008, Tomasz Paterek et al published ingenious research, proving that quantum randomness is the output of measurement experiments, whose input commands a logically independent response. Following up on that work, this paper develops a full mathematical theory of quantum indeterminacy. I explain how, the Paterek experiments imply, that the measurement of pure eigenstates, and the measurement of mixed states, cannot both be isomorphically and faithfully represented by the same single operator. Specifically, unitary representation of pure states is contradicted by the Paterek experiments. Profoundly, this denies the axiomatic status of Quantum Postulates, that state, symmetries are unitary, and observables Hermitian. Here, I show how indeterminacy is the information of transition, from pure states to mixed. I show that the machinery of that transition is unpreventable, logically circular, unitary-generating self-reference: all logically independent. Profoundly, this indeterminate system becomes apparent, as a visible feature of the mathematics, when unitarity --- imposed by Postulate --- is given up and abandoned.
Keywords:
foundations of quantum theory, quantum mechanics, quantum randomness, quantum indeterminacy, quantum information, prepared state, measured state, pure states, mixed states, unitary, redundant unitarity, orthogonal, scalar product, inner product, mathematical logic, logical independence, self-reference, logical circularity, mathematical undecidability.
Abstract: In 2008 Tomasz Paterek et al published experiments demonstrating that quantum randomness results from logical independence. The job of this paper is to derive implications for Matrix Mechanics. Surprisingly (and apparently unwittingly), the Paterek experiments imply that faithful representation of (non-random) pure states contradicts the Quantum Postulate which imposes unitary, Hermitian and Hilbert space mathematics on all states. Paterek's Boolean formalism asserts and demands a non-unitary environment for pure states, which is freely restricted to logically independent unitary structure, wherever the creation of mixed states demands unitarity. Consequently, the Paterek experiments contradict the Quantum Postulate which imposes unitary, Hermitian and Hilbert space structures, axiomatically as blanket ontology, across the whole theory. Examination of the ‘non-unitary to unitary transition’ reveals the machinery of quantum indeterminacy. That machinery involves self-referential circularity, inaccessible history, and the geometrical ambiguity of perfect symmetry. The findings here provide answers for researchers studying Foundations of Quantum Mechanics; they make intuitive good sense of indeterminacy; they provide reason and significance for observable operators and eigenvectors; and they should be helpful for those interested in the Measurement Problem, the EPR paradox and possibly those looking for a method to quantize Gravity. Keywords: foundations of quantum theory, quantum randomness, quantum indeterminacy, logical independence, self-reference, logical circularity, mathematical undecidability.
The homogeneity symmetry is re-examined and shown to be non-unitary. This is motivated by the prospect that logical independence in elementary algebra, entering quantum mathematics, will constitute the basis for a theory explaining quantum randomness. Keywords: foundations of quantum theory, quantum physics, quantum mechanics, wave mechanics, Canonical Commutation Relation, symmetry, homogeneity of space, unitary, non-unitary, unitarity, mathematical logic, formal system, elementary algebra, information, axioms, mathematical propositions, logical independence, quantum indeterminacy, quantum randomness.
Abstract The homogeneity symmetry is re-examined and shown to be non-unitary, with no requirement for the imaginary unit. This removes symmetry, as reason, for imposing unitarity (or self-adjointness) -- by Postulate. The work here is part of a project researching logical independence in quantum mathematics, for the purpose of advancing a full and complete theory of quantum randomness. Keywords foundations of quantum theory, quantum physics, quantum mechanics, wave mechanics, Canonical Commutation Relation, symmetry, homogeneity of space, unitary, non-unitary, unitarity, mathematical logic, formal system, elementary algebra, information, axioms, mathematical propositions, logical independence, quantum indeterminacy, quantum randomness.
Logical foundation for quantum theory is considered. I claim that quantum theory correctly represents Nature when mathematical physics embraces and indeed features, logical anomalies inherent in pure mathematics. This approach links undecidability in arithmetic with the logic of quantum experiments. The undecidablity occupies an algebraic environment which is the missing foundation for the 3-valued logic predicted by Hans Reichenbach, shown by him to resolve `causal anomalies' of quantum mechanics, such as: inconsistency between prepared and measured states, complementarity between pairs of observables, and the EPR-paradox of action at a distance. Arithmetic basic to mathematical physics, is presented formally as a logical system consisting of axioms and propositions. Of special interest are all propositions asserting the existence of particular numbers. All numbers satisfying the axioms permeate the arithmetic indistinguishably, but these logically partition into two distinct sets: numbers whose existence the axioms determine by proof, and numbers whose existence axioms cannot determine, being neither provable nor negatable. Failure of mathematical physics to incorporate this logical distinction is seen as reason for quantum theory being logically at odds with quantum experiments. Nature is interpreted as having rules isomorphic to the abovementioned axioms with these governing arithmetical combinations of necessary and possible values or effects in experiments. Soundness and Completeness theorems from mathematical logic emerge as profoundly fundamental principles for quantum theory, making good intuitive sense of the subject.