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.
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.
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
Abstract Textbook theory says that the Canonical Commutation Relation derives from the homogeneity of space. This paper shows that the Canonical Commutation Relation does not derive from homogeneity of space or the homogeneity symmetry itself, but derives from a duality viewpoint of homogeneity, seen both from the viewpoint of position space, and from the viewpoint of momentum space, combined. Additionally, a specific particular fixed scale factor, relating position space with momentum space is necessary. It is this additional scaling information which enables complementarity between the system variables and makes the system unitary. Without this particular scaling, the Canonical Commutation Relation is left non-unitary and broken. Indeed, unitarity is separate information, unconnected and logically independent of the quantum system's underlying symmetry. This single counter-example contradicts the current consensus that foundational symmetries, underlying quantum systems, are ontologically, intrinsically and unavoidably unitary. And thus removes ‘unitary ontology’, as reason, for axiomatically imposing unitarity (or self-adjointness) — by Postulate — on quantum mechanical systems.
Keywords
foundations of quantum theory, quantum mechanics, wave mechanics, Canonical Commutation Relation, symmetry, homogeneity of space, unitary.
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.
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.
The presence of the imaginary unit i in quantum mechanics is frequently treated as a computational postulate. We demonstrate that the complex structure of Hilbert space is not a decorative choice but a structural necessity derived from the first principles of modern physics: continuous symmetry, the requirement of unitarity to preserve probability, and the correspondence between generators and observables. By examining the Lie algebra of the unitary group, we prove that a real-valued theory fails to map self-adjoint observables to generators of continuous evolution without sacrificing local tomography or the existence of non-trivial phases. We conclude that i is the unique logical bridge between symmetry and observability.