This article is a concise statement of the machinery of quantum indeterminacy — in response to the question: What is indeterminacy; is it something that can be written down?
br>Keywords
foundations of quantum theory, quantum randomness, quantum indeterminacy, logical independence, self-reference, logical circularity, mathematical undecidability, Kurt Godel.
This article is a concise statement of the machinery of quantum indeterminacy — in response to the question: What is indeterminacy; is it something that can be written down?
br>Keywords
foundations of quantum theory, quantum randomness, quantum indeterminacy, logical independence, self-reference, logical circularity, mathematical undecidability, Kurt Godel.
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.
Russell’s theory of descriptions depends essentially on the concept of identity, in particular to eliminate definite descriptions by contextual definitions. But quantum objects, according to a possible metaphysical interpretation I shall assume without discussion here, seem not to satisfy the standard theory of identity, and do not have individuality in a sense. Even so, physicists refer informally to both ‘a quantum so-and-so’ and, worst, to ‘the quantum so-and-so’, as if these expressions were something like descriptions. In this article, we sketch a possible theory of ‘quantum (definite) descriptions’ using a weaker concept of indiscernibility (or indistinguishability), and advance some consequences such a theory may have for the discussions on the logical foundations of quantum theory.
This progress report covers recent developments in the area of quantum randomness, which is an extraordinarily interdisciplinary area that belongs not only to physics, but also to philosophy, mathematics, computer science, and technology. For this reason the article contains three parts that will be essentially devoted to different aspects of quantum randomness, and even directed, although not restricted, to various audiences: a philosophical part, a physical part, and a technological part. For these reasons the article is written on an elementary level, combining simple and non-technical descriptions with a concise review of more advanced results. In this way readers of various provenances will be able to gain while reading the article.
Abstract It is often claimed that quantum mechanics entails that there is indeterminacy in the world, in the sense that physical systems can lack a precise value for a property. Indeed, a radical form of indeterminacy, in which systems have almost none of the properties we normally attribute to them, can be shown to follow from a particular interpretation of quantum mechanics—the bare theory. However, the bare theory is shown to be incoherent. Still, a more limited form of indeterminacy can be located in spontaneous collapse, hidden variables, and many-worlds theories. This indeterminacy is unlike that which arises classically, having nothing to do with ordinary forms of vagueness or the composition of large objects out of smaller parts. Indeterminacy in quantum mechanics can perhaps be avoided, but there seems little reason to do so.
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 Uses a historical approach to explore the extent to which quantum theory mandates indeterminism. Drawing on important examples in the history of physics, it discusses explicitly the crucial distinction – often overlooked in the philosophical literature – between indeterminacy of physical quantities and causal indeterminism. It pursues at a more sophisticated level, issues raised in Ch. 2 concerning determinism and indeterminism in modern science. It concludes that quantum indeterminacy is an objective feature of the world, but that the jury is still out regarding quantum indeterminism.