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Steve Faulkner - One of the best experts on this subject based on the ideXlab platform.
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logical independence of Imaginary and complex numbers in elementary algebra context theory of indeterminacy and quantum randomness
viXra, 2015Co-Authors: Steve FaulknerAbstract:Abstract: As opposed to the classical logic of true and false, when Elementary Algebra is treated as a formal axiomatised system, formulae in that algebra are either provable, disprovable or otherwise, logically independent of axioms. This logical independence is well-known to Mathematical Logic. Here I show that the Imaginary Unit, and by extension, all complex numbers, exist in that algebra, logically independently of the algebra's axioms. The intention is to cover the subject in a way accessible to physicists. This work is part of a project researching logical independence in quantum mathematics, for the purpose of advancing a complete theory of quantum randomness. Elementary Algebra is a theory that cannot be completed and is therefore subject to Godel's Incompleteness Theorems. keywords: mathematical logic, formal system, axioms, mathematical propositions, Soundness Theorem, Completeness Theorem, logical independence, mathematical undecidability, foundations of quantum theory, quantum mechanics, quantum physics, quantum indeterminacy, quantum randomness.
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A Short Note on Why the Imaginary Unit is Inherent in Physics
viXra, 2015Co-Authors: Steve FaulknerAbstract:Abstract: I write out a proof adapted from the work of W E Baylis et al showing how existence of the square root of minus one is inherent in physical theories demanding orthogonal mathematics. Keywords: Imaginary numbers, complex numbers, foundations of quantum theory, quantum physics, quantum mechanics, wave mechanics, Canonical Commutation Relation Unitary, non-Unitary, Unitarity, elementary algebra, quantum indeterminacy, quantum randomness.
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The logical self-reference inside the Fourier transform
viXra, 2015Co-Authors: Steve FaulknerAbstract:Abstract I show that, in general, the Fourier transform is necessarily self-referent and logically circular. Keywords self-reference, logical circularity, mathematical logic, Fourier transform, vector space, orthogonality, orthogonal, Unitarity, Unitary, Imaginary Unit, foundations of quantum theory, quantum mechanics, quantum indeterminacy, quantum information, prepared state, pure state, mixed state, wave packet, scalar product, tensor product.
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Quantum System Symmetry is not the Source of Unitary Information in Wave Mechanics Context Quantum Randomness
viXra, 2015Co-Authors: Steve FaulknerAbstract: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.
Asao Arai - One of the best experts on this subject based on the ideXlab platform.
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Asymptotic analysis of the Fourier transform of a probability measure with application to the quantum Zeno effect
Journal of Mathematical Analysis and Applications, 2013Co-Authors: Asao AraiAbstract:Abstract Let μ be a probability measure on the set R of real numbers and μ ˆ ( t ) : = ∫ R e − i t λ d μ ( λ ) ( t ∈ R ) be the Fourier transform of μ ( i is the Imaginary Unit). Then, under suitable conditions, asymptotic formulae for | μ ˆ ( t / x ) | 2 x in 1 / x as x → ∞ are derived. These results are applied to the so-called quantum Zeno effect to establish asymptotic formulae for its occurrence probability in the inverse of the number N of measurements made in a time interval as N → ∞ .
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Hilbert Space Representations of Generalized Canonical Commutation Relations
Journal of Mathematics, 2013Co-Authors: Asao AraiAbstract:We consider Hilbert space representations of a generalization of canonical commutation relations , where 's are the elements of an algebra with identity , is the Imaginary Unit, and is a real number with antisymmetry . Some basic aspects on Hilbert space representations of the generalized CCR (GCCR) are discussed. We define a Schrödinger-type representation of the GCCR by an analogy with the usual Schrödinger representation of the CCR with degrees of freedom. Also, we introduce a Weyl-type representation of the GCCR. The main result of the present paper is a uniqueness theorem on Weyl representations of the GCCR.
Chu-ryang Wie - One of the best experts on this subject based on the ideXlab platform.
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Bloch sphere model for two-qubit pure states
arXiv: Quantum Physics, 2014Co-Authors: Chu-ryang WieAbstract:The two-qubit pure state is explicitly parameterized by three Unit 2-spheres and a phase factor. For separable states, two of the three Unit spheres are the Bloch spheres of each qubit. The third sphere parameterizes the degree and phase of concurrence, an entanglement measure. This sphere may be considered a variable complex Imaginary Unit where the stereographic projection maps the qubit-A Bloch sphere to a complex plane with this variable Imaginary Unit. This Bloch sphere model gives a consistent description of the two-qubit pure states for both separable and entangled states. We argue that the third sphere (entanglement sphere) parameterizes the nonlocal properties, entanglement and a nonlocal relative phase, while the local relative phases are parameterized by the azimuth angles of the two quasi-Bloch spheres. On these three Unit spheres and a phase factor, the two-qubit pure states and Unitary gates can be geometrically represented. Accomplished by means of Hopf fibration, the complex amplitudes of a two-qubit pure state and the Bloch sphere parameters are related by a single quaternionic relation.
Wie Chu-ryang - One of the best experts on this subject based on the ideXlab platform.
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Bloch sphere model for two-qubit pure states
2019Co-Authors: Wie Chu-ryangAbstract:The two-qubit pure state is explicitly parameterized by three Unit 2-spheres and a phase factor. For separable states, two of the three Unit spheres are the Bloch spheres of each qubit. The third sphere parameterizes the degree and phase of concurrence, an entanglement measure. This sphere may be considered a variable complex Imaginary Unit where the stereographic projection maps the qubit-A Bloch sphere to a complex plane with this variable Imaginary Unit. This Bloch sphere model gives a consistent description of the two-qubit pure states for both separable and entangled states. We argue that the third sphere (entanglement sphere) parameterizes the nonlocal properties, entanglement and a nonlocal relative phase, while the local relative phases are parameterized by the azimuth angles of the two quasi-Bloch spheres. On these three Unit spheres and a phase factor, the two-qubit pure states and Unitary gates can be geometrically represented. Accomplished by means of Hopf fibration, the complex amplitudes of a two-qubit pure state and the Bloch sphere parameters are related by a single quaternionic relation.Comment: 15 pages, 8 figure
Jesús Sánchez - One of the best experts on this subject based on the ideXlab platform.
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Calculation of the Fine-Structure Constant
viXra, 2019Co-Authors: Jesús SánchezAbstract:The fine-structure constant α [1] is a constant in physics that plays a fundamental role in the electromagnetic interaction. It is a dimensionless constant, defined as: α=q^2/(2e_0 hc)=0.0072973525664 (1) Being q the elementary charge, eo the vacuum permittivity, h the Planck constant and c the speed of light in vacuum. The value shown in (1) is according CODATA 2014 [2]. In this paper, it will be explained that the fine-structure constant is one of the roots of the following equation: cos(α^(-1) )=e^(-1) (2) being e the mathematical constant e (the base of the natural logarithm). One of the solutions of this equation is: α=0.0072973520977 (3) This means that it is equal to the CODATA value in nine decimal digits (or the seven most significant ones if you prefer). And therefore, the difference between both values is: Difference=(α(1)-α(3))/(α(1))·100=0.00000642% (4) This coincidence is higher in orders of magnitude than the commonly accepted necessary to validate a theory towards experimentation. As the cosine function is periodical, the equation (2) has infinite roots and could seem the coincidence is just by chance. But as it will be shown in the paper, the separation among the different solutions is sufficiently high to disregard this possibility. It will also be shown that another elegant way to show equation (2) is the following (being i the Imaginary Unit): e^(i/α)-e^(-1)=-e^(-i/α)+e^(-1) (5) Having of course the same root (3). The possible meaning of this other representation (5) will be explained.
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Calculation of the Fine-Structure Constant
2018Co-Authors: Jesús SánchezAbstract:The fine-structure constant α [1] is a constant in physics that plays a fundamental role in the electromagnetic interaction. It is a dimensionless constant, defined as: (1) being q the elementary charge, e0 the vacuum permittivity, h the Planck constant and c the speed of light in vacuum. The value shown in (1) is according CODATA 2014 [2]. In this paper, it will be explained that the fine-structure constant is one of the roots of the following equation: (2) being e the mathematical constant e (the base of the natural logarithm). One of the solutions of this equation is: (3) This means that it is equal to the CODATA value in nine decimal digits (or the seven most significant ones if you prefer). And therefore, the difference between both values is: (4) This coincidence is higher in orders of magnitude than the commonly accepted necessary to validate a theory towards experimentation. As the cosine function is periodical, the Equation (2) has infinite roots and could seem the coincidence is just by chance. But as it will be shown in the paper, the separation among the different solutions is sufficiently high to disregard this possibility. It will also be shown that another elegant way to show Equation (2) is the following (being i the Imaginary Unit): (5) having of course the same root (3). The possible meaning of this other representation (5) will be explained.
