Foundations

14 pages, 617 KiB

Open AccessArticle

Relativistic Time-of-Arrival Measurements: Predictions, Post-Selection and Causality Problems

Charis Anastopoulos

and

Maria-Electra Plakitsi

Foundations 2023, 3(4), 724-737; https://doi.org/10.3390/foundations3040041 - 29 Nov 2023

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Abstract

We analyze time-of-arrival probability distributions for relativistic particles in the context of quantum field theory (QFT). We show that QFT leads to a unique prediction, modulo post-selection that incorporates properties of the apparatus into the initial state. We also show that an experimental distinction of different probability assignments is possible especially in near-field measurements. We also analyze causality in relativistic measurements. We consider a quantum state obtained by a spacetime-localized operation on the vacuum, and we show that detection probabilities are typically characterized by small transient non-causal terms. We explain that these terms originate from Feynman propagation of the initial operation, because the Feynman propagator does not vanish outside the light cone. We discuss possible ways to restore causality, and we argue that this may not be possible in measurement models that involve switching the field–apparatus coupling on and off. Full article

(This article belongs to the Section Physical Sciences)

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64 pages, 648 KiB

Open AccessReview

Ostrowski-Type Fractional Integral Inequalities: A Survey

Muhammad Tariq

Sotiris K. Ntouyas

and

Bashir Ahmad

Foundations 2023, 3(4), 660-723; https://doi.org/10.3390/foundations3040040 - 13 Nov 2023

Viewed by 240

Abstract

This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions,

(ζ, m)

-convex [...] Read more.

(ζ, m)

-convex functions, s-convex functions,

(s, r)

-convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions,

M T

-convex functions, P-convex functions, m-convex functions,

(s, m)

-convex functions, exponentially s-convex functions,

(β, m)

-convex functions, exponential-convex functions,

\bar{ζ}, β, γ, δ

-convex functions, quasi-geometrically convex functions,

s - e

-convex functions and n-polynomial exponentially s-convex functions. Riemann–Liouville fractional integral, Katugampola fractional integral, k-Riemann–Liouville, Riemann–Liouville fractional integrals with respect to another function, Hadamard fractional integral, fractional integrals with exponential kernel and Atagana-Baleanu fractional integrals are included. Results for Ostrowski-Mercer-type inequalities, Ostrowski-type inequalities for preinvex functions, Ostrowski-type inequalities for Quantum-Calculus and Ostrowski-type inequalities of tensorial type are also presented. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions II)

17 pages, 567 KiB

Open AccessArticle

Comparison between Two Competing Newton-Type High Convergence Order Schemes for Equations on Banach Spaces

Ioannis K. Argyros

Manoj K. Singh

and

Samundra Regmi

Foundations 2023, 3(4), 643-659; https://doi.org/10.3390/foundations3040039 - 30 Oct 2023

Viewed by 349

Abstract

We carried out a local comparison between two ninth convergence order schemes for solving nonlinear equations, relying on first-order Fréchet derivatives. Earlier investigations require the existence as well as the boundedness of derivatives of a high order to prove the convergence of these schemes. However, these derivatives are not in the schemes. These assumptions restrict the applicability of the schemes, which may converge. Numerical results along with a boundary value problem are given to examine the theoretical results. Both schemes are symmetrical not only in the theoretical results (formation and convergence order), but the numerical and dynamical results are also similar. We calculated the convergence radii of the nonlinear schemes. Moreover, we obtained the extraneous fixed points for the proposed schemes, which are repulsive and are not part of the solution space. Lastly, the theoretical and numerical results are supported by the dynamic results, where we plotted basins of attraction for a selected test function. Full article

(This article belongs to the Special Issue Iterative Methods with Applications in Mathematical Sciences II)

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9 pages, 271 KiB

Open AccessArticle

The Magical “Born Rule” and Quantum “Measurement”: Implications for Physics

Johan Hansson

Foundations 2023, 3(4), 634-642; https://doi.org/10.3390/foundations3040038 - 29 Sep 2023

Viewed by 384

Abstract

I. The arena of quantum mechanics and quantum field theory is the abstract, unobserved and unobservable, M-dimensional formal Hilbert space ≠ spacetime. II. The arena of observations—and, more generally, of all events (i.e., everything) in the real physical world—is the classical four-dimensional physical spacetime. III. The “Born rule” is the random process “magically” transforming I into II. Wavefunctions are superposed and entangled only in the abstract space I, never in spacetime II. Attempted formulations of quantum theory directly in real physical spacetime actually constitute examples of “locally real” theories, as defined by Clauser and Horne, and are therefore already empirically refuted by the numerous tests of Bell’s theorem in real, controlled experiments in laboratories here on Earth. Observed quantum entities (i.e., events) are never superposed or entangled as they: (1) exclusively “live” (manifest) in real physical spacetime and (2) are not described by entangled wavefunctions after “measurement” effectuated by III. When separated and treated correctly in this way, a number of fundamental problems and “paradoxes” of quantum theory vs. relativity (i.e., spacetime) simply vanish, such as the black hole information paradox, the infinite zero-point energy of quantum field theory and the quantization of general relativity. Full article

(This article belongs to the Section Physical Sciences)

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Foundations, Volume 3, Issue 4 (December 2023) – 4 articles

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