Entropy

18 pages, 850 KiB

Open AccessArticle

A Hybrid Quantum Solver for the Lorenz System

by Sajad Fathi Hafshejani, Daya Gaur, Arundhati Dasgupta, Robert Benkoczi, Narasimha Reddy Gosala and Alfredo Iorio

Entropy 2024, 26(12), 1009; https://doi.org/10.3390/e26121009 - 22 Nov 2024

Abstract

We develop a hybrid classical–quantum method for solving the Lorenz system. We use the forward Euler method to discretize the system in time, transforming it into a system of equations. This set of equations is solved by using the Variational Quantum Linear Solver [...] Read more.

We develop a hybrid classical–quantum method for solving the Lorenz system. We use the forward Euler method to discretize the system in time, transforming it into a system of equations. This set of equations is solved by using the Variational Quantum Linear Solver (VQLS) algorithm. We present numerical results comparing the hybrid method with the classical approach for solving the Lorenz system. The simulation results demonstrate that the VQLS method can effectively compute solutions comparable to classical methods. The method is easily extended to solving similar nonlinear differential equations. Full article

(This article belongs to the Special Issue Quantum Computing for Complex Dynamics, 2nd Edition)

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29 pages, 1916 KiB

Open AccessArticle

Fast Proxy Centers for the Jeffreys Centroid: The Jeffreys–Fisher–Rao Center and the Gauss–Bregman Inductive Center

by Frank Nielsen

Entropy 2024, 26(12), 1008; https://doi.org/10.3390/e26121008 - 22 Nov 2024

Abstract

The symmetric Kullback–Leibler centroid, also called the Jeffreys centroid, of a set of mutually absolutely continuous probability distributions on a measure space provides a notion of centrality which has proven useful in many tasks, including information retrieval, information fusion, and clustering. However, the [...] Read more.

The symmetric Kullback–Leibler centroid, also called the Jeffreys centroid, of a set of mutually absolutely continuous probability distributions on a measure space provides a notion of centrality which has proven useful in many tasks, including information retrieval, information fusion, and clustering. However, the Jeffreys centroid is not available in closed form for sets of categorical or multivariate normal distributions, two widely used statistical models, and thus needs to be approximated numerically in practice. In this paper, we first propose the new Jeffreys–Fisher–Rao center defined as the Fisher–Rao midpoint of the sided Kullback–Leibler centroids as a plug-in replacement of the Jeffreys centroid. This Jeffreys–Fisher–Rao center admits a generic formula for uni-parameter exponential family distributions and a closed-form formula for categorical and multivariate normal distributions; it matches exactly the Jeffreys centroid for same-mean normal distributions and is experimentally observed in practice to be close to the Jeffreys centroid. Second, we define a new type of inductive center generalizing the principle of the Gauss arithmetic–geometric double sequence mean for pairs of densities of any given exponential family. This new Gauss–Bregman center is shown experimentally to approximate very well the Jeffreys centroid and is suggested to be used as a replacement for the Jeffreys centroid when the Jeffreys–Fisher–Rao center is not available in closed form. Furthermore, this inductive center always converges and matches the Jeffreys centroid for sets of same-mean normal distributions. We report on our experiments, which first demonstrate how well the closed-form formula of the Jeffreys–Fisher–Rao center for categorical distributions approximates the costly numerical Jeffreys centroid, which relies on the Lambert W function, and second show the fast convergence of the Gauss–Bregman double sequences, which can approximate closely the Jeffreys centroid when truncated to a first few iterations. Finally, we conclude this work by reinterpreting these fast proxy Jeffreys–Fisher–Rao and Gauss–Bregman centers of Jeffreys centroids under the lens of dually flat spaces in information geometry. Full article

(This article belongs to the Special Issue Information Theory in Emerging Machine Learning Techniques)

21 pages, 3650 KiB

Open AccessArticle

A Real-Time Fault Diagnosis Method for Multi-Source Heterogeneous Information Fusion Based on Two-Level Transfer Learning

by Danmin Chen, Zhiqiang Zhang, Funa Zhou and Chaoge Wang

Entropy 2024, 26(12), 1007; https://doi.org/10.3390/e26121007 - 22 Nov 2024

Abstract

A convolutional neural network can extract features from high-dimensional data, but the convolution operation has a high time complexity and requires a large amount of computation. For equipment with a high sampling frequency, fault diagnosis methods based on convolutional neural networks cannot meet [...] Read more.

A convolutional neural network can extract features from high-dimensional data, but the convolution operation has a high time complexity and requires a large amount of computation. For equipment with a high sampling frequency, fault diagnosis methods based on convolutional neural networks cannot meet the requirements of online fault diagnosis. To solve this problem, this study proposes a fault diagnosis method for multi-source heterogeneous information fusion based on two-level transfer learning. This method aims to fully utilize multi-source heterogeneous information and external domain data, construct a two-level transfer mechanism to fuse multi-source heterogeneous information, avoid convolutional operations, and achieve real-time fault diagnosis. Its main work is to build a feature extraction network model of screenshots, design a mechanism for transfer from the feature extraction model using screenshots to the deep learning model using one-dimensional sequence signals, and complete the transfer from a convolutional neural network to a deep neural network. After two-level transfer, the fault diagnosis model not only integrates the characteristics of one-dimensional sequence signals and screenshots but also avoids convolution operations and has a low time complexity. The effectiveness of the proposed method is verified using a gearbox dataset and a bearing dataset. Full article

(This article belongs to the Section Multidisciplinary Applications)

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28 pages, 1718 KiB

Open AccessArticle

Advancing Continuous Distribution Generation: An Exponentiated Odds Ratio Generator Approach

by Xinyu Chen, Zhenyu Shi, Yuanqi Xie, Zichen Zhang, Achraf Cohen and Shusen Pu

Entropy 2024, 26(12), 1006; https://doi.org/10.3390/e26121006 - 22 Nov 2024

Abstract

This paper presents a new methodology for generating continuous statistical distributions, integrating the exponentiated odds ratio within the framework of survival analysis. This new method enhances the flexibility and adaptability of distribution models to effectively address the complexities inherent in contemporary datasets. The [...] Read more.

This paper presents a new methodology for generating continuous statistical distributions, integrating the exponentiated odds ratio within the framework of survival analysis. This new method enhances the flexibility and adaptability of distribution models to effectively address the complexities inherent in contemporary datasets. The core of this advancement is illustrated by introducing a particular subfamily, the “Type 2 Gumbel Weibull-G family of distributions”. We provide a comprehensive analysis of the mathematical properties of these distributions, including statistical properties such as density functions, moments, hazard rate and quantile functions, Rényi entropy, order statistics, and the concept of stochastic ordering. To test the robustness of our new model, we apply five distinct methods for parameter estimation. The practical applicability of the Type 2 Gumbel Weibull-G distributions is further supported through the analysis of three real-world datasets. These real-life applications illustrate the exceptional statistical precision of our distributions compared to existing models, thereby reinforcing their significant value in both theoretical and practical statistical applications. Full article

(This article belongs to the Special Issue Number Theoretic Methods in Statistics: Theory and Applications)

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14 pages, 2843 KiB

Open AccessArticle

User Real Comments Incentive Mechanism Based on Blockchain in E-Commerce Transactions—A Tripartite Evolutionary Game Analysis

by Chengyi Le, Ran Zheng, Ting Lu and Yu Chen

Entropy 2024, 26(12), 1005; https://doi.org/10.3390/e26121005 - 22 Nov 2024

Abstract

In response to the widespread issue of fake comments on e-commerce platforms, this study aims to analyze and propose a blockchain-based solution to incentivize authentic user feedback and reduce the prevalence of fraudulent reviews. Specifically, this paper constructs a tripartite evolutionary game model [...] Read more.

In response to the widespread issue of fake comments on e-commerce platforms, this study aims to analyze and propose a blockchain-based solution to incentivize authentic user feedback and reduce the prevalence of fraudulent reviews. Specifically, this paper constructs a tripartite evolutionary game model between sellers, buyers, and e-commerce platforms to study the real comment mechanism of blockchain. The strategy evolution under different incentive factors is simulated using replication dynamic equation analysis and Matlab software simulation. The study found that introducing smart contracts and “tokens” for incentives not only increased incentives for real comments but also reduced the negative experiences caused by “speculative” sellers, thereby influencing buyers to opt for authentic reviews. By structuring interactions through blockchain, the mechanism helped lower informational entropy thus reducing disorder and unpredictability in buyer and seller behavior and contributing to system stability. Further, by increasing penalties for dishonest behavior under the “credit on the chain” system, the platform lowered entropy in the system by promoting trust and reducing fraudulent activities. The real comment mechanism based on blockchain proposed in this paper can effectively enhance the order and transparency within the comment ecosystem. These findings contribute to theory and practice by providing strategic insights for e-commerce platforms to encourage genuine feedback, reduce informational entropy, and mitigate fake comments, ultimately fostering a more reliable online marketplace. Full article

(This article belongs to the Section Complexity)

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13 pages, 1453 KiB

Open AccessArticle

The Two-Spin Enigma: From the Helium Atom to Quantum Ontology

by Philippe Grangier, Alexia Auffèves, Nayla Farouki, Mathias Van Den Bossche and Olivier Ezratty

Entropy 2024, 26(12), 1004; https://doi.org/10.3390/e26121004 - 22 Nov 2024

Abstract

The purpose of this article is to provide a novel approach and justification of the idea that classical physics and quantum physics can neither function nor even be conceived without the other—in line with ideas attributed to, e.g., Niels Bohr or Lev Landau. [...] Read more.

The purpose of this article is to provide a novel approach and justification of the idea that classical physics and quantum physics can neither function nor even be conceived without the other—in line with ideas attributed to, e.g., Niels Bohr or Lev Landau. Though this point of view may contradict current common wisdom, we will show that it perfectly fits with empirical evidence, and can be maintained without giving up physical realism. In order to place our arguments in a convenient historical perspective, we will proceed as if we were following the path of a scientific investigation about the demise, or vanishing, of some valuable properties of the two electrons in the helium atom. We will start from experimentally based evidence in order to analyze and explain the physical facts, moving cautiously from a classical to a quantum description, without mixing them up. The overall picture will be that the physical properties of microscopic systems are quantized, as initially shown by Planck and Einstein, and that they are also contextual, i.e., they can be given a physical sense only by embedding a microscopic system within a macroscopic measurement context. Full article

(This article belongs to the Section Quantum Information)

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2 pages, 215 KiB

Open AccessCorrection

Correction: Telesca et al. Visibility Graph Analysis of Reservoir-Triggered Seismicity: The Case of Song Tranh 2 Hydropower, Vietnam. Entropy 2022, 24, 1620

by Luciano Telesca, Anh Tuan Thai, Michele Lovallo and Dinh Trong Cao

Entropy 2024, 26(12), 1003; https://doi.org/10.3390/e26121003 - 22 Nov 2024

Abstract

There was an error in the original publication [...] Full article

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19 pages, 903 KiB

Open AccessArticle

A Contemporary View on Carnot’s Réflexions

by Jan-Peter Meyn

Entropy 2024, 26(12), 1002; https://doi.org/10.3390/e26121002 - 21 Nov 2024

Abstract

Entropy and energy had not yet been introduced to physics by the time Carnot wrote his seminal Réflexions. Scholars continue to discuss what he really had in mind and what misconceptions he might have had. Actually, his work can be read as a [...] Read more.

Entropy and energy had not yet been introduced to physics by the time Carnot wrote his seminal Réflexions. Scholars continue to discuss what he really had in mind and what misconceptions he might have had. Actually, his work can be read as a correct introduction to the physics of heat engines when the term calorique is replaced by entropy and entropy is used as the other fundamental thermal quantity besides temperature. Carnot’s concepts of falling entropy as an analogy to the waterfall, and the separation of real thermal processes into reversible and irreversible processes are adopted. Some details of Carnot’s treatise are ignored, but the principal ideas are quoted and assumed without modification. With only two thermal quantities, temperature and entropy, modern heat engines can be explained in detail. Only after the principal function of heat engines is developed is energy introduced as physical quantity in order to compare thermal engines with mechanical and electrical engines and, specifically, to calculate efficiency. Full article

(This article belongs to the Special Issue 200 Years Anniversary of “Sadi Carnot, Réflexions Sur La Puissance Motrice Du Feu”; Bachelier: Paris, France, 1824)

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3 pages, 169 KiB

Open AccessEditorial

Closing Editorial: Causal Relativistic Hydrodynamics for Viscous Fluids

by Esteban Calzetta

Entropy 2024, 26(12), 1001; https://doi.org/10.3390/e26121001 - 21 Nov 2024

Abstract

Relativistic viscous hydrodynamics [...] Full article

(This article belongs to the Special Issue Causal Relativistic Hydrodynamics for Viscous Fluids)

7 pages, 216 KiB

Open AccessArticle

A Characterization of Optimal Prefix Codes

by Spencer Congero and Kenneth Zeger

Entropy 2024, 26(12), 1000; https://doi.org/10.3390/e26121000 - 21 Nov 2024

Abstract

A property of prefix codes called strong monotonicity is introduced, and it is proven that for a given source, a prefix code is optimal if and only if it is complete and strongly monotone. Full article

(This article belongs to the Special Issue Advances in Information and Coding Theory III)

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11 pages, 11842 KiB

Open AccessArticle

Control of a New Financial Risk Contagion Dynamic Model Based on Finite-Time Disturbance

by Yifeng Wei, Chengrong Xie, Xia Qing and Yuhua Xu

Entropy 2024, 26(12), 999; https://doi.org/10.3390/e26120999 - 21 Nov 2024

Abstract

With the widespread application of chaotic systems in many fields, research on chaotic systems is becoming increasingly in-depth. This article first proposes a new dynamic model of financial risk contagion based on financial principles and discusses some basic dynamic characteristics of the new [...] Read more.

With the widespread application of chaotic systems in many fields, research on chaotic systems is becoming increasingly in-depth. This article first proposes a new dynamic model of financial risk contagion based on financial principles and discusses some basic dynamic characteristics of the new chaotic system, such as equilibrium points, dissipativity, Poincaré diagrams, bifurcation diagrams, etc. Secondly, with the consideration of privacy during data transmission, the method was designed to protect the privacy of controlled systems in finite time based on perturbation. A controller designed for finite time was developed based on Lyapunov stability principles, which achieves system synchronization within a finite time and protects the privacy of the controlled system. The effectiveness was also verified by numerical simulations. Full article

(This article belongs to the Topic Recent Trends in Nonlinear, Chaotic and Complex Systems)

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23 pages, 553 KiB

Open AccessArticle

Inferring About the Average Value of Audit Errors from Sequential Ratio Tests

by Grzegorz Sitek and Mariusz Pleszczyński

Entropy 2024, 26(11), 998; https://doi.org/10.3390/e26110998 - 20 Nov 2024

Abstract

The book amounts are modeled as values of a random variable, represented by a mixture of distributions of both the correct and error-contaminated amounts. The mixing coefficient represents the proportion of items with non-zero error amounts. This study addresses the problem of determining [...] Read more.

The book amounts are modeled as values of a random variable, represented by a mixture of distributions of both the correct and error-contaminated amounts. The mixing coefficient represents the proportion of items with non-zero error amounts. This study addresses the problem of determining the sample size needed for testing statistical hypotheses regarding mean accounting errors. The average sample size is estimated using the Sequential Probability Ratio Test (SPRT), applying the Monte Carlo method. Estimating average audit errors is a common challenge in economic research. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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20 pages, 6191 KiB

Open AccessArticle

How Do Transformers Model Physics? Investigating the Simple Harmonic Oscillator

by Subhash Kantamneni, Ziming Liu and Max Tegmark

Entropy 2024, 26(11), 997; https://doi.org/10.3390/e26110997 - 19 Nov 2024

Abstract

How do transformers model physics? Do transformers model systems with interpretable analytical solutions or do they create an “alien physics” that is difficult for humans to decipher? We have taken a step towards demystifying this larger puzzle by investigating the simple harmonic oscillator [...] Read more.

How do transformers model physics? Do transformers model systems with interpretable analytical solutions or do they create an “alien physics” that is difficult for humans to decipher? We have taken a step towards demystifying this larger puzzle by investigating the simple harmonic oscillator (SHO),

\ddot{x} + 2 γ \dot{x} + ω_{0}^{2} x = 0

, one of the most fundamental systems in physics. Our goal was to identify the methods transformers use to model the SHO, and to do so we hypothesized and evaluated possible methods by analyzing the encoding of these methods’ intermediates. We developed four criteria for the use of a method within the simple test bed of linear regression, where our method was

y = w x

and our intermediate was w: (1) Can the intermediate be predicted from hidden states? (2) Is the intermediate’s encoding quality correlated with the model performance? (3) Can the majority of variance in hidden states be explained by the intermediate? (4) Can we intervene on hidden states to produce predictable outcomes? Armed with these two correlational (1,2), weak causal (3), and strong causal (4) criteria, we determined that transformers use known numerical methods to model the trajectories of the simple harmonic oscillator, specifically, the matrix exponential method. Our analysis framework can conveniently extend to high-dimensional linear systems and nonlinear systems, which we hope will help reveal the “world model” hidden in transformers. Full article

(This article belongs to the Topic AI and Computational Methods for Modelling, Simulations and Optimizing of Advanced Systems: Innovations in Complexity)

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16 pages, 971 KiB

Open AccessArticle

Derangetropy in Probability Distributions and Information Dynamics

by Masoud Ataei and Xiaogang Wang

Entropy 2024, 26(11), 996; https://doi.org/10.3390/e26110996 - 18 Nov 2024

Abstract

We introduce derangetropy, which is a novel functional measure designed to characterize the dynamics of information within probability distributions. Unlike scalar measures such as Shannon entropy, derangetropy offers a functional representation that captures the dispersion of information across the entire support of a [...] Read more.

We introduce derangetropy, which is a novel functional measure designed to characterize the dynamics of information within probability distributions. Unlike scalar measures such as Shannon entropy, derangetropy offers a functional representation that captures the dispersion of information across the entire support of a distribution. By incorporating self-referential and periodic properties, it provides insights into information dynamics governed by differential equations and equilibrium states. Through combinatorial justifications and empirical analysis, we demonstrate the utility of derangetropy in depicting distribution behavior and evolution, providing a new tool for analyzing complex and hierarchical systems in information theory. Full article

(This article belongs to the Special Issue Mathematics in Information Theory and Modern Applications)

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18 pages, 317 KiB

Open AccessArticle

Information-Theoretic Generalization Bounds for Batch Reinforcement Learning

by Xingtu Liu

Entropy 2024, 26(11), 995; https://doi.org/10.3390/e26110995 - 18 Nov 2024

Abstract

We analyze the generalization properties of batch reinforcement learning (batch RL) with value function approximation from an information-theoretic perspective. We derive generalization bounds for batch RL using (conditional) mutual information. In addition, we demonstrate how to establish a connection between certain structural assumptions [...] Read more.

We analyze the generalization properties of batch reinforcement learning (batch RL) with value function approximation from an information-theoretic perspective. We derive generalization bounds for batch RL using (conditional) mutual information. In addition, we demonstrate how to establish a connection between certain structural assumptions on the value function space and conditional mutual information. As a by-product, we derive a high-probability generalization bound via conditional mutual information, which was left open and may be of independent interest. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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19 pages, 1472 KiB

Open AccessArticle

Generalized Filter Bank Orthogonal Frequency Division Multiplexing: Low-Complexity Waveform for Ultra-Wide Bandwidth and Flexible Services

by Yu Xin, Jian Hua, Tong Bao, Yaxing Hao, Ziheng Xiao, Xin Nie and Fanggang Wang

Entropy 2024, 26(11), 994; https://doi.org/10.3390/e26110994 - 18 Nov 2024

Abstract

Terahertz (THz) communication is a crucial technique in sixth generation (6G) mobile networks, which allow for ultra-wide bandwidths to enable ultra-high data rate wireless communication. However, the current subcarrier spacing and the size of fast Fourier transform (FFT) of the orthogonal frequency division [...] Read more.

Terahertz (THz) communication is a crucial technique in sixth generation (6G) mobile networks, which allow for ultra-wide bandwidths to enable ultra-high data rate wireless communication. However, the current subcarrier spacing and the size of fast Fourier transform (FFT) of the orthogonal frequency division multiplexing (OFDM) in 5G NR are insufficient regarding the bandwidth requirements of terahertz scenarios. In this paper, a novel waveform is proposed to address the ultra-wideband issue, namely the generalized filter bank orthogonal frequency division multiplexing (GFB-OFDM) waveform. The main advantages are summarized as follows: (1) The K-point IFFT is implemented by two levels of IFFTs in smaller sizes, i.e, performing M-point IFFT in N times and performing N-point IFFT in M times, where

K = N \times M

. (2) The proposed waveform can accommodate flexible subcarrier spacings and different numbers of the subbands to provide various services in a single GFB-OFDM symbol. (3) Different bandwidths can be supported using a fixed filter since the filtering is performed on each subband. In contrast, the cyclic prefix orthogonal frequency division multiplexing (CP-OFDM) in 4G/5G requires various filters. (4) The existing detection for CP-OFDM can be directly employed as the detector of the proposed waveform. Lastly, the comprehensive simulation results demonstrate that GFB-OFDM outperforms CP-OFDM in terms of the out-of-band leakage, complexity and error performance. Full article

(This article belongs to the Special Issue Advanced New Physical Layer Technologies for Next-Generation Wireless Communications)

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26 pages, 766 KiB

Open AccessArticle

Still No Evidence for an Effect of the Proportion of Non-Native Speakers on Natural Language Complexity

by Alexander Koplenig

Entropy 2024, 26(11), 993; https://doi.org/10.3390/e26110993 - 18 Nov 2024

Abstract

In a recent study, I demonstrated that large numbers of L2 (second language) speakers do not appear to influence the morphological or information-theoretic complexity of natural languages. This paper has three primary aims: First, I address recent criticisms of my analyses, showing that [...] Read more.

In a recent study, I demonstrated that large numbers of L2 (second language) speakers do not appear to influence the morphological or information-theoretic complexity of natural languages. This paper has three primary aims: First, I address recent criticisms of my analyses, showing that the points raised by my critics were already explicitly considered and analysed in my original work. Furthermore, I show that the proposed alternative analyses fail to withstand detailed examination. Second, I introduce new data on the information-theoretic complexity of natural languages, with the estimates derived from various language models—ranging from simple statistical models to advanced neural networks—based on a database of 40 multilingual text collections that represent a wide range of text types. Third, I re-analyse the information-theoretic and morphological complexity data using novel methods that better account for model uncertainty in parameter estimation, as well as the genealogical relatedness and geographic proximity of languages. In line with my earlier findings, the results show no evidence that large numbers of L2 speakers have an effect on natural language complexity. Full article

(This article belongs to the Special Issue Complexity Characteristics of Natural Language)

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24 pages, 1736 KiB

Open AccessArticle

Multi-Label Feature Selection with Feature–Label Subgraph Association and Graph Representation Learning

by Jinghou Ruan, Mingwei Wang, Deqing Liu, Maolin Chen and Xianjun Gao

Entropy 2024, 26(11), 992; https://doi.org/10.3390/e26110992 - 18 Nov 2024

Abstract

In multi-label data, a sample is associated with multiple labels at the same time, and the computational complexity is manifested in the high-dimensional feature space as well as the interdependence and unbalanced distribution of labels, which leads to challenges regarding feature selection. As [...] Read more.

In multi-label data, a sample is associated with multiple labels at the same time, and the computational complexity is manifested in the high-dimensional feature space as well as the interdependence and unbalanced distribution of labels, which leads to challenges regarding feature selection. As a result, a multi-label feature selection method based on feature–label subgraph association with graph representation learning (SAGRL) is proposed to represent the complex correlations of features and labels, especially the relationships between features and labels. Specifically, features and labels are mapped to nodes in the graph structure, and the connections between nodes are established to form feature and label sets, respectively, which increase intra-class correlation and decrease inter-class correlation. Further, feature–label subgraphs are constructed by feature and label sets to provide abundant feature combinations. The relationship between each subgraph is adjusted by graph representation learning, the crucial features in different label sets are selected, and the optimal feature subset is obtained by ranking. Experimental studies on 11 datasets show the superior performance of the proposed method with six evaluation metrics over some state-of-the-art multi-label feature selection methods. Full article

(This article belongs to the Section Multidisciplinary Applications)

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46 pages, 15416 KiB

Open AccessReview

Mathematical Modeling of Physical Reality: From Numbers to Fractals, Quantum Mechanics and the Standard Model

by Marian Kupczynski

Entropy 2024, 26(11), 991; https://doi.org/10.3390/e26110991 - 18 Nov 2024

Abstract

In physics, we construct idealized mathematical models in order to explain various phenomena which we observe or create in our laboratories. In this article, I recall how sophisticated mathematical models evolved from the concept of a number created thousands of years ago, and [...] Read more.

In physics, we construct idealized mathematical models in order to explain various phenomena which we observe or create in our laboratories. In this article, I recall how sophisticated mathematical models evolved from the concept of a number created thousands of years ago, and I discuss some challenges and open questions in quantum foundations and in the Standard Model. We liberated nuclear energy, landed on the Moon and built ‘quantum computers’. Encouraged by these successes, many believe that when we reconcile general relativity with quantum theory we will have the correct theory of everything. Perhaps we should be much humbler. Our perceptions of reality are biased by our senses and by our brain, bending them to meet our priors and expectations. Our abstract mathematical models describe only in an approximate way different layers of physical reality. To describe the motion of a meteorite, we can use a concept of a material point, but the point-like approximation breaks completely when the meteorite hits the Earth. Similarly, thermodynamic, chemical, molecular, atomic, nuclear and elementary particle layers of physical reality are described using specific abstract mathematical models and approximations. In my opinion, the theory of everything does not exist. Full article

(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering III)

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