Knowledge, locality, information

Knowledge, Science, Information theory  · 

Table of content

Introduction and motivation

The universe is characterized by a multitude of connections between its parts and this property can be observed at all levels and in all sectors of knowledge. It seems that there are very few elements whose connections with the rest of reality are not known, that is, elements completely disconnected. Artists and scientists in every era have grasped new connections and better understood the already known ones. Also, it rarely happens that elements considered connected then turn out to be completely independent: the entire Western and Eastern thought (the thought of all humanity) has searched and is constantly searching for new connections. For the first time in human history one begins to perceive a sort of unity, perhaps already present in ancient thought, which was then lost with the progressive specialization of knowledge in recent centuries.

Arguments not very different from this one have slowly led us to think that some of the principles on which human thought and knowledge are built are in fact principles with which nature operates and manifests itself. Speaking in informal terms (the formal treatment will be proposed later) it is possible to identify a fundamental opposition that can be found both in the arts and in science. In physical terms it declines in the problem of locality/non-locality of phenomena and theories. In the context of quantum mechanics (QM), non-locality has taken on a particular meaning that we will analyze later. However it is interesting to note that the theories of classical mechanics, assuming the speed of light as an infinite speed, represent a type of non-local theory since the time elapsed between a cause and its effect is exactly 0. Let consider Newton theory of gravity: if the Sun disappeared at this very moment, at exactly the same instant the Earth would stop rotating around its star and will move in a straight line thus moving away from the solar system which would instantly begin to disintegrate. More recent experimental results and more precise theoretical descriptions motivated by new discoveries starting from Maxwell electromagnetic theory, have shown that any information can only travel at the speed of light in vacuum $c$. In the previous example, if the Sun disappeared at this very moment, Earth would begin to abandon its orbit only when the information “the Sun has disappeared” had reached it and since the information can only travel at $c$ and since $c$ is not infinite then the lapse of time between the disappearance of the Sun and the modification of the trajectory of the Earth will be greather than 0. In these terms, in modern physics, we speak of local theory only when the time elapses between a cause and an effect is such thatn at least a beam of light starting from the cause can reach the effect. Hence, we speak of non-local theory when there are systems connected in such a way that the modification of one of the two elements instantly causes the modification of the element connected to it.

QM is a non-local theory [berkovitz]. BUT this in no way violates the fundamental principles of physics since when non-local phenomena occur, no information has traveled from one element to another: a non-local connection is not-informational. Since there is no information involved in the connection, its speed is not required to be less or equal that $c$. Since there is no information involved in the connection, it is not required that the principle of locality be respected and therefore the constraint that the signal must travel at the maximum speed of light is dropped. At this point the locality/non-locality problem has become an informational/non-informational type problem on which we will dwell at length. Consider the following table:

Principle Property Justification
Locality $v\leq c$ Definition of locality
Information $v\leq c$ GR
Non-locality $v>c$ Negation of definition of locality
Non-information $v>c$ Negation of GR

Hence,

Locality status Property Informative status
Locality $v\leq c$ Information
Non-locality $v>c$ Non-information

Or,

$locality\Leftrightarrow v\leq c\Leftrightarrow information\neq 0\land non-locality\Leftrightarrow v>c\Leftrightarrow information=0$

Information

Information is a key concept whose general definition, shared by all, is currently an open problem. Each discipline adopts a particular definition and allows to compare systems and phenomena. Information is particularly interesting because it allows eliminating the problem of measurement in order to describe the phenomena not on the basis of their quantitative but qualitative properties.

Consider the following interpretation of the information. Classical mechanics, developed in the context of the enlightenment, deterministic and mechanistic western culture, has become the symbol of the predictive accuracy of theories. The “broken dream” of classical mechanics was to describe reality by predicting with absolute certainty the evolution of a system knowing exactly all its previous states. Statistical mechanics and thermodynamics have taught that describing complex systems formed by many bodies is a problem that cannot be solved exactly in the general form. Therefore classical mechanics applied to a multitude of particles predicts the average behaviour of the system as a whole but avoids describing each component individually. Frequently, the general solutions are numerical or simulations produced through a large computing power: it is possible to simulate the behavior of a complex system but not to solve it analytically in the general case.

Proceeding along this road it became clear that, besides being impossible to analytically solve the behavior of a complex system, it is impossible to obtain a complete description of the behavior of a system since it is not known how many parameters are needed to describe it completely [nagy]. This is why it is necessary to introduce the concept of information [gentle]. A research conducted on ETO [eto] prove the relevance of the concept of information in the scientific world: the following figure represents the results of the survey [35006][57652][60834][86103]. In particular we want to draw attention to a particular definition of information called Fisher Information [fisher] which brings the statistical problem closer to the world of algebra and non-Euclidean geometries. At this point begins the research that transversally merges different disciplines. From algebra to atomic physics (statistical mechanics and QM) up to GR.

The following table (updated at 24 january 2023) can be found, always updated, at the link and represents some information that can help understand the scientific interest in Fisher Information, as well as the research perspectives and the application effects.

Cluster ID Most common subject category CSET phrases Cluster size# Growth rating# sorted descending Citation rating# Average paper age# Extreme growth predicted# % U.S.-affiliated# % China-affiliated# % articles in English# % articles in Chinese# AI percentage# Robotics percentage (English only)# Computer vision percentage (English only)# Natural language processing percentage (English only)# Patent impact rating# % articles with patent citation (beta)# Number of citing patents# % industry-affiliated#
86103 social science Sustainable Development, Fisher Information, Strategic Management, sustainability, Urban Systems Design 119 72.02 45.14 9.02 no 36.97 13.45 86.58 5.17 7.00 0.00 2.04 1.02 0.00 0.00 0 1.26
57652 computer science Optimal Sensor Placement, Target Localization, local positioning systems, UAVs, positioning accuracy 243 58.26 46.30 6.69 no 8.23 49.79 87.55 9.93 22.78 15.19 0.84 0.00 61.80 0.92 6 4.26
60834 mathematics Fisher Information Measure, time series, Fisher Shannon analysis, Shannon entropy, cell membrane 50 55.85 43.75 16.93 no 36.00 14.00 92.87 1.50 4.44 0.00 0.00 0.00 54.37 0.64 3 0.97
35006 physics Shannon information entropy, Fisher information, Shannon quantum information, Shannon entropies, Fisher entropy 195 47.76 65.64 13.82 no 3.08 14.87 92.40 1.29 0.54 0.00 0.00 0.00 50.16 0.11 2 0.00
46425 mathematics entropy power inequality, Rényi entropy power, Shannon Entropy, Fisher information, differential entropy 170 14.76 49.27 13.49 no 42.94 14.71 93.48 2.27 5.39 0.00 0.00 0.00 42.96 0.15 1 2.60

Research line

Free research is encouraged in any case, driven by curiosity and the desire for knowledge. The project, in its entirety, aims to find an order that transcends the problem of measurement assuming a scalar field in which the information is inextricably contained by the observer who modifies its state upon observation. Beyond the physical problem, the mathematical problem aims to formulate a coherent model in which it is possible to trace the inconsistencies mentioned in introduction and motivation section. Below is the list (constantly being updated) of problems divided by state and topic.

Id Problem Status Topic Result document
1 Fisher Information (FI) properties Completed Calculus, Statistics Running
2 Fisher Information matrix (FIM) properties Completed Calculus, Statistics, Algebra Running
3 Generalization of linear operators (GO as general operators, LO as linear operators) Running Calculus, Algebra  
4 Is it always possible to reduce a GO to a linear operator? Open Calculus, Algebra  
5 Is it necessary to introduce a Fisher Information Tensor (FIT)? Open Algebra  
6 Formalization of wave equations in terms of GO (or LO according to 4) Open Calculus, Algebra, QM  
7 Formalization of field equations in terms of GO (or LO according to 4) Open Algebra, GR  
8 Eigenvalues and eigenvectors for a GO (or LO according to 4) Open Calculus, Algebra, Computer algebra  
9 FIM (or FIT according to 5) of pdf in the general case Running Calculus, Algebra  
10 FIM (or FIT according to 5) of allowed wave function Running Calculus, QM, Computer algebra  
11 Loss of information in measuring processes Open Computer algebra, CS  
12 Relationship between FIM (or FIT according to 5) and entropy in CS Open Computer algebra, CS  
13 Relation between FIM (or FIT according to 5) and entropy in QM Open QM  

CS: computer science

https://meet.google.com/jau-dhck-aeo

Dirac equation video

  1. https://www.youtube.com/watch?v=OCuaBmAzqek
  2. https://www.youtube.com/watch?v=tR6UebCvFqE
  3. https://www.youtube.com/watch?v=0DL-Xa1f3QI

QFT playlist

https://youtube.com/playlist?list=PLsp_BbZBIk_6_5pi9tHHmoVJzjqpfBkgJ

Bibliograpy

  1. Nagy, Á. Int J Quantum Chem 2022, 122( 8), e26679. https://doi.org/10.1002/qua.26679
  2. Berkovitz, Joseph, “Action at a Distance in Quantum Mechanics”, The Stanford Encyclopedia of Philosophy (Spring 2016 Edition), Edward N. Zalta (ed.), https://plato.stanford.edu/archives/spr2016/entries/qm-action-distance/
  3. Frieden, B. Roy. Physics from Fisher Information: A Unification. Cambridge University Press, 1998, https://www.cambridge.org/core/books/physics-from-fisher-information/EAD061980A31B012779A6BC5D7F93822
  4. Gentle, James E. “Theory of statistics.” George Mason University (2013).
  5. Sciencemap.eto.tech, “ETO Map of Science.”, https://sciencemap.eto.tech/
    1. Sciencemap.eto.tech, “ETO Map of Science.”, https://sciencemap.eto.tech/cluster/?cluster_id=86103
    2. Sciencemap.eto.tech, “ETO Map of Science.”, https://sciencemap.eto.tech/cluster/?cluster_id=60834
    3. Sciencemap.eto.tech, “ETO Map of Science.”, https://sciencemap.eto.tech/cluster/?cluster_id=57652
    4. Sciencemap.eto.tech, “ETO Map of Science.”, https://sciencemap.eto.tech/cluster/?cluster_id=35006