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Philosophy of Science

The Use of the Axiomatic Method in Quantum Physics

Yvon Gauthier

Published Sep 1, 1971• 17 citations
Publisher

Abstract

Although the introduction of the modern axiomatic method in physics is attributed to Hilbert, it is only recently that physicists and mathematicians have applied it significantly, i.e. on a basis extensive enough to promise fruitful results. Carnap, for one, stresses the importance of the axiomatic method, yet he considers its application in physics as a task for the future.

DOI
10.1086/288384

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Although the introduction of the modern axiomatic method in physics is attributed to Hilbert, it is only recently that physicists and mathematicians have applied it significantly, i.e. on a basis extensive enough to promise fruitful results. Carnap, for one, stresses the importance of the axiomatic method, yet he considers its application in physics as a task for the future.
Motivated by Hilbert's sixth problem on axiomatization of physics, the author is proposing a rather provocative abstract axiomatic framework called S - formalization, where S is an arbitrary physical system .The proposal is an attempt to provide a general axiomatic framework , from which mathematical frameworks of new physical theories can be formulated on firm axiomatic basis and the mathematical frameworks of subjects such as standard (nonrelativistic) quantum mechanics are naturally derived as special cases of this general axiomatic framework. Introduction
We discuss the Hilbert program for the axiomatization of physics in the context of what Hilbert and von Neumann came to call the analytical apparatus and its conditions of reality.We suggest that the idea of a physical logic is the basis for a physical mathematics and we use quantum mechanics as a paradigm case for axiomatics in the sense of Hilbert.Finite probability theory requires finite derivations in the measurement theory of QM and we give a polynomial formulation of local complementation for the metric induced on the topology of the Hilbert space.The conclusion hints at a constructivist physics.
The ongoing progress in quantum theory emphasizes the crucial role of the very basic principles of quantum theory. However, this is not properly followed in teaching quantum mechanics on the graduate and undergraduate levels of physics studies. The existing textbooks typically avoid the axiomatic presentation of the theory. We emphasize usefulness of the systematic, axiomatic approach to the basics of quantum theory as well as its importance in the light of the modern scientific-research context.
The issue of using the physical method in economics is no longer an innovative idea. However, nowadays the methods of mathematical quantum mechanics are also applied to economic sciences. This is the natural result of the fact that as applicable in quantum mechanics, mathematical spaces and tools turn out to be useful in other fields of science. Then it is possible to assume that the problem of the choice of the space is a universal question that is associated not only with mathematics and physics but also with economics or social sciences. In this paper the author considers various formulations of Hilbert space in relation to finite-dimensional quantum mechanics which constitutes a fundament to also apply my outcomes in economics. On the basis of mathematical considerations the author puts forward the hypothesis that the complex Hilbert space is characterized with numerous advantages in relation to its real and quaternionic alternativesThe issue of using the physical method in economics is no longer an innovative idea. However, nowadays the methods of mathematical quantum mechanics are also applied to economic sciences. This is the natural result of the fact that as applicable in quantum mechanics, mathematical spaces and tools turn out to be useful in other fields of science. Then it is possible to assume that the problem of the choice of the space is a universal question that is associated not only with mathematics and physics but also with economics or social sciences. In this paper the author considers various formulations of Hilbert space in relation to finite-dimensional quantum mechanics which constitutes a fundament to also apply my outcomes in economics. On the basis of mathematical considerations the author puts forward the hypothesis that the complex Hilbert space is characterized with numerous advantages in relation to its real and quaternionic alternatives
The quantum inverse scattering method is a means of finding exact solutions of two-dimensional models in quantum field theory and statistical physics (such as the sine-Gordon equation or the quantum non-linear Schrödinger equation). These models are the subject of much attention amongst physicists and mathematicians. The present work is an introduction to this important and exciting area. It consists of four parts. The first deals with the Bethe ansatz and calculation of physical quantities. The authors then tackle the theory of the quantum inverse scattering method before applying it in the second half of the book to the calculation of correlation functions. This is one of the most important applications of the method and the authors have made significant contributions to the area. Here they describe some of the most recent and general approaches and include some new results. The book will be essential reading for all mathematical physicists working in field theory and statistical physics.
This volume will act as a guide through the various aspects of quantum mechanics. It not only covers the basics but also addresses new themes developed in the field of quantum mechanics in recent years. These include quantum mechanics in relation to electronics, quantum dots, spintronics, cryptography, and other more theoretical aspects, such as the path integral formulation and supersymmetric quantum mechanics. The volume presents a number of mathematical tools and physical consequences derived from quantum mechanics. The starting point of the volume is a very brief review of the phenomenology associated with the origins of quantum theory, as the branch of science was understood in the beginning of the 20th century, leading to an interpretation of the results at that time. Key features: • Provides understanding and demystification of the quantum theory • Presents applications to information theory and encryption • Introduces applications to medicine, in both treatment and diagnosis • Covers applications to modern communications systems • Looks at the philosophical implications of quantum mechanics, its reality, and its perception • Describes the application to the basics of solid state devices