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Springer eBooks

Principles and Lessons of Quantum Physics. A Review of Interpretations, Mathematical Formalism, and Perspectives

Rudolf Haag

Published Jan 1, 1996• 1 citation
Publisher
DOI
10.1007/978-3-642-61458-3_7

Related papers

Understanding quantum mechanics : an introduction to the abstract formalism and the basic ideas of quantum mechanics

GORDON J. TROUP

•Jan 1, 1968•0 citations

Information and the Reconstruction of Quantum Physics

Gregg S. Jaeger

Abstract The reconstruction of quantum physics has been connected with the interpretation of the quantum formalism, and has continued to be so with the recent deeper consideration of the relation of information to quantum states and processes. This recent form of reconstruction has mainly involved conceiving quantum theory on the basis of informational principles, providing new perspectives on physical correlations and entanglement that can be used to encode information. By contrast to the traditional, interpretational approach to the foundations of quantum mechanics, which attempts directly to establish the meaning of the elements of the theory and often touches on metaphysical issues, the newer, more purely reconstructive approach sometimes defers this task, focusing instead on the mathematical derivation of the theoretical apparatus from simple principles or axioms. In its most pure form, this sort of theory reconstruction is fundamentally the mathematical derivation of the elements of theory from explicitly presented, often operational principles involving a minimum of extra‐mathematical content. Here, a representative series of specifically information‐based treatments—from partial reconstructions that make connections with information to rigorous axiomatizations, including those involving the theories of generalized probability and abstract systems—is reviewed.

Annalen der Physik•Sep 12, 2018•13 citations

An Introduction to the mathematical formalization of quantum mechanics

R. Bonventre

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Arkady Plotnitsky

This article considers the relationships between quantum theory (QT) and quantum-like theories (QLTs), theories using mathematical models based on the formalism of QT, from a reverse perspective, that of QLTs. The article argues that QT is no longer a theory of the behaviour, in particular motion, of physical objects, as was the case in classical physics and relativity. Instead, QT is a form of decision theory, involving a special 'topology' of decisions, using the term topology in part metaphorically, but only in part, because it applies in its proper mathematical sense to the formalism of QT. Part of this topology is the concept of free will, reconsidered through the concept of decision. This character of QT is grounded in a particular type of interpretations of QT, 'reality without realism' (RWR) interpretations. To address the affinities and differences between QT and QLTs, the article introduces two new principles: 'the unambiguity principle', equally applicable in QT and QLTs, or in mathematics and science in general, and 'the free will principle', only applicable in QLTs and not in QT. The article also reflects on the limits of quantum-like sciences (QLSs) and mathematical-experimental science in general in dealing with human thinking and decision making.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.

Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences•Nov 27, 2025•4 citations

Mecánica Cuántica: Una Introducción Matemática para Físico-Matemáticos

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This book presents a concise introduction to the mathematical structure of quantum mechanics from the perspective of operator algebras for systems with a finite number of degrees of freedom. It begins with a review of the mathematical tools required throughout the text, followed by an exposition of the standard Schrödinger formulation based on the Dirac–von Neumann axioms. The operator algebraic formulation is then developed from a set of physically motivated principles, providing an alternative and conceptually transparent description of quantum systems. The book concludes with an application comparing the standard and operator algebraic formalisms. Written in Spanish, this book is intended primarily for Spanish-speaking undergraduate and graduate students in physics, mathematics, and mathematical physics, aiming to provide accessible access to modern mathematical formulations of quantum mechanics.

•Jul 16, 2026•0 citations

A Mathematical Primer on Quantum Mechanics

Alessandro Teta

UNITEXT for physics•Jan 1, 2018•8 citations
Abstract The reconstruction of quantum physics has been connected with the interpretation of the quantum formalism, and has continued to be so with the recent deeper consideration of the relation of information to quantum states and processes. This recent form of reconstruction has mainly involved conceiving quantum theory on the basis of informational principles, providing new perspectives on physical correlations and entanglement that can be used to encode information. By contrast to the traditional, interpretational approach to the foundations of quantum mechanics, which attempts directly to establish the meaning of the elements of the theory and often touches on metaphysical issues, the newer, more purely reconstructive approach sometimes defers this task, focusing instead on the mathematical derivation of the theoretical apparatus from simple principles or axioms. In its most pure form, this sort of theory reconstruction is fundamentally the mathematical derivation of the elements of theory from explicitly presented, often operational principles involving a minimum of extra‐mathematical content. Here, a representative series of specifically information‐based treatments—from partial reconstructions that make connections with information to rigorous axiomatizations, including those involving the theories of generalized probability and abstract systems—is reviewed.
This article considers the relationships between quantum theory (QT) and quantum-like theories (QLTs), theories using mathematical models based on the formalism of QT, from a reverse perspective, that of QLTs. The article argues that QT is no longer a theory of the behaviour, in particular motion, of physical objects, as was the case in classical physics and relativity. Instead, QT is a form of decision theory, involving a special 'topology' of decisions, using the term topology in part metaphorically, but only in part, because it applies in its proper mathematical sense to the formalism of QT. Part of this topology is the concept of free will, reconsidered through the concept of decision. This character of QT is grounded in a particular type of interpretations of QT, 'reality without realism' (RWR) interpretations. To address the affinities and differences between QT and QLTs, the article introduces two new principles: 'the unambiguity principle', equally applicable in QT and QLTs, or in mathematics and science in general, and 'the free will principle', only applicable in QLTs and not in QT. The article also reflects on the limits of quantum-like sciences (QLSs) and mathematical-experimental science in general in dealing with human thinking and decision making.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.
This book presents a concise introduction to the mathematical structure of quantum mechanics from the perspective of operator algebras for systems with a finite number of degrees of freedom. It begins with a review of the mathematical tools required throughout the text, followed by an exposition of the standard Schrödinger formulation based on the Dirac–von Neumann axioms. The operator algebraic formulation is then developed from a set of physically motivated principles, providing an alternative and conceptually transparent description of quantum systems. The book concludes with an application comparing the standard and operator algebraic formalisms. Written in Spanish, this book is intended primarily for Spanish-speaking undergraduate and graduate students in physics, mathematics, and mathematical physics, aiming to provide accessible access to modern mathematical formulations of quantum mechanics.