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Emergence Of The Quantum From The Classical: Mathematical Aspects Of Quantum Processes

Maurice A. de Gosson

Published Nov 10, 2017• 2 citations
OpenAlex

Abstract

The emergence of quantum mechanics from classical world mechanics is now a well-established theme in mathematical physics. This book demonstrates that quantum mechanics can indeed be viewed as a refinement of Hamiltonian mechanics, and builds on the work of George Mackey in relation to their mathematical foundations. Additionally when looking at the differences with classical mechanics, quantum mechanics crucially depends on the value of Planck's constant h. Recent cosmological observations tend to indicate that not only the fine structure constant α but also h might have varied in both time and space since the Big Bang. We explore the mathematical and physical consequences of a variation of h; surprisingly we see that a decrease of h leads to transitions from the quantum to the classical. Emergence of the Quantum from the Classical provides help to undergraduate and graduate students of mathematics, physics and quantum theory looking to advance into research in the field. Readership: Undergraduate and graduate students of mathematics and physics, interested in analysis and differential equations, probability and statistics, geometry and topology and mathematical physics.

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The emergence of quantum mechanics from classical world mechanics is now a well-established theme in mathematical physics. This book demonstrates that quantum mechanics can indeed be viewed as a refinement of Hamiltonian mechanics, and builds on the work of George Mackey in relation to their mathematical foundations. Additionally when looking at the differences with classical mechanics, quantum mechanics crucially depends on the value of Planck's constant h. Recent cosmological observations tend to indicate that not only the fine structure constant α but also h might have varied in both time and space since the Big Bang. We explore the mathematical and physical consequences of a variation of h; surprisingly we see that a decrease of h leads to transitions from the quantum to the classical. Emergence of the Quantum from the Classical provides help to undergraduate and graduate students of mathematics, physics and quantum theory looking to advance into research in the field. Readership: Undergraduate and graduate students of mathematics and physics, interested in analysis and differential equations, probability and statistics, geometry and topology and mathematical physics.
This Element explores the formal and conceptual foundations of phase space formulations of classical and quantum mechanics. It provides an overview of the core mathematical and physical content of Hamiltonian mechanics, stochastic phase space mechanics, contact Hamiltonian mechanics, and open and closed quantum mechanics on phase space. The formal material is unified via three interpretative themes relating to structured possibility spaces, Liouville's theorem and its failure, and the classical and quantum notions of open and closed systems. This Element book is intended for researchers and graduate students in the philosophy and foundations of physics with an interest in the conceptual foundations of physical theory.
This book examines a number of problems of quantum mechanics, most of which are not usually discussed. What is the origin of probabilities in the mechanics of the microworld? What is the nature of Planck's constant h? What is the nature of probability amplitudes? What is the wave function? A system of axioms for quantum theory is formulated. A model is studied according to which a classical oscillator in a thermostat can be interpreted as a quantum one. The measurement problem is discussed in detail. For advanced undergraduate students, graduate students, and specialists interested in the foundations of quantum theory.
Introducing a geometric view of fundamental physics, starting from quantum mechanics and its experimental foundations, this book is ideal for advanced undergraduate and graduate students in quantum mechanics and mathematical physics. Focusing on structural issues and geometric ideas, this book guides readers from the concepts of classical mechanics to those of quantum mechanics. The book features an original presentation of classical mechanics, with the choice of topics motivated by the subsequent development of quantum mechanics, especially wave equations, Poisson brackets and harmonic oscillators. It also presents new treatments of waves and particles and the symmetries in quantum mechanics, as well as extensive coverage of the experimental foundations.
Quantum mechanics is a cornerstone of modern physics, yet its fundamental origin and mathematical foundation remain elusive. In this paper, we derive the complete structure of quantum mechanics from a purely classical field theory. By gauging the double cover of the Lorentz symmetry group of space–time, the resulting equations of motion naturally yield all postulates of quantum mechanics, including the emergence of the Planck constant. This approach provides a coherent interpretation and demonstrates that quantum mechanics can be viewed as a mathematically consistent classical field theory.
The principle of action plays a central role in both classical and quantum mechanics, yet its physical meaning is often introduced in a purely formal manner. In this work, the concept of action is examined from a scale-based perspective that clarifies its role in the quantum–classical correspondence. By emphasizing the dimensionless ratio S/ℏ, we show that the emergence of classical dynamics can be understood as a consequence of quantum phase coherence rather than as an independent postulate. Within the path integral formulation, classical trajectories arise naturally from stationary-phase conditions when the action associated with a physical process becomes large compared to Planck’s constant. This interpretation provides a unified conceptual framework in which variational principles, quantum interference, and classical determinism are linked through the scale of action. The analysis highlights the role of Planck’s constant as a universal parameter governing the physical relevance of alternative histories, analogous to the function of Boltzmann’s constant in statistical physics. Beyond its theoretical implications, the proposed perspective offers pedagogical value by providing a coherent bridge between classical and quantum mechanics suitable for advanced undergraduate and early graduate physics education.
Offering a detailed account of the key concepts and mathematical apparatus of quantum mechanics, this textbook is an ideal companion to both undergraduate and graduate courses. The formal and practical aspects of the subject are explained clearly alongside examples of modern applications, providing students with the tools required to thoroughly understand the theory and apply it. The authors provide an intuitive conceptual framework that is grounded in a coherent physical explanation of quantum phenomena, established over decades of teaching and research in quantum mechanics and its foundations. The book's educational value is enhanced by the inclusion of examples and exercises, with solutions available online, and an extensive bibliography is provided. Notes throughout the text provide fascinating context on the tumultuous history of quantum mechanics, the people that developed it, and the questions that still remain at its center. This title is also available as Open Access on Cambridge Core.