Mathematical Physics of Quantum Mechanics

Neumann, J. v., Wheeler, N. A.

Boris Khots, Dmitriy Khots

Certain results that have been predicted by Quantum Mechanics (QM) theory are not always supported by experiments. This defines a deep crisis in contemporary physics and, in particular, quantum mechanics. We believe that, in fact, the mathematical apparatus employed within today's physics is a possible reason. In particular, we consider the concept of infinity that exists in today's mathematics as the root cause of this problem. We have created Observer's Mathematics that offers an alternative to contemporary mathematics. This paper is an attempt to relay how Observer's Mathematics may explain some of the contradictions in QM theory results. We consider the Hamiltonian Mechanics, Newton equation, Schrodinger equation, two slit interference, wave-particle duality for single photons, uncertainty principle, Dirac equations for free electron in a setting of arithmetic, algebra, and topology provided by Observer's Mathematics (see www.mathrelativity.com). Certain results and communications pertaining to solution of these problems are provided.

Maurice A. de Gosson

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.

Anders Wimo

A. Mangoni

Yuli V. Nazarov, Jeroen Danon

This book has grown out of a one term course on quantum mechanics for third year mathematics students taught in the School of Mathematics at the University of Bristol. The genesis of quantum mechanics is in the field of physics, but from early on the infusion of new mathematics into the field has often proven fruitful (e.g. Hilbert space, matrix theory, the mathematical framework of Hamiltonian mechanics, tensor products, etc.). This course seeks to develop the mathematical framework that will equip the student for further studies in quantum mechanics while, at the same time, providing some insight into “what it means” physically.