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Journal of Physics A Mathematical and General

Quantum mechanics via extended measures

S. K. Srinivasan

Published Dec 7, 1997• 4 citations
Publisher

Abstract

Quantum phenomena are discussed using extended measures. The complex measurable processes introduced earlier are studied in greater depth in the context of harmonic oscillators. Some of the fundamental problems of radiation and interaction of radiation with matter are analysed in the general frame of extended measure theory. The connection to Feynman path integrals is also discussed and it is shown how the present approach circumvents the divergence difficulties particularly in the estimation of Lamb shift.

DOI
10.1088/0305-4470/30/23/026

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Quantum phenomena are discussed using extended measures. The complex measurable processes introduced earlier are studied in greater depth in the context of harmonic oscillators. Some of the fundamental problems of radiation and interaction of radiation with matter are analysed in the general frame of extended measure theory. The connection to Feynman path integrals is also discussed and it is shown how the present approach circumvents the divergence difficulties particularly in the estimation of Lamb shift.
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Quantum measurement problem has existed many years and inspired a large of literature in both physics and philosophy, but there is still no conclusion and consensus on it. We show it can be subsumed into the quantum theory if we extend the Feynman path integral by considering the relativistic effect of Feynman paths. According to this extended theory, we deduce not only the Klein-Gordon equation, but also the wave-function-collapse equation. It is shown that the stochastic and instantaneous collapse of the quantum measurement is due to the “potential noise" of the apparatus or environment and “inner correlation" of wave function respectively. Therefore, the definite-status of the macroscopic matter is due to itself and this does not disobey the quantum mechanics. This work will give a new recognition for the measurement problem.
"Feynman path integrals are ubiquitous in quantum physics, even if a large part of the scientific community still considers them as a heuristic tool that lacks a sound mathematical definition. Our book aims to refute this prejudice, providing an extensive and self-contained description of the mathematical theory of Feynman path integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. This second edition presents a detailed discussion of the general theory of complex integration on infinite dimensional spaces, providing on one hand a unified view of the various existing approaches to the mathematical construction of Feynman path integrals and on the other hand a connection with the classical theory of stochastic processes. Moreover, new chapters containing recent applications to several dynamical systems have been added. This book bridges between the realms of stochastic analysis and the theory of Feynman path integration. It is accessible to both mathematicians and physicists"--
Quantum measurement problem is still unconsensus since it has existed many years and inspired a large of literature in physics and philosophy. We show it can be subsumed into the quantum theory if we extend the Feynman path integral by considering the relativistic effect of Feynman paths. According to this extended theory, we deduce not only the Klein-Gordon equation, but also the wave-function-collapse equation. It is showing that the stochastic and instantaneous collapse of the quantum measurement is due to the "potential noise" of the apparatus or environment and "inner correlation" of wave function respectively. Therefore, the definite-status of the macroscopic matter is due to itself and this does not disobey the quantum mechanics. This work will give a new recognition for the measurement problem.
Path integration methods are of crucial importance to quantum mechanics and quantum field theory. There are multiple ways the path integral can be constructed, one method uses the link between the Fokker-Planck and the Langevin equations, as is covered in this thesis. This is related to the standard derivation of the path integral in physics, in which the time is discretised. We study the mathematical problems related to the Feynman path integral, in particular the impossibility of a Lebesgue-type measure on the space of paths. It is discussed how oscillatory integrals can be used to have a well-defined `integral' on an infinite dimensional space. This formalism is subsequently applied to both the harmonic and the anharmonic oscillator. We prove an infinite dimensional oscillatory integral exists and obtain a convergent expression for both these cases, under conditions. Examples of these conditions are the initial wavefunctions being Schwartz functions as well as conditions on the endtime, angular frequency and coupling constant. Finally, a possible approach to establish an oscillatory integral for the hydrogen atom is discussed. It is proven that the result is no longer independent of the sequence of projection operators, which is a key step towards a rigorous path integral for this system.
The Feynman path-integral formulation of quantum mechanics is developed. The math- ematical framework is worked out from Schrödinger's equation, illustrating the compati- bility of the two formulations. We introduce the sum over all paths, known as the kernel K, and discuss its implications for events occurring in succession. We compute the kernel for quadratic Lagrangians. Then, making use of imaginary time, we consider the discrete version of the kernel, known as the Euclidean path integral, and use it to solve the (quan- tum) harmonic and anharmonic oscillators through numerical Monte Carlo simulations. Lastly, we use the framework to quantize the electromagnetic field, as a motivation to discuss perturbation theory via Feynman diagrams.