Basic quantisation rules of semiclassical analysis

University essay from Uppsala universitet/Analys och sannolikhetsteori

Author: Anders Israelsson; [2016]

Keywords: Semiclassical analysis; Gårding inequality; Borel Theorem; Kohn-Nirenberg; Weyl; Quantization; Symbol; Asympotic expansion; Schwartz space; Tempered distribution; Pseudodifferential operator; Fourier transform; Stationary phase; Microlocal analysis; Oscillatory integral; Poisson Bracket; Lie Bracket;

Abstract: Semiclassical analysis is the study of how to connect classical mechanics with quantummechanics in a mathematically rigorous way. What is crucial for quantum mechanics isthat the operators that occur are self-adjoint and map a subset of L2 into L2. This hasbeen proven in this thesis. Also, a technique of asymptotic expansion of functions interms of Planck's constant h is developed, in order to study the so called semiclassicallimit (i.e. the limit as h approaches 0) We also develop different approaches toquantisation, and a calculus of operators (i.e. quantum observables) based on acalculus for their corresponding symbols (or classical observables).

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