Classical and Quantum Mechanics via Lie algebras

Arnold Neumaier

This is to announce the availability of a draft of a book
Arnold Neumaier and Dennis Westra,
Classical and Quantum Mechanics via Lie algebras,
Cambridge University Press, to appear (2009?).
arXiv:0810.1019
http://lanl.arxiv.org/abs/0810.1019
(333 pages without references)
Abstract and table of contents are given below.

Your comments are welcome.
Please send them to the newsgroup if they are of general interest,
and to me directly otherwise.


Arnold Neumaier


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The goal of this book is to present classical mechanics, quantum
mechanics, and statistical mechanics in an almost completely algebraic
setting, thereby introducing mathematicians, physicists, and
engineers to the ideas relating classical and quantum mechanics with
Lie algebras and Lie groups. The book emphasizes the
closeness of classical and quantum mechanics, and the material is
selected in a way to make this closeness as apparent as possible.

Much of the material covered here is not part of standard
textbook treatments of classical or quantum mechanics (or is only
superficially treated there). For physics students who want to
get a broader view of the subject, this book may therefore serve
as a useful complement to standard treatments of quantum mechanics.

Almost without exception, this book is about precise concepts and
exact results in classical mechanics, quantum mechanics, and
statistical mechanics. The structural properties of
mechanics are discussed independent of computational techniques for
obtaining quantitatively correct numbers from the assumptions made.
The standard approximation machinery for calculating from first
principles explicit thermodynamic properties of materials, or
explicit cross sections for high energy experiments can be found in
many textbooks and is not repeated here.

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Part I An invitation to quantum mechanics

1 Motivation
2 Classical oscillating systems
3 Spectral analysis


Part II Statistical mechanics

4 Phenomenological thermodynamics
5 Quantities, states and statistics
6 The laws of thermodynamics
7 Models, statistics, and measurements


Part III Lie algebras and Poisson algebras

8 Lie algebras
9 Mechanics in Poisson algebras
10 Representation and classification


Part IV Mechanics and differential geometry

11 Fields, forms, and derivatives
12 Conservative mechanics on manifolds
13 Hamiltonian quantum mechanics


Part V Representations and spectroscopy

14 Harmonic oscillators and coherent states
15 Spin and fermions
16 Highest weight representations
17 Spectroscopy and spectra