17 Quantization
Given any Hamiltonian classical mechanical system with phase space , physics textbooks have a standard recipe for producing a quantum system, by a method known as “canonical quantization”. We will see that for linear functions on phase space, this is just the construction we have already seen of a unitary representation of the Heisenberg Lie algebra, the Schr¨odinger representation. The Stone-von Neumann theorem assures us that this is the unique such construction, up to unitary equivalence. We will also see that this recipe can only ever be partially successful: the Schr¨odinger representation gives us a representation of a sub-algebra of the Lie algebra of all functions on phase space (the polynomials of degree two and below), but a no-go theorem shows that this cannot be extended to a representation of the full infinite dimensional Lie algebra. Recipes for quantizing higher-order polynomials will always sufer from a lack of uniqueness, a phenomenon known to physicists as the existence of “operator ordering ambiguities.”
In later chapters we will see that this quantization prescription does give unique quantum systems corresponding to some Hamiltonian systems (in particular the harmonic oscillator and the hydrogen atom), and does so in a manner that allows a description of the quantum system purely in terms of representation theory.
Chapter contents
- 17.1 Canonical quantization
- 17.2 The Groenewold-van Hove no-go theorem
- 17.3 Canonical quantization in d dimensions
- 17.4 Quantization and symmetries
- 17.5 More general notions of quantization
- 17.6 For further reading
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正文：英文 · OCR 机器稿 · 待校对
核对状态：OCR 机器稿 · 待校对
原书 PDF · 印刷页 197、198、199、200、201、202、203
来源版本：2025-10-20
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