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I'm not sure if this question is too mathematical so it could be better on the mathematics forum, but I believe theoretical physicists are equipped to answer this abstractly.

So I haven't found any book that defines the canonical quantization rigorously, so here is my attempt:

The canonical quantization $q$ is a function defined:

$q: \mathbb{P} \to V$ where $\mathbb{P}$ is the phase space, and $$V = \text{span}\left\{ \vec{x}, i\vec{\nabla}\right\}$$ such that $q(\vec{x}) = \hat{\vec{x}}, q(\vec{p}) = i \vec{\nabla}$.

Also, the phase space is an inner product space, so we define: $$q(\vec{a} \cdot \vec{b}) = \frac{q(\vec{a}) \cdot q(\vec{b}) + q(\vec{b}) \cdot q(\vec{a})}{2}$$ in order to keep everything Hermitian and it's easy to show things are well defined. The problem I faced is when I want to define it over the cross product:

How should I define $q(\vec{a}\times \vec{b})$?

Vincent Thacker
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Habouz
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1 Answers1

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There is no rigorous formulation of canonical quantization, among other things due to the Groenewold-van Hove theorem (see e.g. this answer of mine) stating that there is no map from functions on phase space to operators on Hilbert space that has all properties we'd like a quantization map to have.

ACuriousMind
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