# How can a classical phase space be unquantizable?

On page 2 of the paper "2 + 1 dimensional gravity as an exactly soluble system" Witten claims that:

Depending on its topology, a finite-dimensional phase space might be unquantizable,

How a classical phase space might be unquantizable? Is it special to finite dimensional phase space or some infinite dimensional phase spaces also might be unquantizable? What are sufficient and necessary conditions quantizability of a classical phase space?

Is not it a problem for quantum mechanics which some classical system have not quantum counterpart?

• Interesting to see also what Witten means by "quantization of a classical state space". – pglpm Dec 6 '20 at 10:04
• It is maybe related to the Groenewold-van Hove theorem for special phase spaces, but it is a supposition. – Jeanbaptiste Roux Dec 6 '20 at 10:24

1. Ref. 1 is likely referring to the integral quantization condition $$\frac{1}{2\pi\hbar}\omega~\in~H^2(M,\mathbb{Z})\tag{1}$$ in geometric quantization of a symplectic manifold $$(M,\omega)$$, aka. classical phase space, cf. Ref. 2.

2. The condition (1) is related to the Bohr-Sommerfeld quantization condition, cf. e.g. this Phys.SE post.

3. Note that the remainder of Ref. 1 uses covariant phase space quantization, cf. Ref. 3.

References:

1. E. Witten, 2+1D gravity as an exactly soluble system, Nucl. Phys. B311 (1988) 46.

2. N.M.J. Woodhouse, Geometric Quantization, 1992; section 8.3.

3. C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories. Published in Three hundred years of gravitation (Eds. S. W. Hawking and W. Israel), (1987) 676.

• Would you please explain more about it, I could not get the point of your answer. I know classical mechanics is described by $(M, \omega)$, where $\omega$ is a non-degenerate closed two-form. – Arian Dec 6 '20 at 15:07
• I updated the answer. – Qmechanic Dec 6 '20 at 15:35