nLab
geometric quantization of non-integral forms

Context

Geometric quantization

Symplectic geometry

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

Given a presymplectic form (X,ω)(X, \omega) (hence any closed differential 2-form), a prequantization of it in the traditional sense is a choice of circle bundle with connection on XX whose curvature 2-form is ω\omega. Since the circle group U(1)U(1) is equivalent, as a smooth ∞-group to the 2-group coming from the crossed module ()(\mathbb{Z} \hookrightarrow \mathbb{R}), such a lift exists whenever the periods of ω\omega are integral, hence are in the inclusion of the integers into the real numbers \mathbb{Z} \hookrightarrow \mathbb{R}.

But conversely this means that for any closed differential 2-form ω\omega there is a connection on a 2-bundle on a (Γ)(\Gamma \hookrightarrow \mathbb{R})-principal 2-bundle, where Γ\Gamma \hookrightarrow \mathbb{R} is the inclusion of the discrete group of periods of ω\omega.

If here Γ\Gamma \hookrightarrow \mathbb{R} is a global multiple of the canonical inclusion \mathbb{Z} \hookrightarrow \mathbb{R} then there is of course an isomorphism (/Γ)(/)=U(1)(\mathbb{R}/\Gamma) \simeq (\mathbb{R}/\mathbb{Z}) = U(1). This identification coresponds to a choice of Planck's constant (see there).

If however Γ\Gamma is not finitely generated, then the smooth 2-group (Γ)(\Gamma \to \mathbb{R}) is not equivalent to a smooth 1-group, and hence this is a genuine case of higher geometric prequantization. The upshot being that while not every closed 2-form has an ordinary prequantization, it always does have one in higher geometric prequantization, at least if we admit to choose the structure 2-group accordingly.

These considerations are currently mostly motivated purely mathematically. But the claim is that there useful physical applications (… eventually to be added here…).

References

The observation that non-integral closed 2-forms can be prequantized by diffeological principal bundles for the diffeological quotient of \mathbb{R} by the subgroup of periods is due to

and reviewed in section II 2.5 of

The remark that the non-manifold quotient is usefully thought of as regarded instead in higher geometric prequantization by prequantum principal 2-bundles was made in

Mentioning of prequantization of non-integral forms is also in section 3.2.1 of