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geometric quantization of non-integral forms

Contents

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

  • Alan Weinstein, Cohomology of symplectomorphism groups and critical

    values of hamiltonians_, Math. Z. 201 (1989), 75—82

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

  • Yoshiaki Maeda, Peter Michor, Takushiro Ochiai, Akira Yoshioka (eds.), From Geometry to Quantum Mechanics: In Honor of Hideki Omori

Last revised on September 1, 2013 at 20:20:51. See the history of this page for a list of all contributions to it.