In geometric quantization we want to go from a symplectic manifold $\left( M, \omega \right)$ to a Hilbert space $H$. If $M$ is prequantizable, we find a prequantum bundle $L \rightarrow M$ with Hermitian form $H$ and connection $\nabla$ with curvature $-i \omega$. We can then pick a polarization, and take our Hilbert space to be the polarized, square integrable sections over $L$. Here, square integrability is with respect the norm:
$$ \vert f \vert^2 = \int_M H \left( f, f \right) \omega^n .$$
Additionally, we can take our observable mapping from the classical observables in $C^{\infty} \left( M \right)$ to quantum observable in $\text{End} \left( H \right)$ is given by:
$$f \mapsto \hat{f} = - i \nabla_{X_f} + f.$$
However certain problems still exist. As far as I'm aware the (only?) two issues are:
- This quantization gives physically incorrect eigenvalues for some operators (see example below).
- For non-Kähler polarizations, there can be divergence issues. This is because, if the polarization $P$ has non-zero isotropic distribution then the norm defined earlier will be constant along these real directions, leading to a divergence. For instance, see the explanation in the first paragraph of Baykara, Section 7.
The remedy is metaplectic correction. There are various approaches to this: using half-forms, using spin structures, etc. For example, see Carosso, pg 17
Although I see seen many alternative approaches, Carosso gives one example of how metaplectic correction can fix the classical-to-quantum observable mapping. For instance, Carosso modifies the observable mapping to:
$$f \mapsto \hat{f} = - i \nabla_{X_f} + f - \frac{i}{2} \text{Tr} \left( A \right)$$
where, if the polarization is $P = \text{span} \{ X_i \}$ so $[ X_i, X_j ] = A^n_k X_n$, then $A$ is the matrix $A = \left( A^n_k \right)$. Note that Baykara and others explain how Metaplectic Correction resolves this issue in other ways, although they're ultimately equivalent.
My questions are:
- How does Metaplectic Correction fix the vacuum energy for a geometrically quantized theory?
- Are the divergence and eigenvalue issues mentioned earlier equivalent? Namely, are they the same issue, stated in different ways? We know metaplectic correction fixes them both. However are they different, and one could be fixed without the other necessarily being fixed too?
Example of Incorrect Eigenvalues
Consider the following example to see where geometric quantization, before metaplectic correction has been appplied, fails to give operators with the correct eigenvalue. Let us look at the 2-dimensional harmonic oscillator in complex coordinates, where our classical Hamiltonian quantizes to:
$$H = \frac{1}{2} \left( q^2 + p^2 \right) = \frac{1}{2} z \bar{z}$$ $$ \mapsto \hat{H} = z \partial_z $$
This has eigenfunctions $z^n$ with eigenvalues $n$. However for the harmonic oscillator, we expect our eigenvalues to be $n + \frac{1}{2}$. Note that the corrected operator mapping from Carosso restores this additive factor of $\frac{1}{2}$. Specifically, we can see the vacuum energy was not accounted for.
