geometric quantization higher geometric quantization
geometry of physics: Lagrangians and Action functionals + Geometric Quantization
prequantum circle n-bundle = extended Lagrangian
prequantum 1-bundle = prequantum circle bundle, regularcontact manifold,prequantum line bundle = lift of symplectic form to differential cohomology
In the contect of geometric quantization a prequantum operator is a linear operator is a canonical action of a Hamiltonian observable on the space of sections of the prequantum line bundle. After a choice of polarization, if this restricts to the subspace of polarized sections, hence to the actual space of quantum states (“wavefunctions”) then this is the operator that represents the given observable.
More in detail, the quantomorphism group $\mathbf{Aut}(\mathbf{c}_{conn})$ naturally acts on the space of sections $\mathbf{\Gamma}_X(E)$ of the prequantum line bundle.
For $O \in \mathbf{Aut}(\mathbf{c}_{conn})$ a given Hamiltonian symplectomorphism with Hamiltonian, the corresponding map
is the prequantum operator that quantizes $O$.
Given a choice of polarization the actual quantum operator corresponding to this is the restriction, if it exists, to the sub-space of polarized sections.
Typically this is considered for infinitesimal elements of the quantomorphism group, hence for elements of the Poisson bracket Lie algebra (which then typically end up as unbounded operators), see def. below.
Let $(X,\omega)$ be a presymplectic manifold.
Let $\nabla : X \to \mathbf{B} U(1)_{conn}$ be a prequantum line bundle $E \to X$ with connection for $\omega$. Write $\Gamma_X(E)$ for its space of smooth sections, the prequantum space of states.
For $f \in C^\infty(X, \mathbb{C})$ a function on phase space, the corresponding quantum operator is the linear map
given by
where
$v_f$ is the Hamiltonian vector field corresponding to $f$;
$\nabla_{v_f} : \Gamma_X(E) \to \Gamma_X(E)$ is the covariant derivative of sections along $v_f$ for the given choice of prequantum connection;
$f \cdot (-) : \Gamma_X(E) \to \Gamma_X(E)$ is the operation of degreewise multiplication pf sections.
(origin of the formulas for prequantum operators)+
The formula (1) may look a bit mysterious on first sight. The correction term to the covariant derivative appearing in this formula is ultimately due to the fact that with $v$ the Hamiltonian vector field corresponding to a Hamiltonian $H_v$ via
then the Lie derivative of $\theta$ (the symplectic potentiation, related by $d \theta = \omega$) is
for
Here the second term on the right is what yields the covariant derivative in (1), while the first summand is the correction term in (1).
A derivation of these formulas from first principles is given in (Fiorenza-Rogers-Schreiber 13a, example 3.2.3 and remark 3.3.16).
Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher U(1)-gerbe connections in geometric prequantization, Reviews in Mathematical Physics, Volume 28, Issue 06, July 2016 (arXiv:1304.0236)
Domenico Fiorenza, Chris Rogers, Urs Schreiber L-∞ algebras of local observables from higher prequantum bundles, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 142 (arXiv:1304.6292)