Contents

# Contents

## Idea

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.

$\widehat {(-)} : \mathbf{\Gamma}_X(E) \times \mathbf{Aut}(\mathbf{c}_{conn}) \to \mathbf{\Gamma}_X(E) \,.$

For $O \in \mathbf{Aut}(\mathbf{c}_{conn})$ a given Hamiltonian symplectomorphism with Hamiltonian, the corresponding map

$\widehat{O} : \mathbf{\Gamma}_X(E) \to \mathbf{\Gamma}_X(E)$

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.

## Definition

### For ordinary phase spaces

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.

###### Definition

(prequantum operators)

For $f \in C^\infty(X, \mathbb{C})$ a function on phase space, the corresponding quantum operator is the linear map

$\hat f \colon \Gamma_X(E) \to \Gamma_X(E)$

given by

(1)$\psi \mapsto -i \nabla_{v_f} \psi + f \cdot \psi \,,$

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.

###### Remark

(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

$\iota_v \omega = d H_v$

then the Lie derivative of $\theta$ (the symplectic potentiation, related by $d \theta = \omega$) is

$\mathcal{L}_v \theta = d \tilde H_v$

for

$\tilde H_v = H_v + \iota_v \theta \,.$

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).

## References

Last revised on February 8, 2020 at 11:05:18. See the history of this page for a list of all contributions to it.