# nLab quantum operator (in geometric quantization)

for the moment the conent that would go here is rather at quantum observable

# Contents

## Idea

In the contect of geometric quantization a prequantum operator is an operator that presents an observable in quantum mechanics/quantum field theory once a polarization is chosen.

More in detail, the quantomorphism group $\mathrm{Aut}\left({c}_{\mathrm{conn}}\right)$ naturally acts on the space of sections ${\Gamma }_{X}\left(E\right)$ of the prequantum line bundle.

$\stackrel{^}{\left(-\right)}:{\Gamma }_{X}\left(E\right)×\mathrm{Aut}\left({c}_{\mathrm{conn}}\right)\to {\Gamma }_{X}\left(E\right)\phantom{\rule{thinmathspace}{0ex}}.$\widehat {(-)} : \mathbf{\Gamma}_X(E) \times \mathbf{Aut}(\mathbf{c}_{conn}) \to \mathbf{\Gamma}_X(E) \,.

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

$\stackrel{^}{O}:{\Gamma }_{X}\left(E\right)\to {\Gamma }_{X}\left(E\right)$\widehat{O} : \mathbf{\Gamma}_X(E) \to \mathbf{\Gamma}_X(E)

is the prequantum operator that quantizes $O$.

Revised on April 11, 2013 23:59:52 by Urs Schreiber (131.174.41.18)