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 a linear operator that presents an observable in quantum mechanics/quantum field theory once a polarization is chosen.

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

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

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

O^:Γ X(E)Γ X(E) \widehat{O} : \mathbf{\Gamma}_X(E) \to \mathbf{\Gamma}_X(E)

is the prequantum operator that quantizes OO.

Revised on October 12, 2013 21:51:40 by Toby Bartels (98.19.41.253)