nLab
quantum operator (in geometric quantization)

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 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.

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. 1 below.

Definition

For ordinary phase spaces

Let (X,ω)(X,\omega) be a presymplectic manifold.

Let :XBU(1) conn\nabla : X \to \mathbf{B} U(1)_{conn} be a prequantum line bundle EXE \to X with connection for ω\omega. Write Γ X(E)\Gamma_X(E) for its space of smooth sections, the prequantum space of states.

Definition

(prequantum operators)

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

f^:Γ X(E)Γ X(E) \hat f \colon \Gamma_X(E) \to \Gamma_X(E)

given by

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

where

  • v fv_f is the Hamiltonian vector field corresponding to ff;

  • v f:Γ X(E)Γ X(E)\nabla_{v_f} : \Gamma_X(E) \to \Gamma_X(E) is the covariant derivative of sections along v fv_f for the given choice of prequantum connection;

  • f():Γ X(E)Γ X(E)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 vv the Hamiltonian vector field corresponding to a Hamiltonian H vH_v via

ι vω=dH v \iota_v \omega = d H_v

then the Lie derivative of θ\theta (the symplectic potentiation, related by dθ=ωd \theta = \omega) is

vθ=dH˜ v \mathcal{L}_v \theta = d \tilde H_v

for

H˜ v=H v+ι vθ. \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 December 22, 2017 at 08:25:29. See the history of this page for a list of all contributions to it.