nLab quantum operator (in geometric quantization)

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).

Related concepts

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

References

• 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)