quantum observable


Symplectic geometry


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An observable in quantum physics.

Definition in geometric quantization

We consider the notion of quantum observables in the the context of geometric quantization. See also quantum operator (in geometric quantization).

On a symplectic manifold

Let (X,ω)(X, \omega) be a (pre-)symplectic manifold, thought of as the phase space of a physical system.

Assume that ω\omega is prequantizable (integral) and let :XBU(1) conn\nabla : X \to \mathbf{B} U(1)_{conn} be a prequantum bundle EXE \to X with connection for ω\omega, hence with curvature F =ωF_\nabla = \omega. Write Γ X(E)\Gamma_X(E) for the space of smooth sections of the associated complex line bundle. This is the prequantum space of states.


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

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

given by

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


  • 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.


In terms of Higher geometric prequantum theory we may, as discussed there, identify the Poisson bracket Lie algebra 𝔓𝔬𝔦𝔰𝔰𝔬𝔫(X,ω)\mathfrak{Poisson}(X,\omega) with the Lie algebra of the group of automorphism exp(O):\exp(O) \colon \nabla \stackrel{\simeq}{\to} \nabla regarded in the slice over BU(1) conn\mathbf{B}U(1)_{conn}. Moreover, the space of sections is equivalently the space of maps Ψ://U(1) conn\Psi \colon \nabla \to \mathbb{C}//U(1)_{conn} in the slice from \nabla into the differential refinement of the smooth universal line bundle //U(1)BU(1)\mathbb{C}//U(1) \to \mathbf{B}U(1). In this formulation the action of prequantum operators is just the precomposition action

exp(O)^:(Ψ//U(1) conn)(exp(O)Ψ//U(1) conn). \widehat{\exp(O)} \colon (\nabla \stackrel{\Psi}{\to} \mathbb{C}//U(1)_{conn}) \mapsto (\nabla \stackrel{exp(O)}{\to} \nabla \stackrel{\Psi}{\to} \mathbb{C}//U(1)_{conn}) \,.

Now after a choice of polarization a quantum state is a prequantum wave function which is covariantly constant along the Lagrangian submanifolds of the foliation. Not all prequantum operators will respect the space of such quantum states inside all quantum states. Those that do become genuine quantum operators.


Let 𝒫\mathcal{P} be a polarization of the symplectic manifold (X,ω)(X,\omega). then a quantum state or wavefunction is a prequantum state ψ\psi such that Ψ\nabla \Psi vanishes along the leaves of the polarization.

A quantum operator is a prequantum operator which preserves quantum states among all prequantum states.


A prequantum operator given by a Hamiltonian function ff with Hamiltonian vector field v fv_f is a quantum operator, def. 2, with respect to a given polarization 𝒫\mathcal{P} precisely if its flow preserves 𝒫\mathcal{P}, hence precisely if

[v f,𝒫]𝒫. [v_f, \mathcal{P}] \subset \mathcal{P} \,.

If 𝒫\mathcal{P} is a Kähler polarization then its underlying almost complex structure it induces a spin^c structure, as discussed there. If ρ:GQuantMorph(X,) \rho \colon G \to QuantMorph(X,\nabla) is a Hamiltonian action (a homomorphism to the quantomorphism group) such that each prequantum operator ρ(g)\rho(g) is a quantum operator in that it preserves the polarization, by prop. 1, then the corresponding spin^c structure is GG-invariant. Accordingly the index of the spin^c Dirac operator which gives the geometric quantization by cohomological quantization exists not just in K-theory, where it yields the space of quantum states, but even in GG-equivariant K-theory, exhibiting a representation of GG on the Hilbert space. This is the action of the quantum observables given by ρ\rho from the point of view of cohomological quantization.


Over a phase space which is a cotangent bundle and with respect to the corresponding canonical vertical polarization, a Hamiltonian function is a quantum operator precisely if it is at most linear in the canonical momenta.

See for instance (Blau, around p. 35)

On an nn-plectic smooth \infty-groupoid


Irreducible representations and superselection sectors

The space of quantum states forms a linear representation of a given algebra of observables. The decomposition of that into irreducible representations is physically the decomposition into superselection sectors.


See the references at geometric quantization.

Standard facts are recalled for instance around p. 35 of

Computation of quantum observables by index maps in equivariant K-theory is in (see specifically around p. 8 and 9)

Revised on March 1, 2014 05:57:10 by David Corfield (