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An observable in quantum physics.
We consider the notion of quantum observables in the the context of geometric quantization. See also quantum operator (in geometric quantization).
Let $(X, \omega)$ be a (pre-)symplectic manifold, thought of as the phase space of a physical system.
Assume that $\omega$ is prequantizable (integral) and let $\nabla : X \to \mathbf{B} U(1)_{conn}$ be a prequantum bundle $E \to X$ with connection for $\omega$, hence with curvature $F_\nabla = \omega$. Write $\Gamma_X(E)$ for the space of smooth sections of the associated complex line bundle. This is the prequantum space of states.
For $f \in C^\infty(X, \mathbb{C})$ a function on phase space, the corresponding pre-quantum operator is the linear map on prequantum states
given by
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.
In terms of Higher geometric prequantum theory we may, as discussed there, identify the Poisson bracket Lie algebra $\mathfrak{Poisson}(X,\omega)$ with the Lie algebra of the group of automorphism $\exp(O) \colon \nabla \stackrel{\simeq}{\to} \nabla$ regarded in the slice over $\mathbf{B}U(1)_{conn}$. Moreover, the space of sections is equivalently the space of maps $\Psi \colon \nabla \to \mathbb{C}//U(1)_{conn}$ in the slice from $\nabla$ into the differential refinement of the smooth universal line bundle $\mathbb{C}//U(1) \to \mathbf{B}U(1)$. In this formulation the action of prequantum operators is just the precomposition action
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 $\mathcal{P}$ be a polarization of the symplectic manifold $(X,\omega)$. then a quantum state or wavefunction is a prequantum state $\psi$ such that $\nabla \Psi$ 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 $f$ with Hamiltonian vector field $v_f$ is a quantum operator, def. 2, with respect to a given polarization $\mathcal{P}$ precisely if its flow preserves $\mathcal{P}$, hence precisely if
If $\mathcal{P}$ is a Kähler polarization then its underlying almost complex structure it induces a spin^c structure, as discussed there. If $\rho \colon G \to QuantMorph(X,\nabla)$ is a Hamiltonian action (a homomorphism to the quantomorphism group) such that each prequantum operator $\rho(g)$ is a quantum operator in that it preserves the polarization, by prop. 1, then the corresponding spin^c structure is $G$-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 $G$-equivariant K-theory, exhibiting a representation of $G$ on the Hilbert space. This is the action of the quantum observables given by $\rho$ 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)
(…)
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)