Geometric quantization

Suppose ${\displaystyle (M,\omega )}$ is a symplectic manifold with symplectic form ${\displaystyle \omega }$. Suppose at first that ${\displaystyle \omega }$ is exact, meaning that there is a globally defined symplectic potential ${\displaystyle \theta }$ with ${\displaystyle d\theta =\omega }$. We can consider the "prequantum Hilbert space" of square-integrable functions on ${\displaystyle M}$ (with respect to the Liouville volume measure). For each smooth function ${\displaystyle f}$ on ${\displaystyle M}$, we can define the Kostant–Souriau prequantum operator

${\displaystyle Q(f):=-i\hbar \left(X_{f}+{\frac {1}{i\hbar }}\theta (X_{f})\right)+f}$.

where ${\displaystyle X_{f}}$ is the Hamiltonian vector field associated to ${\displaystyle f}$.

More generally, suppose ${\displaystyle (M,\omega )}$ has the property that the integral of ${\displaystyle \omega /(2\pi \hbar )}$ over any closed surface is an integer. Then we can construct a line bundle ${\displaystyle L}$ with connection whose curvature 2-form is ${\displaystyle \omega /\hbar }$. In that case, the prequantum Hilbert space is the space of square-integrable sections of ${\displaystyle L}$, and we replace the formula for ${\displaystyle Q(f)}$ above with

${\displaystyle Q(f)=-i\hbar \nabla _{X_{f}}+f}$,

with ${\displaystyle \nabla }$ the connection. The prequantum operators satisfy

${\displaystyle [Q(f),Q(g)]=i\hbar Q(\{f,g\})}$

for all smooth functions ${\displaystyle f}$ and ${\displaystyle g}$.[4]

The construction of the preceding Hilbert space and the operators ${\displaystyle Q(f)}$ is known as prequantization.

The next step in the process of geometric quantization is the choice of a polarization. A polarization is a choice at each point in ${\displaystyle M}$ a Lagrangian subspace of the complexified tangent space of ${\displaystyle M}$. The subspaces should form an integrable distribution, meaning that the commutator of two vector fields lying in the subspace at each point should also lie in the vector field at each point. The quantum (as opposed to prequantum) Hilbert space is the space of sections of ${\displaystyle L}$ that are covariantly constant in the direction of the polarization.[5][lower-alpha 2] The idea is that in the quantum Hilbert space, the sections should be functions of only ${\displaystyle n}$ variables on the ${\displaystyle 2n}$-dimensional classical phase space.

If ${\displaystyle f}$ is a function for which the associated Hamiltonian flow preserves the polarization, then ${\displaystyle Q(f)}$ will preserve the quantum Hilbert space.[6] The assumption that the flow of ${\displaystyle f}$ preserve the polarization is a strong one. Typically not very many functions will satisfy this assumption.

The half-form correction—also known as the metaplectic correction—is a technical modification to the above procedure that is necessary in the case of real polarizations to obtain a nonzero quantum Hilbert space; it is also often useful in the complex case. The line bundle ${\displaystyle L}$ is replaced by the tensor product of ${\displaystyle L}$ with the square root of the canonical bundle of ${\displaystyle L}$. In the case of the vertical polarization, for example, instead of considering functions ${\displaystyle f(x)}$ of ${\displaystyle x}$ that are independent of ${\displaystyle p}$, one considers objects of the form ${\displaystyle f(x){\sqrt {dx}}}$. The formula for ${\displaystyle Q(f)}$ must then be supplemented by an additional Lie derivative term.[7] In the case of a complex polarization on the plane, for example, the half-form correction allows the quantization of the harmonic oscillator to reproduce the standard quantum mechanical formula for the energies, ${\displaystyle (n+1/2)\hbar \omega }$, with the "${\displaystyle +1/2}$" coming courtesy of the half-forms.[8]

Geometric quantization of Poisson manifolds and symplectic foliations also is developed. For instance, this is the case of partially integrable and superintegrable Hamiltonian systems and non-autonomous mechanics.