Geometric Quantization Geometric quantization

1 geometric quantization

1.1 prequantization
1.2 polarization
1.3 half-form correction
1.4 poisson manifolds

geometric quantization

the geometric quantization procedure falls following 3 steps: prequantization, polarization, , metaplectic correction. prequantization produces natural hilbert space quantization procedure observables transforms poisson brackets on classical side commutators on quantum side. nevertheless, prequantum hilbert space understood big . idea 1 should select poisson-commuting set of n variables on 2n-dimensional phase space , consider functions (or, more properly, sections) depend on these n variables. n variables can either real-valued, resulting in position-style hilbert space, or complex-valued, producing segal–bargmann space. polarization coordinate-independent description of such choice of n poisson-commuting functions. metaplectic correction (also known half-form correction) technical modification of above procedure necessary in case of real polarizations , convenient complex polarizations.

prequantization

suppose



(
m
,
ω
)


{\displaystyle (m,\omega )}

symplectic manifold symplectic form



ω


{\displaystyle \omega }

. suppose @ first



ω


{\displaystyle \omega }

exact, meaning there globally defined symplectic potential



θ


{\displaystyle \theta }





d
θ
=
ω


{\displaystyle d\theta =\omega }

. can consider prequantum hilbert space of square-integrable functions on



m


{\displaystyle m}

(with respect liouville volume measure). each smooth function



f


{\displaystyle f}

on



m


{\displaystyle m}

, can define kostant–souriau prequantum operator

q
(
f
)
:=
i
ℏ

(

x

f


−


i
ℏ


θ
(

x

f


)
)

+
f


{\displaystyle q(f):=i\hbar \left(x_{f}-{\frac {i}{\hbar }}\theta (x_{f})\right)+f}

.

more generally, suppose



(
m
,
ω
)


{\displaystyle (m,\omega )}

has property integral of



ω

/

(
2
π
ℏ
)


{\displaystyle \omega /(2\pi \hbar )}

on closed surface integer. can construct line bundle



l


{\displaystyle l}

connection curvature 2-form



ω

/

ℏ


{\displaystyle \omega /\hbar }

. in case, prequantum hilbert space space of square-integrable sections of



l


{\displaystyle l}

, , replace formula



q
(
f
)


{\displaystyle q(f)}

above with

q
(
f
)
=
i
ℏ

∇


x

f




+
f


{\displaystyle q(f)=i\hbar \nabla _{x_{f}}+f}

,

where




x

f




{\displaystyle x_{f}}

hamiltonian vector field associated



f


{\displaystyle f}

,



∇


{\displaystyle \nabla }

connection. prequantum operators satisfy

[
q
(
f
)
,
q
(
g
)
]
=
i
ℏ
q
(
{
f
,
g
}
)


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

for smooth functions



f


{\displaystyle f}

,



g


{\displaystyle g}

.

the construction of preceding hilbert space , operators



q
(
f
)


{\displaystyle q(f)}

known prequantization.

polarization

the next step in process of geometric quantization choice of polarization. polarization choice @ each point in



m


{\displaystyle m}

lagrangian subspace of complexified tangent space of



m


{\displaystyle m}

. subspaces should form integrable distribution, meaning commutator of 2 vector fields lying in subspace @ each point should lie in vector field @ each point. quantum (as opposed prequantum) hilbert space space of sections of



l


{\displaystyle l}

covariantly constant in direction of polarization. idea in quantum hilbert space, sections should functions of



n


{\displaystyle n}

variables on



2
n


{\displaystyle 2n}

-dimensional classical phase space.

if



f


{\displaystyle f}

function associated hamiltonian flow preserves polarization,



q
(
f
)


{\displaystyle q(f)}

preserve quantum hilbert space. assumption flow of



f


{\displaystyle f}

preserve polarization strong one. typically not many functions satisfy assumption.

half-form correction

the half-form correction—also known metaplectic correction—is technical modification above procedure necessary in case of real polarizations obtain nonzero quantum hilbert space; useful in complex case. line bundle



l


{\displaystyle l}

replaced tensor product of



l


{\displaystyle l}

square root of canonical bundle of



l


{\displaystyle l}

. in case of vertical polarization, example, instead of considering functions



f
(
x
)


{\displaystyle f(x)}

of



x


{\displaystyle x}

independent of



p


{\displaystyle p}

, consider objects of form



f
(
x
)


d
x




{\displaystyle f(x){\sqrt {dx}}}

. formula



q
(
f
)


{\displaystyle q(f)}

must supplemented additional lie derivative term. in case of complex polarization on plane, example, half-form correction allows quantization of harmonic oscillator reproduce standard quantum mechanical formula energies,



(
n
+
1

/

2
)
ℏ
ω


{\displaystyle (n+1/2)\hbar \omega }

,



+
1

/

2


{\displaystyle +1/2}

coming courtesy of half-forms.

poisson manifolds

geometric quantization of poisson manifolds , symplectic foliations developed. instance, case of partially integrable , superintegrable hamiltonian systems , non-autonomous mechanics.