<s>
In	O
mathematical	O
physics	O
,	O
geometric	B-Algorithm
quantization	I-Algorithm
is	O
a	O
mathematical	O
approach	O
to	O
defining	O
a	O
quantum	O
theory	O
corresponding	O
to	O
a	O
given	O
classical	O
theory	O
.	O
</s>
<s>
The	O
modern	O
theory	O
of	O
geometric	B-Algorithm
quantization	I-Algorithm
was	O
developed	O
by	O
Bertram	O
Kostant	O
and	O
Jean-Marie	O
Souriau	O
in	O
the	O
1970s	O
.	O
</s>
<s>
The	O
geometric	B-Algorithm
quantization	I-Algorithm
procedure	O
falls	O
into	O
the	O
following	O
three	O
steps	O
:	O
prequantization	B-Algorithm
,	O
polarization	O
,	O
and	O
metaplectic	O
correction	O
.	O
</s>
<s>
Prequantization	B-Algorithm
produces	O
a	O
natural	O
Hilbert	O
space	O
together	O
with	O
a	O
quantization	O
procedure	O
for	O
observables	O
that	O
exactly	O
transforms	O
Poisson	O
brackets	O
on	O
the	O
classical	O
side	O
into	O
commutators	O
on	O
the	O
quantum	O
side	O
.	O
</s>
<s>
The	O
n	O
variables	O
can	O
be	O
either	O
real-valued	O
,	O
resulting	O
in	O
a	O
position-style	O
Hilbert	O
space	O
,	O
or	O
complex	O
analytic	O
,	O
producing	O
something	O
like	O
the	O
Segal	B-Algorithm
–	I-Algorithm
Bargmann	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
The	O
construction	O
of	O
the	O
preceding	O
Hilbert	O
space	O
and	O
the	O
operators	O
is	O
known	O
as	O
prequantization	B-Algorithm
.	O
</s>
<s>
The	O
next	O
step	O
in	O
the	O
process	O
of	O
geometric	B-Algorithm
quantization	I-Algorithm
is	O
the	O
choice	O
of	O
a	O
polarization	O
.	O
</s>
<s>
Geometric	B-Algorithm
quantization	I-Algorithm
of	O
Poisson	O
manifolds	O
and	O
symplectic	O
foliations	O
also	O
is	O
developed	O
.	O
</s>
<s>
Assuming	O
that	O
the	O
area	O
of	O
the	O
sphere	O
is	O
an	O
integer	O
multiple	O
of	O
,	O
we	O
can	O
perform	O
geometric	B-Algorithm
quantization	I-Algorithm
and	O
the	O
resulting	O
Hilbert	O
space	O
carries	O
an	O
irreducible	O
representation	O
of	O
SU(2 )	O
.	O
</s>
