<s>
In	O
mathematics	O
,	O
the	O
Segal	B-Algorithm
–	I-Algorithm
Bargmann	I-Algorithm
space	I-Algorithm
(	O
for	O
Irving	O
Segal	O
and	O
Valentine	O
Bargmann	O
)	O
,	O
also	O
known	O
as	O
the	O
Bargmann	O
space	O
or	O
Bargmann	O
–	O
Fock	O
space	O
,	O
is	O
the	O
space	O
of	O
holomorphic	O
functions	O
F	O
in	O
n	O
complex	O
variables	O
satisfying	O
the	O
square-integrability	O
condition	O
:	O
</s>
<s>
is	O
known	O
as	O
the	O
reproducing	O
kernel	O
for	O
the	O
Segal	B-Algorithm
–	I-Algorithm
Bargmann	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
One	O
may	O
interpret	O
a	O
unit	O
vector	O
in	O
the	O
Segal	B-Algorithm
–	I-Algorithm
Bargmann	I-Algorithm
space	I-Algorithm
as	O
the	O
wave	O
function	O
for	O
a	O
quantum	O
particle	O
moving	O
in	O
In	O
this	O
view	O
,	O
plays	O
the	O
role	O
of	O
the	O
classical	O
phase	O
space	O
,	O
whereas	O
is	O
the	O
configuration	O
space	O
.	O
</s>
<s>
This	O
connection	O
will	O
be	O
made	O
more	O
precise	O
below	O
,	O
after	O
we	O
introduce	O
the	O
Segal	O
–	O
Bargmann	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Indeed	O
,	O
Bargmann	O
was	O
led	O
to	O
introduce	O
the	O
particular	O
form	O
of	O
the	O
inner	O
product	O
on	O
the	O
Segal	B-Algorithm
–	I-Algorithm
Bargmann	I-Algorithm
space	I-Algorithm
precisely	O
so	O
that	O
the	O
creation	O
and	O
annihilation	O
operators	O
would	O
be	O
adjoints	O
of	O
each	O
other	O
.	O
</s>
<s>
These	O
operators	O
satisfy	O
the	O
ordinary	O
canonical	O
commutation	O
relations	O
,	O
and	O
it	O
can	O
be	O
shown	O
that	O
that	O
they	O
act	O
irreducibly	O
on	O
the	O
Segal	B-Algorithm
–	I-Algorithm
Bargmann	I-Algorithm
space	I-Algorithm
;	O
see	O
Section	O
14.4	O
of	O
.	O
</s>
<s>
Since	O
the	O
operators	O
and	O
from	O
the	O
previous	O
section	O
satisfy	O
the	O
Weyl	O
relations	O
and	O
act	O
irreducibly	O
on	O
the	O
Segal	B-Algorithm
–	I-Algorithm
Bargmann	I-Algorithm
space	I-Algorithm
,	O
the	O
Stone	O
–	O
von	O
Neumann	O
theorem	O
applies	O
.	O
</s>
<s>
Thus	O
,	O
there	O
is	O
a	O
unitary	O
map	O
from	O
the	O
position	O
Hilbert	O
space	O
to	O
the	O
Segal	B-Algorithm
–	I-Algorithm
Bargmann	I-Algorithm
space	I-Algorithm
that	O
intertwines	O
these	O
operators	O
with	O
the	O
usual	O
position	O
and	O
momentum	O
operators	O
.	O
</s>
<s>
The	O
map	O
may	O
be	O
computed	O
explicitly	O
as	O
a	O
modified	O
double	O
Weierstrass	B-Algorithm
transform	I-Algorithm
,	O
</s>
<s>
One	O
can	O
also	O
describe	O
as	O
the	O
inner	O
product	O
of	O
with	O
an	O
appropriately	O
normalized	O
coherent	O
state	O
with	O
parameter	O
,	O
where	O
,	O
now	O
,	O
we	O
express	O
the	O
coherent	O
states	O
in	O
the	O
position	O
representation	O
instead	O
of	O
in	O
the	O
Segal	B-Algorithm
–	I-Algorithm
Bargmann	I-Algorithm
space	I-Algorithm
.	O
</s>
<s>
We	O
may	O
now	O
be	O
more	O
precise	O
about	O
the	O
connection	O
between	O
the	O
Segal	B-Algorithm
–	I-Algorithm
Bargmann	I-Algorithm
space	I-Algorithm
and	O
the	O
Husimi	O
function	O
of	O
a	O
particle	O
.	O
</s>
<s>
The	O
claim	O
is	O
then	O
that	O
the	O
above	O
density	O
is	O
the	O
Husimi	O
function	O
of	O
,	O
which	O
may	O
be	O
obtained	O
from	O
the	O
Wigner	O
function	O
of	O
by	O
convolving	O
with	O
a	O
double	O
Gaussian	O
(	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
)	O
.	O
</s>
<s>
Thus	O
,	O
there	O
can	O
be	O
many	O
different	O
inversion	O
formulas	O
for	O
the	O
Segal	O
–	O
Bargmann	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
There	O
are	O
various	O
generalizations	O
of	O
the	O
Segal	B-Algorithm
–	I-Algorithm
Bargmann	I-Algorithm
space	I-Algorithm
and	O
transform	O
.	O
</s>
<s>
The	O
various	O
Gaussians	O
appearing	O
in	O
the	O
ordinary	O
Segal	B-Algorithm
–	I-Algorithm
Bargmann	I-Algorithm
space	I-Algorithm
and	O
transform	O
are	O
replaced	O
by	O
heat	O
kernels	O
.	O
</s>
<s>
This	O
generalized	O
Segal	O
–	O
Bargmann	B-Algorithm
transform	I-Algorithm
could	O
be	O
applied	O
,	O
for	O
example	O
,	O
to	O
the	O
rotational	O
degrees	O
of	O
freedom	O
of	O
a	O
rigid	O
body	O
,	O
where	O
the	O
configuration	O
space	O
is	O
the	O
compact	O
Lie	O
groups	O
SO(3 )	O
.	O
</s>
<s>
This	O
generalized	O
Segal	O
–	O
Bargmann	B-Algorithm
transform	I-Algorithm
gives	O
rise	O
to	O
a	O
system	O
of	O
coherent	O
states	O
,	O
known	O
as	O
heat	O
kernel	O
coherent	O
states	O
.	O
</s>
