<s>
In	O
physics	O
,	O
a	O
spin	B-Application
network	I-Application
is	O
a	O
type	O
of	O
diagram	O
which	O
can	O
be	O
used	O
to	O
represent	O
states	O
and	O
interactions	O
between	O
particles	O
and	O
fields	O
in	O
quantum	O
mechanics	O
.	O
</s>
<s>
From	O
a	O
mathematical	O
perspective	O
,	O
the	O
diagrams	O
are	O
a	O
concise	O
way	O
to	O
represent	O
multilinear	O
functions	O
and	O
functions	O
between	O
representations	O
of	O
matrix	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
Roger	O
Penrose	O
described	O
spin	B-Application
networks	I-Application
in	O
1971	O
.	O
</s>
<s>
Spin	B-Application
networks	I-Application
have	O
since	O
been	O
applied	O
to	O
the	O
theory	O
of	O
quantum	O
gravity	O
by	O
Carlo	O
Rovelli	O
,	O
Lee	O
Smolin	O
,	O
Jorge	O
Pullin	O
,	O
Rodolfo	O
Gambini	O
and	O
others	O
.	O
</s>
<s>
Spin	B-Application
networks	I-Application
can	O
also	O
be	O
used	O
to	O
construct	O
a	O
particular	O
functional	O
on	O
the	O
space	O
of	O
connections	O
which	O
is	O
invariant	O
under	O
local	O
gauge	O
transformations	O
.	O
</s>
<s>
A	O
spin	B-Application
network	I-Application
,	O
as	O
described	O
in	O
Penrose	O
(	O
1971	O
)	O
,	O
is	O
a	O
kind	O
of	O
diagram	O
in	O
which	O
each	O
line	O
segment	O
represents	O
the	O
world	O
line	O
of	O
a	O
"	O
unit	O
"	O
(	O
either	O
an	O
elementary	O
particle	O
or	O
a	O
compound	O
system	O
of	O
particles	O
)	O
.	O
</s>
<s>
Diagrams	O
whose	O
line	O
segments	O
are	O
all	O
joined	O
at	O
vertices	O
are	O
called	O
closed	O
spin	B-Application
networks	I-Application
.	O
</s>
<s>
Time	O
may	O
be	O
viewed	O
as	O
going	O
in	O
one	O
direction	O
,	O
such	O
as	O
from	O
the	O
bottom	O
to	O
the	O
top	O
of	O
the	O
diagram	O
,	O
but	O
for	O
closed	O
spin	B-Application
networks	I-Application
the	O
direction	O
of	O
time	O
is	O
irrelevant	O
to	O
calculations	O
.	O
</s>
<s>
For	O
bosons	O
,	O
such	O
as	O
photons	B-Application
and	O
gluons	O
,	O
n	O
is	O
an	O
even	O
number	O
.	O
</s>
<s>
For	O
fermions	O
,	O
such	O
as	O
electrons	O
and	O
quarks	B-Operating_System
,	O
n	O
is	O
odd	O
.	O
</s>
<s>
Given	O
any	O
closed	O
spin	B-Application
network	I-Application
,	O
a	O
non-negative	O
integer	O
can	O
be	O
calculated	O
which	O
is	O
called	O
the	O
norm	O
of	O
the	O
spin	B-Application
network	I-Application
.	O
</s>
<s>
However	O
,	O
they	O
imply	O
that	O
for	O
a	O
spin	B-Application
network	I-Application
to	O
have	O
nonzero	O
norm	O
,	O
two	O
requirements	O
must	O
be	O
met	O
at	O
each	O
vertex	O
.	O
</s>
<s>
Formally	O
,	O
a	O
spin	B-Application
network	I-Application
may	O
be	O
defined	O
as	O
a	O
(	O
directed	O
)	O
graph	O
whose	O
edges	O
are	O
associated	O
with	O
irreducible	O
representations	O
of	O
a	O
compact	O
Lie	O
group	O
and	O
whose	O
vertices	O
are	O
associated	O
with	O
intertwiners	O
of	O
the	O
edge	O
representations	O
adjacent	O
to	O
it	O
.	O
</s>
<s>
A	O
spin	B-Application
network	I-Application
,	O
immersed	O
into	O
a	O
manifold	B-Architecture
,	O
can	O
be	O
used	O
to	O
define	O
a	O
functional	O
on	O
the	O
space	O
of	O
connections	O
on	O
this	O
manifold	B-Architecture
.	O
</s>
<s>
In	O
loop	O
quantum	O
gravity	O
(	O
LQG	O
)	O
,	O
a	O
spin	B-Application
network	I-Application
represents	O
a	O
"	O
quantum	O
state	O
"	O
of	O
the	O
gravitational	O
field	O
on	O
a	O
3-dimensional	O
hypersurface	O
.	O
</s>
<s>
The	O
set	O
of	O
all	O
possible	O
spin	B-Application
networks	I-Application
(	O
or	O
,	O
more	O
accurately	O
,	O
"	O
s-knots	O
"	O
that	O
is	O
,	O
equivalence	O
classes	O
of	O
spin	B-Application
networks	I-Application
under	O
diffeomorphisms	O
)	O
is	O
countable	O
;	O
it	O
constitutes	O
a	O
basis	O
of	O
LQG	O
Hilbert	O
space	O
.	O
</s>
<s>
where	O
the	O
sum	O
goes	O
over	O
all	O
intersections	O
i	O
of	O
Σ	O
with	O
the	O
spin	B-Application
network	I-Application
.	O
</s>
<s>
ji	O
=	O
0	O
,	O
1/2	O
,	O
1	O
,	O
3/2	O
,	O
...	O
is	O
the	O
spin	O
associated	O
with	O
the	O
link	O
i	O
of	O
the	O
spin	B-Application
network	I-Application
.	O
</s>
<s>
The	O
two-dimensional	O
area	O
is	O
therefore	O
"	O
concentrated	O
"	O
in	O
the	O
intersections	O
with	O
the	O
spin	B-Application
network	I-Application
.	O
</s>
<s>
The	O
volume	O
of	O
a	O
3D	O
submanifold	O
that	O
contains	O
part	O
of	O
a	O
spin	B-Application
network	I-Application
is	O
given	O
by	O
a	O
sum	O
of	O
contributions	O
from	O
each	O
node	O
inside	O
it	O
.	O
</s>
<s>
One	O
can	O
think	O
that	O
every	O
node	O
in	O
a	O
spin	B-Application
network	I-Application
is	O
an	O
elementary	O
"	O
quantum	O
of	O
volume	O
"	O
and	O
every	O
link	O
is	O
a	O
"	O
quantum	O
of	O
area	O
"	O
surrounding	O
this	O
volume	O
.	O
</s>
<s>
Over	O
a	O
manifold	B-Architecture
however	O
,	O
assumptions	O
like	O
diffeomorphism	O
invariance	O
are	O
needed	O
to	O
make	O
the	O
duality	O
exact	O
(	O
smearing	O
Wilson	O
loops	O
is	O
tricky	O
)	O
.	O
</s>
<s>
Michael	O
A	O
.	O
Levin	O
and	O
Xiao-Gang	O
Wen	O
have	O
also	O
defined	O
string-nets	O
using	O
tensor	O
categories	O
that	O
are	O
objects	O
very	O
similar	O
to	O
spin	B-Application
networks	I-Application
.	O
</s>
<s>
However	O
the	O
exact	O
connection	O
with	O
spin	B-Application
networks	I-Application
is	O
not	O
clear	O
yet	O
.	O
</s>
<s>
In	O
mathematics	O
,	O
spin	B-Application
networks	I-Application
have	O
been	O
used	O
to	O
study	O
skein	O
modules	O
and	O
character	B-Algorithm
varieties	I-Algorithm
,	O
which	O
correspond	O
to	O
spaces	O
of	O
connections	O
.	O
</s>
