<s>
In	O
quantum	O
mechanics	O
and	O
statistical	O
mechanics	O
,	O
parastatistics	B-Algorithm
is	O
one	O
of	O
several	O
alternatives	O
to	O
the	O
better	O
known	O
particle	O
statistics	O
models	O
(	O
Bose	O
–	O
Einstein	O
statistics	O
,	O
Fermi	O
–	O
Dirac	O
statistics	O
and	O
Maxwell	O
–	O
Boltzmann	O
statistics	O
)	O
.	O
</s>
<s>
Herbert	O
S	O
.	O
Green	O
is	O
credited	O
with	O
the	O
creation	O
of	O
parastatistics	B-Algorithm
in	O
1953	O
.	O
</s>
<s>
Consider	O
the	O
operator	B-Algorithm
algebra	I-Algorithm
of	O
a	O
system	O
of	O
N	O
identical	O
particles	O
.	O
</s>
<s>
There	O
is	O
an	O
SN	O
group	O
(	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
order	O
N	O
)	O
acting	O
upon	O
the	O
operator	B-Algorithm
algebra	I-Algorithm
with	O
the	O
intended	O
interpretation	O
of	O
permuting	B-Algorithm
the	O
N	O
particles	O
.	O
</s>
<s>
Quantum	O
mechanics	O
requires	O
focus	O
on	O
observables	O
having	O
a	O
physical	O
meaning	O
,	O
and	O
the	O
observables	O
would	O
have	O
to	O
be	O
invariant	O
under	O
all	O
possible	O
permutations	B-Algorithm
of	O
the	O
N	O
particles	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
observable	O
algebra	O
would	O
have	O
to	O
be	O
a	O
*	O
-subalgebra	O
invariant	O
under	O
the	O
action	O
of	O
SN	O
(	O
noting	O
that	O
this	O
does	O
not	O
mean	O
that	O
every	O
element	O
of	O
the	O
operator	B-Algorithm
algebra	I-Algorithm
invariant	O
under	O
SN	O
is	O
an	O
observable	O
)	O
.	O
</s>
<s>
For	O
N	O
identical	O
parabosons	B-Algorithm
of	O
order	O
p	O
(	O
where	O
p	O
is	O
a	O
positive	O
integer	O
)	O
,	O
permissible	O
Young	O
diagrams	O
are	O
all	O
those	O
with	O
p	O
or	O
fewer	O
rows	O
.	O
</s>
<s>
For	O
N	O
identical	O
parafermions	B-Algorithm
of	O
order	O
p	O
,	O
permissible	O
Young	O
diagrams	O
are	O
all	O
those	O
with	O
p	O
or	O
fewer	O
columns	O
.	O
</s>
<s>
A	O
paraboson	B-Algorithm
field	O
of	O
order	O
p	O
,	O
where	O
if	O
x	O
and	O
y	O
are	O
spacelike-separated	O
points	O
,	O
and	O
if	O
where	O
 [  ,  ] 	O
is	O
the	O
commutator	O
and	O
 {  ,  } 	O
is	O
the	O
anticommutator	O
.	O
</s>
<s>
Note	O
that	O
this	O
disagrees	O
with	O
the	O
spin-statistics	O
theorem	O
,	O
which	O
is	O
for	O
bosons	O
and	O
not	O
parabosons	B-Algorithm
.	O
</s>
<s>
There	O
might	O
be	O
a	O
group	O
such	O
as	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
Sp	O
acting	O
upon	O
the	O
φ(i )	O
s	O
.	O
Observables	O
would	O
have	O
to	O
be	O
operators	O
which	O
are	O
invariant	O
under	O
the	O
group	O
in	O
question	O
.	O
</s>
<s>
A	O
parafermion	B-Algorithm
field	O
of	O
order	O
p	O
,	O
where	O
if	O
x	O
and	O
y	O
are	O
spacelike-separated	O
points	O
,	O
and	O
if	O
.	O
</s>
<s>
corresponds	O
to	O
creating	O
n	O
identical	O
parabosons	B-Algorithm
at	O
x1	O
,...,	O
xn	O
.	O
</s>
<s>
corresponds	O
to	O
creating	O
n	O
identical	O
parafermions	B-Algorithm
.	O
</s>
<s>
gives	O
distinct	O
states	O
for	O
each	O
permutation	B-Algorithm
π	O
in	O
Sn	O
.	O
</s>
<s>
It	O
is	O
also	O
unitary	B-Algorithm
.	O
</s>
<s>
Moreover	O
,	O
is	O
an	O
operator-valued	O
representation	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
Sn	O
and	O
as	O
such	O
,	O
we	O
can	O
interpret	O
it	O
as	O
the	O
action	O
of	O
Sn	O
upon	O
the	O
n-particle	O
Hilbert	O
space	O
itself	O
,	O
turning	O
it	O
into	O
a	O
unitary	B-Algorithm
representation	O
.	O
</s>
<s>
QCD	O
can	O
be	O
reformulated	O
using	O
parastatistics	B-Algorithm
with	O
the	O
quarks	O
being	O
parafermions	B-Algorithm
of	O
order	O
3	O
and	O
the	O
gluons	O
being	O
parabosons	B-Algorithm
of	O
order	O
8	O
.	O
</s>
