In physics, the eightfold way is a theory
organizing subatomic hadrons. The name was
coined by American physicist Murray Gell-Mann
as an allusion to the Noble Eightfold Path
of Buddhism. It led to the development of
the quark model.
== Organization ==
=== 
Octets and decuplets ===
The eightfold way organizes the mesons and
spin-1/2 baryons into an octet. An equivalent
theory was independently proposed by Israeli
physicist Yuval Ne'eman. The principles of
the eightfold way also applied to the spin-3/2
baryons, forming a decuplet. However, one
of the particles of this decuplet had never
been previously observed. Gell-Mann called
this particle the Ω− and predicted in 1962
that it would have a strangeness −3, electric
charge −1 and a mass near 1680 MeV/c2. In
1964, a particle closely matching these predictions
was discovered by a particle accelerator group
at Brookhaven. Gell-Mann received the 1969
Nobel Prize in Physics for his work on the
theory of elementary particles.
== Historical development ==
=== 
Development ===
Historically, quarks were motivated by an
understanding of flavour symmetry. First,
it was noticed (1961) that groups of particles
were related to each other in a way that matched
the representation theory of SU(3). From that,
it was inferred that there is an approximate
symmetry of the universe which is parametrized
by the group SU(3). Finally (1964), this led
to the discovery of three light quarks (up,
down, and strange) interchanged by these SU(3)
transformations.
=== Modern interpretation ===
The eightfold way may be understood in modern
terms as a consequence of flavor symmetries
between various kinds of quarks. Since the
strong nuclear force affects quarks the same
way regardless of their flavor, replacing
one flavor of quark with another in a hadron
should not alter its mass very much, provided
the respective quark masses are smaller than
the strong interaction scale—which holds
for the three lights quarks. Mathematically,
this replacement may be described by elements
of the SU(3) group. The octets and other hadron
arrangements are representations of this group.
== Flavor symmetry ==
=== SU(3) ===
There is an abstract three-dimensional vector
space:
up quark
→
(
1
0
0
)
,
down quark
→
(
0
1
0
)
,
strange 
quark
→
(
0
0
1
)
{\displaystyle {\text{up quark}}\rightarrow
{\begin{pmatrix}1\\0\\0\end{pmatrix}},\qquad
{\text{down quark}}\rightarrow {\begin{pmatrix}0\\1\\0\end{pmatrix}},\qquad
{\text{strange quark}}\rightarrow {\begin{pmatrix}0\\0\\1\end{pmatrix}}}
and the laws of physics are approximately
invariant under applying a determinant-1 unitary
transformation to this space (sometimes called
a flavour rotation):
(
x
y
z
)
↦
A
(
x
y
z
)
,
where
A
is in
S
U
(
3
)
{\displaystyle {\begin{pmatrix}x\\y\\z\end{pmatrix}}\mapsto
A{\begin{pmatrix}x\\y\\z\end{pmatrix}},\quad
{\text{where }}A{\text{ is in }}SU(3)}
Here, SU(3) refers to the Lie group of 3×3
unitary matrices with determinant 1 (special
unitary group). For example, the flavour rotation
A
=
(
0
1
0
−
1
0
0
0
0
1
)
{\displaystyle A={\begin{pmatrix}0&1&0\\-1&0&0\\0&0&1\end{pmatrix}}}
is a transformation that simultaneously turns
all the up quarks in the universe into down
quarks and vice versa. More specifically,
these flavour rotations are exact symmetries
if only strong force interactions are looked
at, but they are not truly exact symmetries
of the universe because the three quarks have
different masses and different electroweak
interactions.
This approximate symmetry is called flavour
symmetry, or more specifically flavour SU(3)
symmetry.
=== Connection to representation theory ===
Assume we have a certain particle—for example,
a proton—in a quantum state
|
ψ
⟩
{\displaystyle |\psi \rangle }
. If we apply one of the flavour rotations
A to our particle, it enters a new quantum
state which we can call
A
|
ψ
⟩
{\displaystyle A|\psi \rangle }
. Depending on A, this new state might be
a proton, or a neutron, or a superposition
of a proton and a neutron, or various other
possibilities. The set of all possible quantum
states spans a vector space.
Representation theory is a mathematical theory
that describes the situation where elements
of a group (here, the flavour rotations A
in the group SU(3)) are automorphisms of a
vector space (here, the set of all possible
quantum states that you get from flavour-rotating
a proton). Therefore, by studying the representation
theory of SU(3), we can learn the possibilities
for what the vector space is and how it is
affected by flavour symmetry.
Since the flavour rotations A are approximate,
not exact, symmetries, each orthogonal state
in the vector space corresponds to a different
particle species. In the example above, when
a proton is transformed by every possible
flavour rotation A, it turns out that it moves
around an 8-dimensional vector space. Those
8 dimensions correspond to the 8 particles
in the so-called "baryon octet" (proton, neutron,
Σ+, Σ0, Σ−, Ξ−, Ξ0, Λ). This corresponds
to an 8-dimensional ("octet") representation
of the group SU(3). Since A is an approximate
symmetry, all the particles in this octet
have similar mass.Every Lie group has a corresponding
Lie algebra, and each group representation
of the Lie group can be mapped to a corresponding
Lie algebra representation on the same vector
space. The Lie algebra
s
u
{\displaystyle {\mathfrak {su}}}
(3) can be written as the set of 3×3 traceless
Hermitian matrices. Physicists generally discuss
the representation theory of the Lie algebra
s
u
{\displaystyle {\mathfrak {su}}}
(3) instead of the Lie group SU(3), since
the former is simpler and the two are ultimately
equivalent
