There is a natural connection between particle
physics and representation theory, as first
noted in the 1930s by Eugene Wigner. It links
the properties of elementary particles to
the structure of Lie groups and Lie algebras.
According to this connection, the different
quantum states of an elementary particle give
rise to an irreducible representation of the
Poincaré group. Moreover, the properties
of the various particles, including their
spectra, can be related to representations
of Lie algebras, corresponding to "approximate
symmetries" of the universe.
== General picture ==
=== Symmetries of a quantum system ===
In quantum mechanics, any particular one-particle
state is represented as a vector in a Hilbert
space
H
{\displaystyle {\mathcal {H}}}
. To help understand what types of particles
can exist, it is important to classify the
possibilities for
H
{\displaystyle {\mathcal {H}}}
allowed by symmetries, and their properties.
In a relativistic quantum system, for example,
G
{\displaystyle G}
might be the Poincaré group, while for the
hydrogen atom,
G
{\displaystyle G}
might be the rotation group SO(3). The particle
state is more precisely characterized by the
associated projective Hilbert space
P
H
{\displaystyle \mathrm {P} {\mathcal {H}}}
, also called ray space, since two vectors
that differ by a nonzero scalar factor correspond
to the same physical quantum state represented
by a ray in Hilbert space, which is an equivalence
class in
H
{\displaystyle {\mathcal {H}}}
and, under the natural projection map
H
→
P
H
{\displaystyle {\mathcal {H}}\rightarrow \mathrm
{P} {\mathcal {H}}}
, an element of
P
H
{\displaystyle \mathrm {P} {\mathcal {H}}}
.
Let
H
{\displaystyle {\mathcal {H}}}
be a Hilbert space describing a particular
quantum system and let
G
{\displaystyle G}
be a group of symmetries of the quantum system.
By definition of a symmetry of a quantum system,
there is a group action on
P
H
{\displaystyle \mathrm {P} {\mathcal {H}}}
. For each
g
∈
G
{\displaystyle g\in G}
, there is a corresponding transformation
V
(
g
)
{\displaystyle V(g)}
of
P
H
{\displaystyle \mathrm {P} {\mathcal {H}}}
. More specifically, if
g
{\displaystyle g}
is some symmetry of the system (say, rotation
about the x-axis by 12°), then the corresponding
transformation
V
(
g
)
{\displaystyle V(g)}
of
P
H
{\displaystyle \mathrm {P} {\mathcal {H}}}
is a map on ray space. For example, when rotating
a stationary (zero momentum) spin-5 particle
about its center,
g
{\displaystyle g}
is a rotation in 3D space (an element of
S
O
(
3
)
{\displaystyle \mathrm {SO(3)} }
), while
V
(
g
)
{\displaystyle V(g)}
is an operator whose domain and range are
each the space of possible quantum states
of this particle, in this example the projective
space
P
H
{\displaystyle \mathrm {P} {\mathcal {H}}}
associated with an 11-dimensional complex
Hilbert space
H
{\displaystyle {\mathcal {H}}}
.
Each map
V
(
g
)
{\displaystyle V(g)}
preserves, by definition of symmetry, the
ray product on
P
H
{\displaystyle \mathrm {P} {\mathcal {H}}}
induced by the inner product on
H
{\displaystyle {\mathcal {H}}}
; according to Wigner's theorem, this transformation
of
P
H
{\displaystyle \mathrm {P} {\mathcal {H}}}
comes from a unitary or anti-unitary transformation
U
(
g
)
{\displaystyle U(g)}
of
H
{\displaystyle {\mathcal {H}}}
. Note, however, that 
the
U
(
g
)
{\displaystyle U(g)}
associated to a given
V
(
g
)
{\displaystyle V(g)}
is not unique, but only unique up to a phase
factor. The composition of the operators
U
(
g
)
{\displaystyle U(g)}
should, therefore, reflect the composition
law in
G
{\displaystyle G}
, but only up to a phase factor:
U
(
g
h
)
=
e
i
θ
U
(
g
)
U
(
h
)
{\displaystyle U(gh)=e^{i\theta }U(g)U(h)}
,where
θ
{\displaystyle \theta }
will depend on
g
{\displaystyle g}
and
h
{\displaystyle h}
. Thus, the map sending
g
{\displaystyle g}
to
U
(
g
)
{\displaystyle U(g)}
is a projective unitary representation of
G
{\displaystyle G}
, or possibly a mixture of unitary and anti-unitary,
if
G
{\displaystyle G}
is disconnected. In practice, anti-unitary
operators are always associated with time-reversal
symmetry.
=== Ordinary versus projective representations
===
It is important physically that in general
U
(
⋅
)
{\displaystyle U(\cdot )}
does not have to be an ordinary representation
of
G
{\displaystyle G}
; it may not be possible to choose the phase
factors in the definition of
U
(
g
)
{\displaystyle U(g)}
to eliminate the phase factors in their composition
law. An electron, for example, is a spin-one-half
particle; its Hilbert space consists of wave
functions on
R
3
{\displaystyle \mathbb {R} ^{3}}
with values in a two-dimensional spinor space.
The action of
S
O
(
3
)
{\displaystyle \mathrm {SO(3)} }
on the spinor space is only projective: It
does not come from an ordinary representation
of
S
O
(
3
)
{\displaystyle \mathrm {SO(3)} }
. There is, however, an associated ordinary
representation of the universal cover
S
U
(
2
)
{\displaystyle \mathrm {SU(2)} }
of
S
O
(
3
)
{\displaystyle \mathrm {SO(3)} }
on spinor space. For many interesting classes
of groups
G
{\displaystyle G}
, Bargmann's theorem tells us that every projective
unitary representation of
G
{\displaystyle G}
comes from an ordinary representation of the
universal cover
G
~
{\displaystyle {\tilde {G}}}
of
G
{\displaystyle G}
. Actually, if
H
{\displaystyle {\mathcal {H}}}
is finite dimensional, then regardless of
the group
G
{\displaystyle G}
, every projective unitary representation
of
G
{\displaystyle G}
comes from an ordinary unitary representation
of
G
~
{\displaystyle {\tilde {G}}}
. If
H
{\displaystyle {\mathcal {H}}}
is infinite dimensional, then to obtain the
desired conclusion, some algebraic assumptions
must be made on
G
{\displaystyle G}
(see below). In this setting the result is
a theorem of Bargmann. Fortunately, the crucial
case of the Poincaré group, Bargmann's theorem
applies. (See Wigner's classification of the
representations of the universal cover of
the Poincaré group.)
The requirement referred to above is that
the Lie algebra
g
{\displaystyle {\mathfrak {g}}}
does not admit a nontrivial one-dimensional
central extension. This is the case if and
only if the second cohomology group of
g
{\displaystyle {\mathfrak {g}}}
is trivial. In this case, it may still be
true that the group admits a central extension
by a discrete group. But extensions of
G
{\displaystyle G}
by a discrete group are nothing but a covers
of
G
{\displaystyle G}
. For instance, the universal cover
G
~
{\displaystyle {\tilde {G}}}
is related to
G
{\displaystyle G}
through the quotient
G
≈
G
~
/
Γ
{\displaystyle G\approx {\tilde {G}}/\Gamma
}
with the central subgroup
Γ
{\displaystyle \Gamma }
being the center of
G
~
{\displaystyle {\tilde {G}}}
itself, isomorphic to the fundamental group
of the covered group.
Thus, in favorable cases, the quantum system
will carry a unitary representation of the
universal cover
G
~
{\displaystyle {\tilde {G}}}
of the symmetry group
G
{\displaystyle G}
. This is desirable because
H
{\displaystyle {\mathcal {H}}}
is much easier to work with than the non-vector
space
P
H
{\displaystyle \mathrm {P} {\mathcal {H}}}
. If the representations of
G
~
{\displaystyle {\tilde {G}}}
can be classified, much more information about
the possibilities and properties of
H
{\displaystyle {\mathcal {H}}}
are available.
=== The Heisenberg case ===
An example in which Bargmann's theorem does
not apply comes from a quantum particle moving
in
R
n
{\displaystyle \mathbb {R} ^{n}}
. The group of translational symmetries of
the associated phase space,
R
2
n
{\displaystyle \mathbb {R} ^{2n}}
, is the commutative group
R
2
n
{\displaystyle \mathbb {R} ^{2n}}
. In the usual quantum mechanical picture,
the
R
2
n
{\displaystyle \mathbb {R} ^{2n}}
symmetry is not implement by a unitary representation
of
R
2
n
{\displaystyle \mathbb {R} ^{2n}}
. After all, in the quantum setting, translations
in position space and translations in momentum
space do not commute. This failure to commute
reflects the failure of the position and momentum
operators—which are the infinitesimal generators
of translations in momentum space and position
space, respectively—to commute. Nevertheless,
translations in position space and translations
in momentum space do commute up to a phase
factor. Thus, we have a well-defined projective
representation of
R
2
n
{\displaystyle \mathbb {R} ^{2n}}
, but it does not come from an ordinary representation
of
R
2
n
{\displaystyle \mathbb {R} ^{2n}}
, even though
R
2
n
{\displaystyle \mathbb {R} ^{2n}}
is simply connected.
In this case, to obtain an ordinary representation,
one has to pass to the Heisenberg group, which
is a nontrivial one-dimensional central extension
of
R
2
n
{\displaystyle \mathbb {R} ^{2n}}
.
== Poincaré group ==
The group of translations and Lorentz transformations
form the Poincaré group, and this group should
be a symmetry of a relativistic quantum system
(neglecting general relativity effects, or
in other words, in flat space). Representations
of the Poincaré group are in many cases characterized
by a nonnegative mass and a half-integer spin
(see Wigner's classification); this can be
thought of as the reason that particles have
quantized spin. (Note that there are in fact
other possible representations, such as tachyons,
infraparticles, etc., which in some cases
do not have quantized spin or fixed mass.)
== 
Other symmetries ==
While the spacetime symmetries in the Poincaré
group are particularly easy to visualize and
believe, there are also other types of symmetries,
called internal symmetries. One example is
color SU(3), an exact symmetry corresponding
to the continuous interchange of the three
quark colors.
== Lie algebras versus Lie groups ==
Many (but not all) symmetries or approximate
symmetries form Lie groups. Rather than study
the representation theory of these Lie groups,
it is often preferable to study the closely
related representation theory of the corresponding
Lie algebras, which are usually simpler to
compute.
Now, representations of the Lie algebra correspond
to representations of the universal cover
of the original group. In the finite-dimensional
case—and the infinite-dimensional case,
provided that Bargmann's theorem applies—irreducible
projective representations of the original
group correspond to ordinary unitary representations
of the universal cover. In those cases, computing
at the Lie algebra level is appropriate. This
is the case, notably, for studying the irreducible
projective representations of the rotation
group SO(3). These are in one-to-one correspondence
with the ordinary representations of the universal
cover SU(2) of SO(3). The representations
of the SU(2) are then in one-to-one correspondence
with the representations of its Lie algebra
su(2), which is isomorphic to the Lie algebra
so(3) of SO(3).
Thus, to summarize, the irreducible projective
representations of SO(3) are in one-to-one
correspondence with the irreducible ordinary
representations of its Lie algebra so(3).
The two-dimensional "spin 1/2" representation
of the Lie algebra so(3), for example, does
not correspond to an ordinary (single-valued)
representation of the group SO(3). (This fact
is the origin of statements to the effect
that "if you rotate the wave function of an
electron by 360 degrees, you get the negative
of the original wave function.") Nevertheless,
the spin 1/2 representation does give rise
to a well-defined projective representation
of SO(3), which is all that is required physically.
== Approximate symmetries ==
Although the above symmetries are believed
to be exact, other symmetries are only approximate.
=== Hypothetical example ===
As an example of what an approximate symmetry
means, suppose an experimentalist lived inside
an infinite ferromagnet, with magnetization
in some particular direction. The experimentalist
in this situation would find not one but two
distinct types of electrons: one with spin
along the direction of the magnetization,
with a slightly lower energy (and consequently,
a lower mass), and one with spin anti-aligned,
with a higher mass. Our usual SO(3) rotational
symmetry, which ordinarily connects the spin-up
electron with the spin-down electron, has
in this hypothetical case become only an approximate
symmetry, relating different types of particles
to each other.
=== General definition ===
In general, an approximate symmetry arises
when there are very strong interactions that
obey that symmetry, along with weaker interactions
that do not. In the electron example above,
the two "types" of electrons behave identically
under the strong and weak forces, but differently
under the electromagnetic force.
=== Example: isospin symmetry ===
An example from the real world is isospin
symmetry, an SU(2) group corresponding to
the similarity between up quarks and down
quarks. This is an approximate symmetry: While
up and down quarks are identical in how they
interact under the strong force, they have
different masses and different electroweak
interactions. Mathematically, there is an
abstract two-dimensional vector space
up quark
→
(
1
0
)
,
down 
quark
→
(
0
1
)
{\displaystyle {\text{up quark}}\rightarrow
{\begin{pmatrix}1\\0\end{pmatrix}},\qquad
{\text{down quark}}\rightarrow {\begin{pmatrix}0\\1\end{pmatrix}}}
and the laws of physics are approximately
invariant under applying a determinant-1 unitary
transformation to this space:
(
x
y
)
↦
A
(
x
y
)
,
where
A
is in
S
U
(
2
)
{\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}\mapsto
A{\begin{pmatrix}x\\y\end{pmatrix}},\quad
{\text{where }}A{\text{ is in }}SU(2)}
For example,
A
=
(
0
1
−
1
0
)
{\displaystyle A={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}
would turn all up quarks in the universe into
down quarks and vice versa. Some examples
help clarify the possible effects of these
transformations:
When these unitary transformations are applied
to a proton, it can be transformed into a
neutron, or into a superposition of a proton
and neutron, but not into any other particles.
Therefore, the transformations move the proton
around a two-dimensional space of quantum
states. The proton and neutron are called
an "isospin doublet", mathematically analogous
to how a spin-½ particle behaves under ordinary
rotation.
When these unitary transformations are applied
to any of the three pions (π0, π+, and π−),
it can change any of the pions into any other,
but not into any non-pion particle. Therefore,
the transformations move the pions around
a three-dimensional space of quantum states.
The pions are called an "isospin triplet",
mathematically analogous to how a spin-1 particle
behaves under ordinary rotation.
These transformations have no effect at all
on an electron, because it contains neither
up nor down quarks. The electron is called
an isospin singlet, mathematically analogous
to how a spin-0 particle behaves under ordinary
rotation.In general, particles form isospin
multiplets, which correspond to irreducible
representations of the Lie algebra SU(2).
Particles in an isospin multiplet have very
similar but not identical masses, because
the up and down quarks are very similar but
not identical.
=== Example: flavour symmetry ===
Isospin symmetry can be generalized to flavour
symmetry, an SU(3) group corresponding to
the similarity between up quarks, down quarks,
and strange quarks. This is, again, an approximate
symmetry, violated by quark mass differences
and electroweak interactions—in fact, it
is a poorer approximation than isospin, because
of the strange quark's noticeably higher mass.
Nevertheless, particles can indeed be neatly
divided into groups that form irreducible
representations of the Lie algebra SU(3),
as first noted by Murray Gell-Mann and independently
by Yuval Ne'eman.
== See also ==
Charge (physics)
Representation theory:
Of Lie algebras
Of Lie groups
Projective representation
Special unitary group
== Notes
