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 group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
s 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
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
. 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 . To help understand what types of particles can exist, it is important to classify the possibilities for
allowed by
symmetries
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, and their properties. Let
be a Hilbert space describing a particular quantum system and let
be a group of symmetries of the quantum system. In a relativistic quantum system, for example,
might be the
Poincaré group
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
, while for the hydrogen atom,
might be the
rotation group SO(3). The particle state is more precisely characterized by the associated
projective Hilbert space , 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
and, under the natural projection map
, an element of
.
By definition of a symmetry of a quantum system, there is a
group action on
. For each
, there is a corresponding transformation
of
. More specifically, if
is some symmetry of the system (say, rotation about the x-axis by 12°), then the corresponding transformation
of
is a map on ray space. For example, when rotating a ''stationary'' (zero momentum) spin-5 particle about its center,
is a rotation in 3D space (an element of
), while
is an operator whose domain and range are each the space of possible quantum states of this particle, in this example the projective space
associated with an 11-dimensional complex Hilbert space
.
Each map
preserves, by definition of symmetry, the
ray product on
induced by the inner product on
; according to
Wigner's theorem, this transformation of
comes from a unitary or anti-unitary transformation
of
. Note, however, that the
associated to a given
is not unique, but only unique ''up to a phase factor''. The composition of the operators
should, therefore, reflect the composition law in
, but only up to a phase factor:
:
,
where
will depend on
and
. Thus, the map sending
to
is a
''projective unitary representation'' of
, or possibly a mixture of unitary and anti-unitary, if
is disconnected. In practice, anti-unitary operators are always associated with
time-reversal symmetry
T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal,
: T: t \mapsto -t.
Since the second law of thermodynamics states that entropy increases as time flows toward the futur ...
.
Ordinary versus projective representations
It is important physically that in general
does not have to be an ordinary representation of
; it may not be possible to choose the phase factors in the definition of
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
with values in a two-dimensional spinor space. The action of
on the spinor space is only projective: It does not come from an ordinary representation of
. There is, however, an associated ordinary representation of the universal cover
of
on spinor space.
For many interesting classes of groups
,
Bargmann's theorem tells us that every projective unitary representation of
comes from an ordinary representation of the universal cover
of
. Actually, if
is finite dimensional, then regardless of the group
, every projective unitary representation of
comes from an ordinary unitary representation of
. If
is infinite dimensional, then to obtain the desired conclusion, some algebraic assumptions must be made on
(see below). In this setting the result is a
theorem of Bargmann. Fortunately, in 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
does not admit a nontrivial one-dimensional central extension. This is the case if and only if the
second cohomology group of
is trivial. In this case, it may still be true that the group admits a central extension by a ''discrete'' group. But extensions of
by discrete groups are covers of
. For instance, the universal cover
is related to
through the quotient
with the central subgroup
being the center of
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
of the symmetry group
. This is desirable because
is much easier to work with than the non-vector space
. If the representations of
can be classified, much more information about the possibilities and properties of
are available.
The Heisenberg case
An example in which Bargmann's theorem does not apply comes from a quantum particle moving in
. The group of translational symmetries of the associated phase space,
, is the commutative group
. In the usual quantum mechanical picture, the
symmetry is not implemented by a unitary representation of
. 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
, but it does not come from an ordinary representation of
, even though
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
.
Poincaré group
The group of translations and
Lorentz transformations
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
form the
Poincaré group
The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
, 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
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally b ...
(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
tachyon
A tachyon () or tachyonic particle is a hypothetical particle that always travels faster than light. Physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such partic ...
s,
infraparticle
An infraparticle is an electrically charged particle and its surrounding cloud of soft photons—of which there are infinite number, by virtue of the infrared divergence of quantum electrodynamics. That is, it is a dressed particle rather than a ...
s, 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
Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are associa ...
SU(3), an exact symmetry corresponding to the continuous interchange of the three
quark
A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
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.
[ Section 5.7] 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 force
Weak may refer to:
Songs
* "Weak" (AJR song), 2016
* "Weak" (Melanie C song), 2011
* "Weak" (SWV song), 1993
* "Weak" (Skunk Anansie song), 1995
* "Weak", a song by Seether from '' Seether: 2002-2013''
Television episodes
* "Weak" (''Fear t ...
s, but differently under the
electromagnetic force
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
.
Example: isospin symmetry
An example from the real world is
isospin symmetry, an
SU(2) group corresponding to the similarity between
up quark
The up quark or u quark (symbol: u) is the lightest of all quarks, a type of elementary particle, and a significant constituent of matter. It, along with the down quark, forms the neutrons (one up quark, two down quarks) and protons (two up quark ...
s and
down quark
The down quark or d quark (symbol: d) is the second-lightest of all quarks, a type of elementary particle, and a major constituent of matter. Together with the up quark, it forms the neutrons (one up quark, two down quarks) and protons (two up ...
s. This is an approximate symmetry: while up and down quarks are identical in how they interact under the
strong force
The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
, they have different masses and different electroweak interactions. Mathematically, there is an abstract two-dimensional vector space
:
and the laws of physics are ''approximately'' invariant under applying a determinant-1
unitary transformation
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Formal definition
More precisely, ...
to this space:
[Lecture notes by Prof. Mark Thomson]
/ref>
:
For example, 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 (, , and ), 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
In particle physics, flavour or flavor refers to the ''species'' of an elementary particle. The Standard Model counts six flavours of quarks and six flavours of leptons. They are conventionally parameterized with ''flavour quantum numbers'' th ...
, an SU(3) group corresponding to the similarity between up quark
The up quark or u quark (symbol: u) is the lightest of all quarks, a type of elementary particle, and a significant constituent of matter. It, along with the down quark, forms the neutrons (one up quark, two down quarks) and protons (two up quark ...
s, down quark
The down quark or d quark (symbol: d) is the second-lightest of all quarks, a type of elementary particle, and a major constituent of matter. Together with the up quark, it forms the neutrons (one up quark, two down quarks) and protons (two up ...
s, 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
Yuval Ne'eman ( he, יובל נאמן, 14 May 1925 – 26 April 2006) was an Israeli theoretical physicist, military scientist, and politician. He was Minister of Science and Development in the 1980s and early 1990s. He was the President o ...
.
See also
*Charge (physics)
In physics, a charge is any of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges correspond to the time-invariant generators of a symmetry group, and specifically, ...
* Representation theory:
** Of Lie algebras
** Of Lie groups
* Projective representation
*Special unitary group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
Notes
References
*Coleman, Sidney (1985) ''Aspects of Symmetry: Selected Erice Lectures of Sidney Coleman''. Cambridge Univ. Press. .
*Georgi, Howard (1999) ''Lie Algebras in Particle Physics''. Reading, Massachusetts: Perseus Books. .
*.
* .
*Sternberg, Shlomo (1994) ''Group Theory and Physics''. Cambridge Univ. Press. . Especially pp. 148–150.
* Especially appendices A and B to Chapter 2.
External links
*
{{Industrial and applied mathematics
Lie algebras
Representation theory of Lie groups
Theoretical physics
Conservation laws
Quantum field theory