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There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner.[1] 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

In quantum mechanics, any particular particle (with a given momentum distribution, location distribution, spin state, etc.) is written as a vector (or "ket") in a Hilbert space H. To help understand what types of particles can exist, it is important to classify the possibilities for H, and their properties. The particle is more precisely characterized by the associated projective Hilbert space PH, since two vectors that differ by a scalar factor (or in physics terminology, two "kets" that differ by a "phase factor") correspond to the same physical quantum state.

Let G be the symmetry group of the universe – that is, the set of symmetries under which the laws of physics are invariant. (For example, one element of G is the simultaneous translation of all particles and fields forward in time by five seconds.) Starting with a particular particle in the state ket \( |p_0\rangle \) , and a symmetry transformation g in G, it is possible to apply the symmetry transformation to the particle to get a new state ket \( |p_g\rangle=g|p_0\rangle\) . For this picture to be consistent, it is necessary that PH holds a projective group representation of G. (For example, this condition guarantees that applying a symmetry transformation, then applying its inverse transformation, will restore the original quantum state.)

Therefore, any given particle is associated with a unique (possibly trivial) representation of G on a projective vector space PH. (We say the particle "lies in", or "transforms as" the representation.) In many important cases, it can be shown that the particle is also (more specifically) associated with a group representation of G on the underlying (non-projective) space H.[2] Wigner's Theorem proves that it is a unitary representation, or possibly anti-unitary.[2]

So we conclude that each type of particle corresponds to a representation of G, and if we can classify the group representations of G, we will have much more information about the possibilities and properties of H, and hence what types of particles can exist.

Poincaré group

The group of translations and Lorentz transformations form the Poincaré group, and this group is certainly a subgroup of G (neglecting general relativity effects, or in other words, in flat space). Hence, any representation of G will in particular be a representation of the Poincaré group. 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
The pattern of weak isospins, weak hypercharges, and color charges (weights) of all known elementary particles in the Standard Model, rotated by the weak mixing angle to show electric charge roughly along the vertical.

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.
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 we lived inside an infinite ferromagnet, with magnetization in some particular direction. An 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.

Lie algebras versus Lie groups

Many (but not all) symmetries or approximate symmetries, for example the ones above, 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.
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
Main article: Isospin

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

\( \text{up quark} \rightarrow \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad \text{down quark} \rightarrow \begin{pmatrix} 0 \\ 1 \end{pmatrix}\)

and the laws of physics are approximately invariant under applying a determinant-1 unitary transformation to this space:[3]

\( \begin{pmatrix} x \\ y \end{pmatrix} \mapsto A \begin{pmatrix} x \\ y \end{pmatrix}, \quad \text{where } A \text{ is in } SU(2)\)

For example, \( A=\begin{pmatrix} 0&1 \\ -1&0 \end{pmatrix}\) 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.[3] 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 the eightfold way).

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Particle physics and representation theory