Topic

How do particle types relate to mathematical objects?

Mathematics Standard Model Representation Theory

Short Answer

Particles correspond to irreducible representations of the Poincare group (for spacetime properties) and the gauge group (for internal properties).

Spacetime Structure: Poincare Group

The Poincare group $\mathcal{P}$ is the symmetry group of special relativity:

$\mathcal{P} = \mathbb{R}^{1,3} \rtimes SO(1,3)$

It consists of:

Translations in spacetime
Lorentz transformations (rotations + boosts)

Wigner Classification

Particles are classified by irreducible representations labeled by:

Mass $m \geq 0$
Spin $s = 0, \frac{1}{2}, 1, \frac{3}{2}, 2, \ldots$

Type	Mass	Spin	Examples
Massive	$m > 0$	$s$	Electron, W/Z bosons, Higgs
Massless	$m = 0$	$\pm s$ (helicity)	Photon, gluon, graviton

Internal Structure: Gauge Groups

Beyond spacetime, particles carry charges under gauge groups.

Representation Theory

A representation of a Lie group $G$ is a homomorphism:

$\rho: G \to GL(V)$

where $V$ is a vector space. Particles “living in” $V$ transform according to $\rho$ .

Standard Model Gauge Group

$G_{SM} = SU(3)_C \times SU(2)_L \times U(1)_Y$

Each factor corresponds to a fundamental force.

Specific Examples

Electron

The left-handed electron transforms as:

Poincare: Spin-1/2 massive representation
$SU(3)_C$ : Trivial (singlet 1) - no color
$SU(2)_L$ : Doublet 2 (with neutrino)
$U(1)_Y$ : Hypercharge $Y = -1/2$

Quark

Left-handed up quark transforms as:

Poincare: Spin-1/2 massive
$SU(3)_C$ : Fundamental 3 (three colors)
$SU(2)_L$ : Doublet 2 (with down quark)
$U(1)_Y$ : Hypercharge $Y = +1/6$

Photon

Poincare: Spin-1 massless (helicity $\pm 1$ )
Gauge: Singlet under all (after symmetry breaking)

Gluon

Poincare: Spin-1 massless
$SU(3)_C$ : Adjoint 8 (carries color)

Mathematical Objects Summary

Mathematical Object	Physical Meaning
Lie group $G$	Symmetry of the theory
Lie algebra $\mathfrak{g}$	Infinitesimal symmetries
Representation $\rho$	How particles transform
Irreducible rep	Single particle type
Dimension of rep	Number of components
Casimir operators	Invariant labels (mass, spin)

Fiber Bundle Perspective

Mathematically, gauge theories are described by:

Principal bundle $P \to M$ with structure group $G$
Associated vector bundle $E = P \times_\rho V$
Matter fields are sections of $E$
Gauge fields are connections on $P$

References

E. P. Wigner, “On Unitary Representations of the Inhomogeneous Lorentz Group,” Annals of Mathematics 40, 149–204 (1939) — Original classification paper
S. Weinberg, The Quantum Theory of Fields, Vol. 1 (Cambridge University Press, 1995), Chapters 2 and 5
H. Georgi, Lie Algebras in Particle Physics, 2nd ed. (Westview Press, 1999)
W. Fulton and J. Harris, Representation Theory: A First Course (Springer, 1991)
B. C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed. (Springer, 2015)