Symmetry Groups in Physics

Group theory provides the mathematical language for symmetry in physics. The symmetries of a physical system -- rotations, translations, permutations, gauge transformations -- form groups whose representation theory determines the structure of quantum states, selection rules, and conservation laws.

Symmetry Groups and Their Actions

Definition8.1Symmetry Group of a Physical System

A symmetry of a physical system is a transformation that leaves the system's laws (Lagrangian, Hamiltonian, or equations of motion) invariant. The set of all symmetries forms a group $G$ under composition. Key examples: the rotation group $\mathrm{SO}(3)$ for rotationally invariant systems, the Poincare group $\mathrm{ISO}(3,1)$ for relativistic systems, the symmetric group $S_n$ for identical particle permutations, and gauge groups $\mathrm{U}(1)$ , $\mathrm{SU}(2)$ , $\mathrm{SU}(3)$ for internal symmetries.

ExampleThe Rotation Group SO(3)

$\mathrm{SO}(3) = \{R \in \mathrm{GL}(3,\mathbb{R}) : R^TR = I,\, \det R = 1\}$ is a 3-dimensional Lie group. Every rotation is specified by an axis $\hat{n}$ and angle $\theta$ : $R(\hat{n},\theta) = e^{-i\theta\hat{n}\cdot\mathbf{J}}$ where $\mathbf{J} = (J_1,J_2,J_3)$ are the generators satisfying $[J_i,J_j] = i\varepsilon_{ijk}J_k$ . The topology of $\mathrm{SO}(3) \cong \mathbb{RP}^3$ is non-simply connected: $\pi_1(\mathrm{SO}(3)) = \mathbb{Z}_2$ . Its universal cover $\mathrm{SU}(2) \cong S^3$ is the spin group, with the double cover $\mathrm{SU}(2) \to \mathrm{SO}(3)$ explaining spinor representations.

Lie Algebras and Generators

Definition8.2Lie Algebra of a Symmetry Group

The Lie algebra $\mathfrak{g}$ of a Lie group $G$ consists of tangent vectors at the identity with the commutator bracket $[X,Y] = XY - YX$ . For matrix Lie groups, $\mathfrak{g} = \{X : e^{tX} \in G \text{ for all } t\}$ . The structure is encoded in structure constants: $[T_a,T_b] = if_{abc}T_c$ where $\{T_a\}$ is a basis. The Killing form $\kappa(X,Y) = \mathrm{tr}(\mathrm{ad}_X \circ \mathrm{ad}_Y)$ classifies the algebra: $\mathfrak{g}$ is semisimple if and only if $\kappa$ is non-degenerate.

ExampleLie Algebras in Physics

$\mathfrak{su}(2)$ : generators $\{J_1,J_2,J_3\}$ with $[J_i,J_j] = i\varepsilon_{ijk}J_k$ . Basis: $J_\pm = J_1 \pm iJ_2$ , $J_3$ with $[J_3,J_\pm] = \pm J_\pm$ , $[J_+,J_-] = 2J_3$ . $\mathfrak{su}(3)$ : 8 generators (Gell-Mann matrices $\lambda_1,\ldots,\lambda_8$ ), rank 2. Poincare algebra: 10 generators ( $P_\mu$ translations, $M_{\mu\nu}$ Lorentz transformations) with $[P_\mu,P_\nu] = 0$ , $[M_{\mu\nu},P_\rho] = i(\eta_{\mu\rho}P_\nu - \eta_{\nu\rho}P_\mu)$ , $[M_{\mu\nu},M_{\rho\sigma}] = i(\eta_{\mu\rho}M_{\nu\sigma} - \cdots)$ . Casimir operators: $P^2 = P_\mu P^\mu$ (mass) and $W^2 = W_\mu W^\mu$ (spin), where $W^\mu = \frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}M_{\nu\rho}P_\sigma$ is the Pauli-Lubanski vector.

Discrete Symmetries

Definition8.3Discrete Symmetry Groups

Discrete symmetries cannot be continuously connected to the identity. In quantum mechanics: parity $P$ (spatial inversion $\mathbf{x}\to-\mathbf{x}$ ), time reversal $T$ (complex conjugation and $t\to-t$ , an antiunitary operator), and charge conjugation $C$ (particle $\leftrightarrow$ antiparticle). The CPT theorem states that any Lorentz-invariant local quantum field theory is invariant under the combined $CPT$ transformation. Individual symmetries $C$ , $P$ , $T$ can be violated (weak interactions violate $P$ and $CP$ ).

RemarkPoint Groups and Crystal Symmetry

In solid-state physics, the symmetry of a crystal is described by its space group (230 types in 3D) combining point group operations (rotations, reflections) with lattice translations. The point group (32 crystallographic types) determines selection rules for optical transitions, Raman activity, and phonon symmetries. The irreducible representations of the point group label the electronic band structure at high-symmetry points in the Brillouin zone.