The Lorentz and Poincaré Groups

The transformations preserving the interval form a Lie group whose structure organizes all of relativistic physics: the Lorentz group $O(1,3)$ and, with translations, the Poincaré group. This page collects its components, generators, algebra, the $SL(2,\mathbb{C})$ cover, and the bridge to Wigner's classification of particles. Abstract Lie-group background is in math/group-theory/00-README.md; the QFT use is in QFT/preliminaries.md. Convention: $\eta =\mathrm{diag}(+,-,-,-)$, $c=1$.

The four components

Lorentz transformations satisfy ${\Lambda }^{\mathsf{T}}\eta \Lambda =\eta$, giving $\det \Lambda =\pm 1$ and $|{\Lambda }^0{}_0|\ge 1$. The two signs split $O(1,3)$ into four disconnected pieces:

Physics builds on $S{O}^{+}(1,3)$, the proper orthochronous group of boosts and rotations; $P$, $T$ are discrete symmetries.

Generators and algebra

The six generators are rotations $J_i$ and boosts $K_i$, satisfying

$$[J_i,J_j]=i{ϵ}_{ijk}{J}_k,[J_i,K_j]=i{ϵ}_{ijk}{K}_k,[K_i,K_j]=-i{ϵ}_{ijk}{J}_k.$$

The boost–boost commutator's minus sign is the source of the Thomas–Wigner rotation (Lorentz transformations). Defining $A_i=\frac{1}{2}(J_i+i{K}_i)$, $B_i=\frac{1}{2}(J_i-i{K}_i)$ decouples the algebra into two $SU(2)$'s: $\mathfrak{so}(1,3{)}_{\mathbb{C}}\cong \mathfrak{su}(2)\oplus \mathfrak{su}(2)$, so reps are labeled $(j_A,j_B)$ — scalars $(0,0)$, Weyl spinors $(\frac{1}{2},0)$, $(0,\frac{1}{2})$, vectors $(\frac{1}{2},\frac{1}{2})$.

Universal cover

The cover $SL(2,\mathbb{C})\to S{O}^{+}(1,3)$ has kernel $\{\pm 1\}$ (a 2-to-1 map), exactly why half-integer spin is allowed: physical states are rays, so projective reps of the Lorentz group are honest reps of its cover. See group-theory/README.md § projective reps.

Poincaré group and particles

Adjoining translations gives the 10-parameter Poincaré group $\mathcal{P}={\mathbb{R}}^{1,3}⋊S{O}^{+}(1,3)$, a semidirect product. Its two Casimirs are $P^2=m^2$ (mass) and $W^2$ (Pauli–Lubanski, spin). Wigner's classification labels particles by $(m,s)$ via unitary irreps — the postulate the QFT section starts from. Conserved charges: translations→energy-momentum, rotations→angular momentum, boosts→center-of-energy.

This is the symmetry the QFT axioms assume; continue to bridge to QFT.