Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 302 / 2

With $\eta =\mathrm{diag}(1,-1,-1)$, the defining condition for the $(1+2)$-dimensional Lorentz group isTaking determinants gives $(\det \Lambda {)}^2=1$, hence $\det \Lambda =\pm 1$. The norm of the zeroth column givesso $|{\Lambda }^0{}_0|\ge 1$. The Proper orthochronous Lorentz group is the component with $\det \Lambda =1$ and ${\Lambda }^0{}_0\ge 1$.

Identify $x=(x^0,x^1,x^2)$ with the real symmetric matrixFor $A\in SL(2,\mathbb{R})$, define $X\mapsto AX{A}^T$. This linear action preserves $\det X$, so it defines $\rho (A)\in S{O}^{+}(1,2)$, and $\rho (AB)=\rho (A)\rho (B)$. Its kernel is $\{\pm I\}$ and it is onto, hence it is a two-to-one covering homomorphism rather than an isomorphism. Consequently $PSL(2,\mathbb{R})=SL(2,\mathbb{R})/\{\pm I\}\cong S{O}^{+}(1,2)$.

The connected $(1+2)$-dimensional Poincare group isThe semidirect product records that Lorentz transformations act nontrivially on translations, which is what organizes momentum orbits. For a representative momentum $p$, its little group is the stabilizer $G_p=\{\Lambda \in S{O}^{+}(1,2):\Lambda p=p\}$. Wigner's classification constructs irreducible particle representations by inducing a unitary irreducible representation of $G_p$ along the Lorentz orbit of $p$.

For a massive positive-energy orbit, choose $p_{\ast }=(m,0,0)$ with $m>0$. Its little group is $SO(2)$, whose unitary irreducible representations are the characters $e^{is\theta }$. Induction gives one-particle states $|p,s⟩$ on the mass shell $p^2=m^2$, $p^0>0$, labeled by mass and spin. For the Poincare group itself, single-valuedness gives $s\in \mathbb{Z}$; its double cover permits half-integers, and its universal cover permits any real $s$, producing anyonic spin in $(2+1)$ dimensions.

For the positive-energy massless orbit choose $k=(E,E,0)$. Its connected little group is the one-parameter group of null rotations, isomorphic to $(\mathbb{R},+)$. Its unitary irreducible representations are the characters $t\mapsto e^{i\rho t}$, $\rho \in \mathbb{R}$, and induction gives the massless one-particle representations. The physically usual finite-component representation has trivial little-group action, $\rho =0$; there is no $SO(2)$ helicity subgroup for a null momentum in $(2+1)$ dimensions.