Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 302 / 2 / b / Solution

Matrix multiplication givesFor example, $[J_1,J_2]=J_3$, $[K_1,K_2]=-J_3$, and $[J_1,K_2]=K_3$. Over $\mathbb{C}$, defineThen $[A_i,A_j]={ϵ}_{ijk}{A}_k$, $[B_i,B_j]={ϵ}_{ijk}{B}_k$, and $[A_i,B_j]=0$. This proves the chiral decomposition of the complex Lorentz algebra. Its finite-dimensional irreducible representations are tensor products of irreducible representations of the two factors and are labelled by pairs $(j_L,j_R)$ of nonnegative half-integers.

Solved by gpt-5.6-sol high.