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

Differentiating ${\Lambda }^T\eta \Lambda =\eta$ at the identity gives $X^T\eta +\eta X=0$. The six matricesobey this condition and form a basis of the Lorentz algebra. With cyclic indices,The $J_i$ mix the two spatial coordinates perpendicular to $x^i$ and therefore generate rotations about that axis; $K_i$ mixes $x^0$ with $x^i$ and generates a Lorentz boost in the $x^i$ direction.

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.

A coordinate transformation $x\mapsto Px$ conjugates an infinitesimal generator, hence $M^{\mu \nu }\mapsto P{M}^{\mu \nu }P$. It fixes the rotation generators $J_i$ and negates the boosts $K_i$. Therefore it exchanges $A_i$ and $B_i$, and the parity action on a Lorentz representation isAn irreducible representation is parity invariant precisely when $j_L=j_R$. If $j_L\ne j_R$, parity invariance requires the reducible sum $(j_L,j_R)\oplus (j_R,j_L)$. Thus the two Weyl-spinor representations $(1/2,0)$ and $(0,1/2)$ are exchanged, while their direct sum is the parity-invariant Dirac spinor.

Parity preserves the Minkowski metric but reverses orientation. Hence ordinary tensor contraction makesparity even, while the Levi-Civita pseudotensor changes sign and makesparity odd. Equivalently, $C_1$ is proportional to ${\mathbf{J}}^2-{\mathbf{K}}^2$ and $C_2$ to $\mathbf{J}\cdot \mathbf{K}$. The combinations $C_1\pm i{C}_2$ are proportional to the two quadratic Lorentz Casimir invariants, and parity exchanges them exactly as it exchanges the two factors in the complexified algebra.