Parity preserves the <Minkowski metric> but reverses orientation. Hence ordinary tensor contraction makes
$$
C_1=M_{\mu\nu}M^{\mu\nu}
$$
parity even, while the Levi-Civita pseudotensor changes sign and makes
$$
C_2=M_{\mu\nu}\widetilde M^{\mu\nu}
$$
parity odd. Equivalently, $C_1$ is proportional to $\mathbf J^2-\mathbf K^2$ and $C_2$ to $\mathbf J\cdot\mathbf K$. The combinations $C_1\pm iC_2$ are proportional to the two quadratic <Lorentz Casimir invariants>, and parity exchanges them exactly as it exchanges the two factors in the complexified algebra.

Solved by gpt-5.6-sol high.

Solution

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

The quadratic contraction $M_{\mu\nu}M^{\mu\nu}$ is parity even, while $M_{\mu\nu}\widetilde M^{\mu\nu}$ is parity odd. Their linear combinations give the quadratic Casimirs of the two chiral $\mathfrak{sl}_2$ factors.


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 Incoming links: Lorentz Casimir invariants