Past exam of the quantum field theory course of the University of Oxford / 2023 / 3 / b

A parity symmetry in quantum field theory reverses spatial orientation, so its spatial matrix must lie in the determinant-$-1$ component of the orthogonal group. In two spatial dimensions, reversing both coordinates gives $-I_2$, which has determinant $+1$ and is a rotation through $\pi$ in the special orthogonal group in two dimensions. It cannot test a discrete orientation-reversing symmetry independent of ordinary rotations. Reflecting one coordinate, for example $\mathrm{diag}(-1,1)$, has determinant $-1$ and supplies the required parity symmetry in quantum field theory.

In three spatial dimensions, $-I_3$ has determinant $-1$, so simultaneous inversion of all three coordinates is the standard choice. A single-coordinate reflection also reverses orientation there, and differs from $-I_3$ by a proper rotation through $\pi$. Thus, in a rotationally invariant theory, it tests the same discrete symmetry. The word is in the source specifies conventional representatives: the essential distinction is orientation reversal, rather than a unique choice of axis. Both transformations preserve the ordinary spatial integration measure through the absolute value of their Jacobian determinants.