A. Zee, Quantum Field Theory in a Nutshell · Problem I.9.2

The key insight is that dimensional regularization in $d$ dimensions treats the vacuum energy density as a function of $d$, and the force between plates emerges from how this energy scales with plate separation. Since the Casimir effect is fundamentally a boundary-value problem with mirror symmetry in the separation coordinate, the energy—and hence force—must be an even function of the displacement. The $1/d$ expansion respects this symmetry, so only even powers appear in the perturbative series.

The key insight is that dimensional regularization in $d$ dimensions treats the vacuum energy density as a function of $d$, and the force between plates emerges from how this energy scales with plate separation. Since the Casimir effect is fundamentally a boundary-value problem with mirror symmetry in the separation coordinate, the energy—and hence force—must be an even function of the displacement. The $1/d$ expansion respects this symmetry, so only even powers appear in the perturbative series.