Proof that Casimir force does not originate from vacuum energy

Understand

We present a simple general proof that Casimir force cannot originate from the vacuum energy of electromagnetic (EM) field.

• The full QED Hamiltonian consists of 3 terms: the pure electromagnetic term $H_{\rm em}$, the pure matter term $H_{\rm matt}$ and the interaction term $H_{\rm int}$.
• The $H_{\rm em}$-term commutes with all matter fields because it does not have any explicit dependence on matter fields.
• As a consequence, $H_{\rm em}$ cannot generate any forces on matter.