The zero-point energy of the field arises formally from the non-commutativity of a and a. This is true for any harmonic oscillator: the zero-point energy ⁠ ħω / 2 ⁠ appears when we write the Hamiltonian: H c l = p 2 2 m + 1 2 m ω ω 2 q 2 = 1 2 ℏ ℏ ω ω ( a a † † + a † † a ) = ℏ ℏ ω ω ( a † † a + 1 2 )