It is often argued that the entire universe is completely bathed in the zero-point electromagnetic field, and as such it can add only some constant amount to expectation values. Physical measurements will therefore reveal only deviations from the vacuum state. Thus the zero-point energy can be dropped from the Hamiltonian by redefining the zero of energy, or by arguing that it is a constant and therefore has no effect on Heisenberg equations of motion. Thus we can choose to declare by fiat that the ground state has zero energy and a field Hamiltonian, for example, can be replaced by: H F − − ⟨ 0 | H F | 0 ⟩ = 1 2 ℏ ℏ ω ω ( a a † † + a † † a ) − − 1 2 ℏ ℏ ω ω = ℏ ℏ ω ω ( a † † a + 1 2 ) − − 1 2 ℏ ℏ ω ω = ℏ ℏ ω ω a † † a without affecting any physical predictions of the theory. The new Hamiltonian is said to be normally ordered (or Wick ordered) and is denoted by a double-dot symbol. The normally ordered Hamiltonian is denoted : H F, i.e.: : H F :≡ ≡ ℏ ℏ ω ω ( a a † † + a † † a ) :≡ ≡ ℏ ℏ ω ω a † † a