Editing Zero-point energy (section)

==== Redefining the zero of energy ====
In the quantum theory of the electromagnetic field, classical wave amplitudes {{mvar|α}} and {{math|''α''*}} are replaced by operators {{mvar|a}} and {{math|''a''<sup>†</sup>}} that satisfy:
<math display="block">\left[a,a^\dagger\right] = 1</math>

The classical quantity {{math|{{abs|''α''}}<sup>2</sup>}} appearing in the classical expression for the energy of a field mode is replaced in quantum theory by the photon number operator {{math|''a''<sup>†</sup>''a''}}. The fact that:
<math display="block">\left[a,a^\dagger a\right] \ne 1</math>
implies that quantum theory does not allow states of the radiation field for which the photon number and a field amplitude can be precisely defined, i.e., we cannot have simultaneous eigenstates for {{math|''a''<sup>†</sup>''a''}} and {{mvar|a}}. The reconciliation of wave and particle attributes of the field is accomplished via the association of a probability amplitude with a classical mode pattern. The calculation of field modes is entirely classical problem, while the quantum properties of the field are carried by the mode "amplitudes" {{math|''a''<sup>†</sup>}} and {{mvar|a}} associated with these classical modes.

The zero-point energy of the field arises formally from the non-commutativity of {{mvar|a}} and {{math|''a''<sup>†</sup>}}. This is true for any harmonic oscillator: the zero-point energy {{math|{{sfrac|''ħω''|2}}}} appears when we write the Hamiltonian:
<math display="block">\begin{align}
H_{cl} &= \frac{p^2}{2m} + \tfrac{1}{2} m \omega^2 {q}^2 \\
&= \tfrac{1}{2} \hbar \omega \left(a a^\dagger + a^\dagger a\right) \\
&=\hbar \omega \left(a^\dagger a +\tfrac{1}{2}\right)
\end{align}</math>

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:{{sfnp|Itzykson|Zuber|1980|p=111}}
<math display="block">\begin{align}
H_F - \left\langle 0|H_F|0\right\rangle &=\tfrac{1}{2} \hbar \omega \left(a a^\dagger + a^\dagger a\right)-\tfrac{1}{2}\hbar \omega \\
&= \hbar \omega \left(a^\dagger a + \tfrac{1}{2} \right)-\tfrac{1}{2}\hbar \omega \\
&= \hbar \omega a^\dagger a
\end{align}</math>
without affecting any physical predictions of the theory. The new Hamiltonian is said to be [[Normal order|normally ordered]] (or Wick ordered) and is denoted by a double-dot symbol. The normally ordered Hamiltonian is denoted {{math|:''H<sub>F</sub>''}}, i.e.:
<math display="block">:H_F : \equiv \hbar \omega \left(a a^\dagger + a^\dagger a\right) : \equiv \hbar \omega a^\dagger a</math>

In other words, within the normal ordering symbol we can commute {{mvar|a}} and {{math|''a''<sup>†</sup>}}. Since zero-point energy is intimately connected to the non-commutativity of {{mvar|a}} and {{math|''a''<sup>†</sup>}}, the normal ordering procedure eliminates any contribution from the zero-point field. This is especially reasonable in the case of the field Hamiltonian, since the zero-point term merely adds a constant energy which can be eliminated by a simple redefinition for the zero of energy. Moreover, this constant energy in the Hamiltonian obviously commutes with {{mvar|a}} and {{math|''a''<sup>†</sup>}} and so cannot have any effect on the quantum dynamics described by the Heisenberg equations of motion.

However, things are not quite that simple. The zero-point energy cannot be eliminated by dropping its energy from the Hamiltonian: When we do this and solve the Heisenberg equation for a field operator, we must include the vacuum field, which is the homogeneous part of the solution for the field operator. In fact we can show that the vacuum field is essential for the preservation of the commutators and the formal consistency of QED. When we calculate the field energy we obtain not only a contribution from particles and forces that may be present but also a contribution from the vacuum field itself i.e. the zero-point field energy. In other words, the zero-point energy reappears even though we may have deleted it from the Hamiltonian.{{sfnp|Milonni|1994|pp=73–74}}