# Do we chose an operator ordering (explicit or implicit) all the time?

When I choose an ordering prescription for a function depending on operators (for example $\hat{H}$ being a functional of $\hat{\vec{E}}(x)$ and $\hat{\vec{B}}(x)$), I provide additional informations to make such a function unambigous: While for a classical function defined on numbers, giving the formula $g(x,y) = x*y = y*x$ is enough, for the operator function we additionally have to make a statement on the ordering of the operators.

First Question: Do we choose an operator ordering prescription all the time?

I will explain what I mean by that with an example:

The "usual" Hamiltonian of the free electromagnetic field, $\hat{H} = \epsilon_0 \int \hat{\vec{E}}^2 + c^2\hat{\vec{B}}^2$ will (in terms of creation and annihilation operators) lead to $\hat{H} = \sum_{\vec{k}, \lambda} \hbar \omega_{vec{k}} (a_{\vec{k}, \lambda}^\dagger a_{\vec{k}, \lambda} + \frac{1}{2})$, while the "normal ordered" Hamiltonian will lead to $\hat{H} = \sum_{\vec{k}, \lambda} \hbar \omega_{vec{k}} a_{\vec{k}, \lambda}^\dagger a_{\vec{k}, \lambda}$.

Is the "usual" way to express the hamiltonian just another ordering prescription, that hasn't a name given to it (yet)?

A follow up Question would be when which prescriptions are allowed, since for example for the elctromagnetic field inside a pair of mirrors once can't choose the normal ordering.

• – Qmechanic Oct 16 '17 at 22:11