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As I checked in my last question here, something like "normal ordering" always goes hand in hand with the definition of an operatorvalued function on the space of operators, like (for example) the hamilton operator, or the electromagnetic field strengh tensor. If I choose, for example for my hamiltonian, such an prescription, like "normal ordering", do I have to stick to it with all other operators that I define? Let my hamiltonian be $:\int E^2 + B^2: = \Sigma_k \omega_k a^\dagger_k a_k $

Does that I mean that the number operator $a^\dagger_k a_k$ and the "uncertainty-operator" $\Delta N^2 = :(a^\dagger_k a_k - \langle N \rangle)^2:$ do also have to be normal ordered?

Quantumwhisp
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No, you absolutely don't have to. In the quantum theory physical observables are self-adjoint operators, and different choices for these operators generally lead to different quantum theories. You have to provide an operator for any physical observable, and you have a choice of ordering in any of them.

Of course there's a requirement that in the classical limit the theory agrees with its classical counterpart, but generally this poses no restriction on operator ordering, since all relevant commutators are proportional to $\hbar$ and thus vanish in the classical limit.

However in most interaction picture QFTs there's only one valid choice of ordering which is normal ordering for reasons of consistency. Note also that these aren't strictly speaking quantum mechanical theories though.

Prof. Legolasov
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