Why is normal ordering even a valid operation in the first place? I mean it can give us some nice results, but why can we do the ordering for the operators like that?
Is its definition motivated by the relation between the normal ordering and the time-ordered product, which is basically the content of Wick's theorem?
Best Answer
In classical physics, quantities are ordinary, commuting $c$-numbers. The order in which we write terms in expressions is of no consequence. In quantum field theory (QFT), on the other hand, quantities are described by operators that, in general, don't commute.
Classical physics is a low-energy approximation of quantum physics - the road from quantum to classical physics ought to be unambiguous - and this is way the way nature goes, from high to low energies. The inverse - the road from classical to quantum, that we take to try and reconstruct the high-energy physics - however, is ambiguous, because of ordering ambiguities in non-commuting quantities.
When we normal order expressions after canonical quantization, we are correcting those ambiguities.
This occurs for the zero-point energy in the Hamiltonian $$ H = \int \frac{d^3p}{(2\pi)^3} E_p \left(a_p^\dagger a_p + \frac12[a_p^\dagger,a_p]\right) $$ You might hear it argued that the since the vacuum energy is unobservable, we are free to throw away the divergent piece (the commutator). Such an argument doesn't work for the charge operator, $$ Q = \int \frac{d^3p}{(2\pi)^3} E_p \left(a_p^\dagger a_p - b_p b_p^\dagger\right) $$ A charged vacuum would have observable effects. The best argument for normal ordering is that it is a rule for removing ordering ambiguities that results in e.g. a neutral vacuum.
Ordering ambiguties also occur in general relativity, when one promotes commuting ordinary derivatives $\partial_\mu$ to non-commuting covariant derivatives $\nabla_\mu$.