Why is normal ordering a valid operation?

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?

• 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? – M. Zeng Sep 1 '14 at 7:49
• I think to remember that there is a conection between ordering and the discretisation (combination of forward and backward derivative) you choose in Feynmann path integral. – arivero Sep 1 '14 at 9:34
• What do you mean by "valid operation"? Given a string of symbols, it is surely a valid operation to reorder the symbols. I don't understand what you're trying to ask. – ACuriousMind Jan 14 '16 at 16:19
• because it first appeared to me that people just reorder the terms, which will give a different result from the original one if we use the commutation relations – M. Zeng Jan 14 '16 at 17:11
• Normal ordering does indeed change the operator - it's not just a canonical way of rewriting it. $:a a^\dagger:\ =\ a^\dagger a \neq a a^\dagger = a^\dagger a + 1$. – tparker Jul 15 '17 at 7:23

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 $\nabla_\mu$.

• well, if normal ordering is solely for removing ambiguities, then any other ordering can do the same job because we just need to choose one from the many possibilities to set as the rule. what I intended to ask is how to verify the consistency of normal ordering with the big picture. – M. Zeng Sep 1 '14 at 8:39
• A good comment! You are right - the ambiguity is resolved by required e.g. the vacuum charge to vanish. – innisfree Sep 1 '14 at 8:46
• When I calculate the charge operator, I immediately get a normally ordered answer. Where did you get that expression for the charge operator? – Sigh_at_psi May 29 '19 at 18:44
• probably A Modern Introduction to Quantum Field Theory, Maggiore, but it's some time ago – innisfree May 30 '19 at 7:20
• we have normal ordering for a and $a^\dagger$ which do not commute. Positions and momenta do not commute. why is there no simple normal ordering between them? – Naima Mar 11 '20 at 8:35

The choice of normal ordering prescription $:~:$ is usually adjusted to the choices of bra vacuum state $\langle \Omega|$ and ket vacuum state $|\Omega\rangle$, so that

$$\langle \Omega|:\hat{\cal O}_1\ldots \hat{\cal O}_{n>0} : |\Omega\rangle~~=~~0 , \qquad\qquad \langle \Omega|\Omega\rangle~~=~~1 .$$

The relation of normal ordering prescription to Wick theorem and other operator ordering prescriptions are typically only secondary. For motivation, see also e.g. this Phys.SE post.

normal ordering is a valid operation provided one can undo it by an appropriate choice of counterterms (of existing couplings or field renormalisations). (How this is done in practice is explained here: http://arxiv.org/abs/1512.02604.)

• It would be great if this could be expanded a bit – innisfree Aug 30 '18 at 12:49

As other answers mention, it was originally (in QED) about getting a neutral vacuum. It is useful to go back to Schwinger's old version of QED, before Dyson's approach became accepted. See Pauli: Selected topics in field quantization.

Pauli presents both ways of looking at it: 1) define the electric current as sum of two terms (p.20 [6.4]), such that the vacuum expectation number operators give zero [6.7-6.9]; and later see that 2) normal ordering, i.e. dropping the infinite vacuum constant, has the same effect when taking the ordinary definition of the current, as far as the S-matrix is concerned (see p.141: "Not grouping together terms with the same argument is a substitute for the neglected subtraction of the vacuum current...").

This is how the 'negative energy sea' was eliminated. There is nothing incorrect about it, other than that the true content of QFT, i.e. renormalization, reaffirms the S-matrix point of view, 2).

The first approach tried to work with operators without reference to the S-matrix, as long as possible, whereas the second concedes from the start that only S-matrix results will be useful.

(Caution: Pauli's book is interesting but not easy to read.)