As Lubos Motl mentions in a comment, for all practical purposes, the sought-for equation (v1) is proved via Wick's Theorem.

It is interesting to try to generalize Wick's Theorem and to try to minimize the number of assumptions that goes into it. Here we will outline one possible approach.

I) Assume that a family $(\hat{A}_i)_{i\in I}$ of operators $\hat{A}_i\in{\cal A}$ lives in a (super) operator algebra ${\cal A}$

with (super) commutator $[\cdot,\cdot]$, and

with center $Z({\cal A})$.

Here

the index $i\in I$ runs over an index set $I$ (it could be continuous), and

the index $i$ contains information, such as, e.g., position $x$, time instant $t$, annihilation/creation label, type of field, etc., of the operator $\hat{A}_i$.

II) Assume that
$$
\forall i,j\in I~: \qquad [\hat{A}_i,\hat{A}_j] ~\in~Z({\cal A}).
$$

III) Assume that there are given two ordering prescriptions, say $T$ and $::$. Here $T$ and $::$ could in principle denote any two ordering prescriptions, e.g. time order, normal order, radial order, etc. This means that the index set $I$ is endowed with two strict total orders, say, $<$ and $\prec$, respectively, such that

The $T$ symbol is (graded) multilinear wrt. supernumbers.

$ T(\hat{A}_{\pi(i_1)}\ldots\hat{A}_{\pi(i_n)})~=~\pm T(\hat{A}_{i_1}\ldots\hat{A}_{i_n} )$ is (graded) symmetric, where $\pi\in S_n$ is a permutation of $n$ elements.

$ T(\hat{A}_{i_1}\ldots\hat{A}_{i_n} )~=~\hat{A}_{i_1}\ldots\hat{A}_{i_n}$ if $i_1 > \ldots > i_n$. [A similar condition should hold for the second ordering $(::,\prec)$.]

In the special case where some of the $ i_1 , \ldots , i_n$ are equal${}^{\dagger}$ (wrt. the order <), then one should symmetrize in appropriate (graded) sense over the corresponding subsets. For instance,
$$ T(\hat{A}_{i_1}\ldots\hat{A}_{i_n} )~=~\hat{A}_{i_1}\ldots\hat{A}_{i_{k-1}}\frac{\hat{A}_{i_k}\hat{A}_{i_{k+1}}\pm\hat{A}_{i_{k+1}}\hat{A}_{i_k}}{2}\hat{A}_{i_{k+2}}\ldots\hat{A}_{i_n}$$ if $i_1 > \ldots > i_k=i_{k+1}> \ldots > i_n$. [A similar condition should hold for the second ordering $(::,\prec)$.]

IV) It then follows from assumptions I-III that the (generalized) contractions
$$
\hat{C}_{ij}~=~T(\hat{A}_i\hat{A}_j)~-~:\hat{A}_i\hat{A}_j:~\in~Z({\cal A})
$$
belong to the center $Z({\cal A})$. [By the way, physicists will often casually refer to the operators in the center $Z({\cal A})$ as *$c$-numbers.*]

V) It is now a straightforward exercise to establish a corresponding Wick's Theorem, meaning a rule for how to re-express one ordering prescription $T(\hat{A}_{i_1}\ldots\hat{A}_{i_n})$ in terms of the other ordering prescription $::$ and (multiple) contractions $\hat{C}_{ij}$. And vice-versa with the roles of the two orderings $T$ and $::$ interchanged. Such Wick's Theorems can now be applied successively to establish nested${}^{\ddagger}$ Wick's Theorems. These Wick's Theorems may be extended to a larger class of operators than just the $(\hat{A}_i)_{i\in I}$ family through (graded) multilinearity.

VI) Let us now assume that the operators $\hat{A}_i$ are Bosonic for simplicity.
A particular consequence of a nested Wick's Theorem is the following version of the sought-for equation

$$T(:\hat{A}^2_i::\hat{A}^2_j:) ~=~ 2\hat{C}_{ij}^2 + 4 \hat{C}_{ij}:\hat{A}_i\hat{A}_j: + :\hat{A}^2_i\hat{A}^2_j:~.$$

Finally, let us mention that Wick's Theorem, radial order, OPE, etc., are also discussed in this and this Phys.SE posts.

--

${}^{\dagger}$ Being *equal wrt. an order* is in general an equivalence relation, and it is often a weaker condition than being *equal* as elements of $I$.

${}^{\ddagger}$
A nested Wick's Theorem (between radial order and normal order) is briefly stated on p. 39 in J. Polchinski, *String Theory,* Vol. 1. Beware that radial order is often only implicitly written in CFT texts. I'll update the answer if I find a better reference.

This post imported from StackExchange Physics at 2015-02-02 13:25 (UTC), posted by SE-user Qmechanic