As an Amazon Associate, I earn from qualifying purchases. For more practice problems like this, see Schaum’s Outline of College Physics, 12th Edition.

1. What the question is asking (in plain language)

A student has seen two different ways of writing the product of two fields

  • In ordinary (canonical) QFT the product is split into a contraction (the two‑point function) plus a normal‑ordered piece

    [ X(z)Y(w)=\langle X(z)Y(w)\rangle+\;:!X(z)Y(w)!: \tag{1} ]

  • In 2‑dimensional conformal field theory (CFT) the same product is written as a sum of a singular part (the terms that diverge when (z\to w)) and a regular part (the Taylor‑expanded non‑singular terms)

    [ X(z)Y(w)=\underbrace{\sum_{n=1}^{N}\frac{{XY}n(w)}{(z-w)^n}}{\text{contraction}} \;+\; \underbrace{\sum_{n=0}^{\infty}\frac{{XY}{-n}(w)}{(z-w)^{-n}}}{\text{regular (``normal‑ordered’’) terms}} . \tag{2} ]

The student wonders:

  • How does Wick’s theorem look when we use the CFT definitions of contraction (the singular part of the OPE) and normal ordering (the regular part)?
  • Can we simply replace “normal‑ordered product’’ in the textbook formula by “regular part’’ and keep the same combinatorial structure?
  • If so, how do we actually obtain the non‑singular terms that appear in an OPE?

Below we give a step‑by‑step answer.

2. Detailed solution

2.1 Review: contraction and normal ordering in a free (Gaussian) CFT

In a free (or Gaussian) 2‑d CFT the fields satisfy Wick’s theorem exactly as in ordinary QFT.
The basic objects are

  • Contraction (or propagator)

    [ \contraction{}{X}{(z)}{Y} X(z)Y(w)\equiv\langle X(z)Y(w)\rangle . ]

    By definition this is the singular part of the OPE; it contains every pole (or logarithm) that diverges when (z\to w).

  • Normal ordering (radial‑ordered product)

    [ :!X(z)Y(w)!:\;\equiv\;X(z)Y(w)-\langle X(z)Y(w)\rangle . \tag{3} ]

    In a CFT the notation (\operatorname{Reg}{X(z)Y(w)}) used in di Francesco is simply another way of writing the same object: [ \operatorname{Reg}{X(z)Y(w)}\equiv :!X(z)Y(w)!: ]

Thus the two formulae (1) and (2) are identical; they just use different terminology.

Key point: In a free CFT the regular part of an OPE is the normal‑ordered product of the two fields.

2.2 Wick’s theorem in the CFT language

Consider a collection of (free) fields (\phi_i(z_i)) with (i=1,\dots,N).
The standard Wick theorem reads

[ \phi_1(z_1)\phi_2(z_2)\cdots\phi_N(z_N) =\sum_{\text{all pairings }P} \Bigl(\prod_{(a,b)\in P}\langle\phi_a(z_a)\phi_b(z_b)\rangle\Bigr)\, :!!\prod_{k\notin P}\phi_k(z_k)!!:\; . \tag{4} ]

All possible ways of pairing the fields are summed over.
Every pair contributes a contraction (the singular part of the OPE), and the remaining un‑contracted fields stay inside a normal‑ordered product.

If we now adopt the CFT notation of di Francesco we simply replace

[ :!!\prod_{k\notin P}\phi_k(z_k)!!:\;\;\longrightarrow\;\; \operatorname{Reg}\Bigl{\prod_{k\notin P}\phi_k(z_k)\Bigr}. ]

Hence the theorem becomes

[ \boxed{ \phi_1(z_1)\cdots\phi_N(z_N)= \sum_{P} \Bigl(\prod_{(a,b)\in P}\underbrace{\contraction{}{\phi_a}{(z_a)}{\phi_b} \phi_a(z_a)\phi_b(z_b)}{\text{singular part}}\Bigr)\; \operatorname{Reg}\Bigl{\prod{k\notin P}\phi_k(z_k)\Bigr} } \tag{5} ]

No extra terms appear; the combinatorial structure is unchanged.

2.3 Extracting the regular (non‑singular) part of an OPE

Suppose we are interested in the OPE of two composite operators, e.g.

[ A(z)=:!\phi_1(z)\phi_2(z)!:\,,\qquad
B(w)=:!\phi_3(w)\phi_4(w)!:\, . ]

We want the expansion of (A(z)B(w)) around (z=w).
Step‑by‑step:

  1. Write each composite operator as a product of elementary fields.
    [ A(z)B(w)=\phi_1(z)\phi_2(z)\,\phi_3(w)\phi_4(w) . ]

  2. Apply Wick’s theorem (5).
    Every contraction is the singular part of the corresponding two‑point function.
    For a free boson (X) we have

    [ \contraction{}{X}{(z)}{X}X(z)X(w)=\langle X(z)X(w)\rangle =-\alpha’\,\ln(z-w)\; . ]

    For fermions or other primaries the singular part is a simple pole or higher‑order pole.

  3. Collect the terms with a given number of contractions.
    Zero contractions give the regular part of the product:

    [ \operatorname{Reg}{\phi_1(z)\phi_2(z)\phi_3(w)\phi_4(w)} =:!\phi_1(z)\phi_2(z)\phi_3(w)\phi_4(w)!: \tag{6} ]

    One contraction gives a term proportional to the two‑point function times a normal‑ordered product of the remaining three fields, etc.

  4. Expand the remaining normal‑ordered fields around (w).
    Because the normal‑ordered product is regular at (z=w), we may Taylor expand each field:

    [ :!\phi_i(z)!:=\sum_{n=0}^{\infty}\frac{(z-w)^n}{n!}\,\partial^n\phi_i(w) . \tag{7} ]

    Substituting (7) into the regular piece (6) yields a series of the form

    [ \operatorname{Reg}{A(z)B(w)} =\sum_{n=0}^{\infty}\frac{(z-w)^n}{n!}\,C_n(w) , \tag{8} ]

    where each coefficient (C_n(w)) is a composite primary built from normal‑ordered products of the elementary fields and their derivatives.
    These (C_n(w)) are precisely the non‑singular terms that appear in the OPE.

  5. Combine singular and regular pieces.

    The full OPE reads

    [ A(z)B(w)=\underbrace{\sum_{\text{singular}} \frac{c_{k}(w)}{(z-w)^{k}}}{\displaystyle\text{contractions}} \;+\; \underbrace{\sum{n=0}^{\infty}\frac{(z-w)^n}{n!}\,C_n(w)}_{\displaystyle\text{regular (normal‑ordered) part}} . \tag{9} ]

    The coefficients (c_k(w)) are obtained from the various ways of contracting fields; the coefficients (C_n(w)) are obtained from the Taylor expansion of the regular product.

2.4 Example: free boson (\partial X(z)\,\partial X(w))

The elementary field (X(z)) has the OPE

[ X(z)X(w)=-\alpha’\,\ln(z-w)+:X(z)X(w): . ]

Differentiating gives

[ \partial X(z)\,\partial X(w)=\frac{-\alpha’}{(z-w)^2}+:!\partial X(z)\partial X(w)!: \tag{10} ]

Now expand the regular part:

[ :!\partial X(z)\partial X(w)!: =:!\partial X(w)\partial X(w)!:+(z-w):!\partial^2 X(w)\partial X(w)!:+\cdots . \tag{11} ]

Thus the OPE is

[ \boxed{ \partial X(z)\,\partial X(w)=\frac{-\alpha’}{(z-w)^2} +\sum_{n=0}^{\infty}\frac{(z-w)^n}{n!}\;:!\partial^{n+1}X(w)\,\partial X(w)!: } \tag{12} ]

The first term is the contraction (singular part).
All the terms in the infinite sum are the regular (normal‑ordered) contributions, obtained exactly by the Taylor expansion described above.

2.5 Summary of the answer to the original question

  • Yes, Wick’s theorem works in a CFT exactly as in ordinary QFT provided the theory is free (Gaussian).
  • The contraction is the singular part of the OPE, i.e. the two‑point function.
  • The regular part of the OPE is by definition the normal‑ordered product of the fields that remain after the contractions have been removed.

  • Therefore the formula

    [ \text{product} = \text{sum over all contractions}\times\bigl(\text{regular part}\bigr) ]

    is completely legitimate; you may replace “normal‑ordered’’ by “regular’’ without changing anything.

  • The non‑singular terms in an OPE are obtained by (i) performing all possible contractions, (ii) writing the remaining fields inside a normal‑ordered product, and (iii) expanding that normal‑ordered product in a Taylor series around the point where the OPE is taken.

3. Final answer (concise statement)

In a free (Gaussian)

Original question: Wick theorem and OPE on Physics Stack Exchange, licensed CC BY-SA.