The "definition" of the normal ordering in CFT looks a bit vague to me.

I found the definition in terms of exponentiated functional derivative pretty opaque.

Also in this context it might help if someone can give a reference or if there is a short explanation to understand how the Operator Product Expansion is derived using products of normal ordered operators.

I don't see the conceptual framework in which these ideas fit together. Some of the books I looked at gave a very disparate view as a collection of some complicated formulas.

Let me give a precise example of the kind of calculation that I am stuck with,

Refer to these lecture notes

I can understand equation 4.26 of this but not the next 4 equations that seem to follow from it leading to 4.28.

It would be helpful if someone can decrypt the calculation.

In light of the kinds of references that came in as responses, I think it would help if I make the problematic calculation a little more explicit.

This has to do with what are called "Vertex Operators" in CFT given as $:e^{ikX(z)}:$ where $::$ is the notation for normal ordering and $k$ is some scalar and $X$ is a conformally invariant free Bosonic field. Then I would like to understand the derivation of this equality,

(all expressions are understood to be valid under the Feynman Path Integral)

$:\partial X(z)\partial X(z)::e^{ikX(w)}: = -\frac{k^2\alpha ^2}{4}\frac{:e^{ikX(w)}:}{(z-w)^2}-ik\alpha\frac{:\partial X(z) e^{ikX(w)}:}{(z-w)}$

where we have $X(z)X(w) = -\frac{\alpha}{2}ln \vert z - w \vert$

and what would be the similar simplification of

$:e^{ikX(z)}::e^{ikX(w)}: = ?$

Some more elaboration on what about normal ordering I am concerned about.

The problem is that I can't these books give an honest definition of what it means to "normal order" operators in CFT. Like there is a very clean definition in rest of QFT whose relation to time-ordering is given by the Wick's Theorem. Here in CFT one is supposed to understand that while normal ordering a string of operators inserted at different points on the space-time one is subtracting away from the product every possible way in which one or more pairs of insertion points can coincide and produce a singularity

Like if A,B,C,D are 4 different Bosonic operators say inserted at 4 different space-time points. Then one would define normal ordering as,

$:ABCD: = ABCD - (AB):CD: - (AC):BD: - (AD):BC:-(BC):AD:-(BD):AC:$ $$-(CD):AB:-(AB)(CD)-(AC)(BD)-(AD)(BC)$$

where () denotes the correlation function of the operators.

Now the point is whether one is supposed to take the above kind of equations as being just well-motivated definition or is there is anything more fundamental from which it is derivable?

There is definitely an issue about defining difference of two divergent expressions here.

  • 3
    $\begingroup$ What are "some books" that you've tried? One axiomatic approach derives from associativity and commutativity of Frenkel, Lepowski, Meurman. A related alternative is in Kac's "Vertex algebras for beginners". $\endgroup$ – Victor Protsak Jun 3 '10 at 9:58
  • $\begingroup$ To emphasize and reiterate the comments by José below: one has to distinguish between (a) the Operator Product Expansion (OPE), which is one of the features of the mathematical formulation of CFT and (b) Wick's theorem, which explains how to rewrite certain products of free boson or fermion fields in the normal form. The $::$ notation used in (a) and (b) is the same, for historical reasons and because (a) is more general. But the definition $:AB:$ in (a) is direct. Provided OPE is local, i.e. commutativity and associativity that I mentioned, you can do calculations like the one you wanted. $\endgroup$ – Victor Protsak Jun 10 '10 at 1:01

The best reference I know for this sort of calculations is the PhD thesis of Kris Thielemans. Chapter 2 is all about the calculations of operator product expansions in two-dimensional conformal field theory. In particular in Section 2.6.1 you will find the kind of calculation you are interested in explained in detail. The formalism is clean and not very difficult at all.


Here's a sketch of the calculation of the operator product expansion between two vertex operators $V_k(z)$ and $V_\ell(w)$ where $$ V_k(z) = :e^{i k X(z)}:,$$ where $$ X(z) = x - i p \log z + i \sum_{n\neq 0} \tfrac1n a_n z^{-n},$$ with canonical commutation relations (CCRs) $[p,x]=1$ and $[a_n,a_m] = n \delta_{n+m,0}$.

The normal ordering prescription consists of writing the creation operators to the left of the annihilation operators: $$ V_k(z) = e^{\sum_{n>0} \frac{k}{n} a_{-n} z^n} e^{ikx} z^{kp} e^{-\sum_{n>0} \frac{k}{n} a_n z^{-n}}.$$

To compute the operator product expansion $V_k(z) V_\ell(w)$ one simply has to take the formal product of the above expressions for the vertex operators and commute the operators through using the CCRs in order to bring the creation operators to the left. The basic calculations are simple Weyl identities of the form $$ e^{-\frac{k}{n}a_n z^{-n}} e^{\frac{\ell}{n} a_{-n} w^n} = e^{\frac{\ell}{n} a_{-n} w^n} e^{-\frac{k}{n}a_n z^{-n}} e^{\frac{k\ell}{n} (\frac{w}{z})^n} $$ and a similar one for $x$ and $p$.

The resulting power series converges provided that the fields are radially ordered so that $|z|>|w|$, but the result can be analytically continued with either a pole at $z=w$ or a branch cut singularity depending on the values of $k$ and $\ell$.

The final step is to expand the $z$-dependent fields around $z=w$.

  • $\begingroup$ Thanks for this reference. I had a look through it but it doesn't seem to address the specific calculation I am looking for. I have now more explicitly written out the equation whose derivation I am trying to understand. $\endgroup$ – Anirbit Jun 4 '10 at 13:59
  • $\begingroup$ I've added a sketch of the calculation. $\endgroup$ – José Figueroa-O'Farrill Jun 5 '10 at 1:52
  • $\begingroup$ Thanks for this detailed reply. But this language of doing CFT is starkly different from the one seen in the books by Polchinski or the book by Francesco et. al. A clear definition of how products of normal ordered operators is defined is not easily found from these books. They mention that it is "analogous" to Wick's Theorem as in QFT but the analogy is far from obvious. Can you give a reference which explains this technology better? $\endgroup$ – Anirbit Jun 8 '10 at 5:54
  • $\begingroup$ Starkly different?! I don't think so. It's at most a notational difference. The above formulae are strictly speaking true inside radially ordered correlation functions, but some people (e.g., me) never write them explicitly. In addition to Thieleman's PhD thesis (linked above), I have found Paul Ginsparg's "Applied Conformal field theory" and the "Lecture notes in string theory" by Dieter Lüst and Stefan Theisen to be good references. $\endgroup$ – José Figueroa-O'Farrill Jun 8 '10 at 10:33
  • 3
    $\begingroup$ The normal-ordered product between two fields $A,B$ say, is unambiguously defined as follows. You compute the operator product expansion $A(z)B(w)$. This is a Laurent series in $z-w$ whose coefficients are fields at $w$. You discard the polar piece and take the limit $z\to w$. The resulting field is by definition the normal ordered product $:AB:(w)$. The normal order product of any number of fields can be done by iterating the above. It's best not to think in term of Wick contractions, with which it agrees only for free fields. $\endgroup$ – José Figueroa-O'Farrill Jun 9 '10 at 23:03

The field $L=:\partial X\partial X:$ is Virasoro and your first formula says that a vertex operator is a primary field (highest weight vector) with respect to the Virasoro action, with appropriate weight. This is a basic computation with vertex operators done in nearly any textbook on conformal field theory (Kaku, Di Francesco, etc). You may also want to look at one of the first mathematical treatments of vertex operators, very clearly written:

Frenkel, I. B.; Kac, V. G. Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62 (1980/81), no. 1, 23--66

By the way, the proper way to think about OPE for bosonic fields is

$$\partial X(z) \partial X(w) = \frac{1}{(z-w)^2}.$$

The "raw" $X(z)$ occurs only as a part of the exponential expression defining the vertex operator, never by itself.


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