Stringy Things

Notes on String Theory – Further Introduction to Operator Product Expansions

1. Generalising the Formula for OPEs

In the last post we continued a review of Chapter 2 in Polchinski, focusing on building understanding of conformal field theories from the perspective of local operator insertions. We finally also arrived at the basic formula for operator product expansions (OPEs). What follows in this post is a continuation of that discussion. That is to say, the following review will also necessarily reference equations in the previous entry. To avoid confusion, equation numbers from the last post will be explicitly stated.

Recall that, in an introduction to the basic formula for OPEs, it was mentioned that because it is an operator statement this means it holds inside a general expectation value. It follows that the operator equation of the form that we considered can have additional operator insertions. This implies that we may write the formula for OPEs in a more general way,

\displaystyle  \langle \mathcal{O}_{i}(z, \bar{z})\mathcal{O}_{j}(z^{\prime}, \bar{z}^{\prime}) ... \rangle = \sum_{k} C_{ij}^{k}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime}) \langle \mathcal{O}_{k}(z^{\prime}, \bar{z}^{\prime}) ... \rangle \ \ (1)

Where ‘…’ again denotes additional insertions and is often left implicit. One can also work out quite simply the equivalent description in the path integral formalism for {n-1} fields.

1.1. OPEs – Generalise for an Infinite Set of Operators

There are a number of other caveats and subtleties about OPEs that we have not yet explored. It will be our aim to do so in this section by reviewing the remaining contents of section 2.2 in Polchinski’s textbook, before progressing toward more advanced topics that will then aid in our understanding of stringy CFTs and the procedure for how to compute OPEs.

Moreover, at this point in Polchinski’s introduction to OPEs, a number of results and definitions are given which may not make complete sense until later. This is because there are a number of key interrelated concepts that have not yet been formally introduced, such as radial ordering, Wick’s theorem, conformal invariance, and the necessary mode expansions that we must consider. These are important conceptual tools in establishing a wider understanding of CFTs and how we may think of OPEs in string theory. So what follows in this section may be considered more in the way of definition, introducing some ideas that relate to OPEs as we work toward more advanced topics that will clarify and enrich some of these ideas.

For instance, let us recall that in the last entry we discussed a normal ordered product that was defined in such a way that it satisfies the naive equation of motion [equation (17) from previous post]. What it is telling us is how the operator product is a harmonic function of {(z_{1}, \bar{z}_{1})}. This statement already offers a hint of what is to come both in this section and other future parts of our study on CFTs, particularly when we more explicitly discuss Wick’s theorem and mode expansions in relation to computing OPEs. For now, we may maintain an introductory tone and say that this statement leads us to an important insight early in Polchinski’s discussion in Section 2.2 of his textbook: notably that from the theory of complex variables a harmonic function may be decomposed locally as the sum of holomorphic and antiholomorphic functions. To begin to explain what this means, and to explain Polchinski’s discussion on pp.37-38 let us consider more deeply (17) from the last post. We can think of it this way,

\displaystyle  \bar{\partial}_{1} [\partial_{1} :X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):] = 0

\displaystyle \bar{\partial}_{1} [:\partial_{1} X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):] = 0 \ \ (2)

The point of (2) is to show that we now have a holomorphic derivative inside the normal ordering. But notice also that this holomorphic derivative will get annihilated by the antiholomorphic derivative acting on it. In other words, by the equation of motion mixed {\partial \bar{\partial}} derivatives vanish. This is telling us something we may perhaps already know or suspect, namely as we continue to think in terms of operators {:\partial_{1} X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):} is in fact a holomorphic function. Now, as Polchinski explains, from the theory of complex analysis it is within the rules that we can Taylor expand such holomorphic (and antiholomorphic) functions. This use of Taylor expansion may be considered one of the first tools in understanding how to compute OPEs. Consider, for example, only the holormorphic case. When we proceed with Taylor expansion in {z_{12}} it is implied that we have nonsingularity as {z_{1} \rightarrow z_{2}} and we obtain the following infinite series,

\displaystyle  :\partial_{1 \xi} X^{\mu}(z_{1} + \xi, \bar{z}_{1} + \xi)X^{\nu}(z_{2}, \bar{z}_{2}): = \sum_{k=1}^{\infty} \frac{\xi^{k}}{k!} :X^{\nu} \partial^{k}X^{\mu}: \ \ (3)

Where {\xi = z_{12}}. We can rewrite (3) as follows, including also the antiholomorphic series,

\displaystyle  = \sum_{k=1}^{\infty} [\frac{z_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}(z_{2}, \bar{z}_{2}): + \frac{\bar{z}_{12}^{k}}{k!} :X^{\nu}\bar{\partial}^{k}X^{\mu}(z_{2}, \bar{z}_{2}):] \ \ (4)

Which is now written only as a function of {z_{2}}. What this is telling us is that if we have some normal ordered product, we may write more generally for this product,

\displaystyle  :X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):

\displaystyle = :X^{\mu}(z_{2}, \bar{z}_{2})X^{\nu}(z_{2}, \bar{z}_{2}): + \sum_{k=1}^{\infty} [\frac{z_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}: + \frac{\bar{z}_{12}^{k}}{k!} :X^{\nu}\bar{\partial}^{k}X^{\mu}:] \ \ (5)

This is exactly the result that Polchinski describes in equation (2.2.4), with the exception that we have simplified the equation by dropping the {\alpha^{\prime}} term. Keeping the {\alpha^{\prime}} term explicit we arrive precisely at Polchinski’s equation,

\displaystyle  :X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):

\displaystyle = - \frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \ln \mid z_{12} \mid^{2} + :X^{\mu}(z_{2}, \bar{z}_{2})X^{\nu}(z_{2}, \bar{z}_{2}) + \sum_{k=1}^{\infty} [\frac{z_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}: + \frac{\bar{z}_{12}^{k}}{k!} :X^{\nu}\partial^{k}X^{\mu}:] \ \ (6)

In which {- \frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \ln \mid z_{12} \mid^{2}} is the regular part of the OPE that one may remember from the two-point function {\langle X^{\mu}(z, \bar{z})X^{\nu}(z^{\prime}, \bar{z}^{\prime}) \rangle}. Again, this is something we will become more familiar with as we progress. Furthermore, notice in general that (6) looks very much like an OPE as given in (1). In fact, it will become increasingly clear, especially toward the end of our present study, that we may think of this as the free field OPE hence the inclusion of the regular piece. Later, we will show explicitly the computation to achieve this result. In the meantime, since it is simply given in Polchinski’s textbook, it has also been stated here with addition of a few more comments as follows.

Note that like its equation of motion, (6) is an operator statement. Secondly, as previously alluded, OPEs in quantum field theory are very much like the analogue of Taylor expansions in calculus. When Taylor expanding some general function {\mathcal{G}(z_{1}, \bar{z}_{1}; z_{2}, \bar{z}_{2}) = :X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):} as above, note that one will obtain terms of the form {\partial^{k}:X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2}):} in which the derivative is outside the normal ordering as opposed to inside the normal ordering. But differentiation and normal ordering commute, which can be proven using some basic identities of functional derivatives, hence the structure of the normal ordering in the OPE (6). Also, for any arbitrary expectation value that involves some product {X^{\mu}(z_{1}, \bar{z}_{1})X^{\nu}(z_{2}, \bar{z}_{2})} multiplied by a number of fields at other points, we have been building (and will continue to build) the intuition to understand exactly why the OPE describes the behaviour for when {z_{1} \rightarrow z_{2}} as an infinite series. In the case of (6), as we deepen our study of CFTs we will come to understand more clearly why it has `a radius of convergence in any given expectation value which is equal to the distance to the nearest other insertion in the path integral’ and why `The operator product is harmonic except at the positions of operators’ (p.38).

Although how we arrive at (6) may not yet make complete sense, the key idea at this point in Polchinski’s discussion is simply that we have a product of two operators and we have described this product as an infinite sum of some coefficients {C_{k}} of some basis operators {A_{k}}. As asymptotic expansions, we will come to write OPEs up to nonsingular terms.

1.2. Subtractions and Cross-contractions

To conclude a review of Section 2.2 in Polchinski, let us consider another example where we have an arbitrary number of fields. As we discussed earlier, the sum then runs over all of the different ways we might choose pairs of fields from the product. We then replace each pair with the expectation value as mentioned in the description of the definition (16) in the last post – i.e., what we have also termed to be the regular part of the OPE. So, if for instance we have three fields, the computation generally takes the following form,

\displaystyle  :X^{\mu_{1}}(z_{1}, \bar{z}_{1})X^{\mu_{2}}(z_{2}, \bar{z}_{2})X^{\mu_{3}}(z_{3}, \bar{z}_{3}):

\displaystyle =X^{\mu_{1}}(z_{1}, \bar{z}_{1})X^{\mu_{2}}(z_{2}, \bar{z}_{2})X^{\mu_{3}}(z_{3}, \bar{z}_{3}) + (\frac{\alpha^{\prime}}{2} \eta^{\mu_{1} \mu_{2}} \ln \mid z_{12} \mid^{2} X^{\mu_{3}}(z_{3}, \bar{z}_{3}) + 2 \ \text{permutations}) \ \ (7)

Now, consider again (16) from the previous entry. It can now be seen how we may write this definition in a more compact and general way. Consider, for instance, the arbitrary functional {\mathcal{F} = \mathcal{F}[\partial X^{\mu_{1} ... \mu_{n}}]}. The terms in brackets represent a combination of an arbitrary number of fields. If, as before, we Taylor expand and make this expression an expansion of polynomials of {X}, it follows that we may then write the normal ordering for each monomial. This leads directly to the equation (2.2.7) in Polchinski,

\displaystyle :\mathcal{F}: = \exp (\frac{\alpha^{\prime}}{4} \int d^{2}z_{1}d^{2}z_{2} \ln \mid z_{12} \mid^{2} \frac{\delta}{\delta X^{\mu}(z_{1}, \bar{z}_{1})} \frac{\delta}{\delta X_{\mu}(z_{2}, \bar{z}_{2})}) \mathcal{F} \ \ (8)

Where {\mathcal{F}} is any functional of {X}. It can be shown that (8) is equivalent to (16) from the previous post. Again, this may not yet make complete sense. But for now notice that there is a double derivative in the exponent. This double derivative contracts each pair of fields. What this means is that, every time we compute the expansion we will effectively kill two {X} terms. Instead of these {X} terms, we then insert {\ln \mid z_{12} \mid^{2}} which is, of course, the subtraction. Now, reversely, if we act with the inverse exponential, we obtain the opposite of a sum of subtractions in the form of a sum of contractions,

\displaystyle  \mathcal{F} = \exp (-\frac{\alpha^{\prime}}{4} \int d^{2}z_{1}d^{2}z_{2} \ln \mid z_{12} \mid^{2} \frac{\delta}{\delta X^{\mu}()z_{1}, \bar{z}_{1}} \frac{\delta}{\delta X_{\mu}(z_{2}, \bar{z}_{2})}) :\mathcal{F}:

\displaystyle = :\mathcal{F}: + \ \text{contractions} \ \ (9)

As it will become increasingly clear when we compute some detailed examples, this means we are now summing over all of the ways of choosing pairs of fields from {:\mathcal{F}:} instead of {\mathcal{F}}. We then replace each pair with the contraction {-\frac{1}{2} \alpha^{\prime}\eta^{\mu_{i} \mu_{j}} \ln \mid z_{ij} \mid^{2}}. It follows that for any pair of operators, we can generate the respective OPE

\displaystyle :\mathcal{F}: :\mathcal{G}: = :\mathcal{F} \mathcal{G}: + \sum \ \text{cross-contractions} \ \ (10)

What (10) is saying is that we are now summing over all of contracting pairs with one field in {\mathcal{F}} and one field in {\mathcal{G}}, where, again, {\mathcal{F}} and {\mathcal{G}} are arbitrary functionals of {X}. It is this construction of the cross-contractions that enables the following formal expression,

\displaystyle : \mathcal{F}: :\mathcal{G}: = \exp (-\frac{\alpha^{\prime}}{2} \int d^{2}z_{1}d^{2}z_{2} \ln \mid z_{12} \mid^{2} \frac{\delta}{\delta X_{F}^{\mu}(z_{1}, \bar{z}_{1})} \frac{\delta}{\delta X_{G \mu}(z_{2}, \bar{z}_{2})}) : \mathcal{F} \mathcal{G}: \ \ (11)

In which the entire operation now acts on the normal ordering {: \mathcal{F} \mathcal{G}:}.

This concludes the opening discussion on OPEs in Polchinski’s textbook, from which he goes on to consider two examples of computing normal ordering (p.40) before focusing on the important study of Ward identities and Noether’s theorem. It will prove beneficial to review in the future the computation of the two examples that Polchinski offers (see the Appendix of this chapter). In the meantime, it may aid one’s understanding if we instead pause and first explore other concepts integral to stringy CFTs and their OPEs. This will enable us to introduce more notation and more deeply explicate mathematical procedure. Taking such an approach has its obvious advantages, but it also has its disadvantages. The way in which Chapter 2 is structured in Polchinski’s textbook means that, in a few instances, it will be required that we advance our study of CFTs to include a number of other key concepts before making better sense of what we have already discussed, particular in why OPEs have the structure that they do and how we may think about their computational procedure in a more exemplified way. So at this point we bracket the definitions given above to discuss other related topics, before ultimately returning specifically to the subject of OPEs and computing a number of different examples step by step.

References

Joseph Polchinski. (2005). ‘String Theory: An Introduction to the Bosonic String’, Vol. 1.

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