Stringy Things

# Notes on String Theory: Ward Identities, Noether’s Theorem, and OPEs

1.1. Example 1

In the last entry we derived both the quantum version of Noether’s theorem and the Ward identity given in Polchinski’s textbook. This means we obtained the idea of the existence of conserved currents and how Ward identities in general constrain the operator products of these same currents. Let us now elaborate on some examples. The solutions to these examples are given in Polchinski (p.43); however, a more detailed review of the computation and of some of the key concepts will be provided below.

We start with the simplest example, where we once again invoke the theory of free massless scalars. Following Polchinski, the idea is that we want to perform a simple spacetime translation ${\delta X^{\mu} = \epsilon a^{\mu}}$. The action will be left invariant under worldsheet symmetry. But as what we want to derive is the current, given what we have been discussing, this means we should pay special attention to the fact that we are required to add ${\rho (\sigma)}$ to the above translation. Recall that we defined ${\rho(\sigma)}$ in our derivation of the Ward identity. The important point to note is that, again, the action is still invariant and from past discussions we already understand ${\rho(\sigma)}$ has a compact or finite support. From this set-up, let us now rewrite the action for massless scalars,

$\displaystyle S_{P} = \frac{1}{4\pi \alpha^{\prime}} \int d^2\sigma \partial X^{\mu}\partial X_{\mu} \ \ (1)$

When we vary (1) we obtain the following,

$\displaystyle \delta S = \frac{\epsilon a_{\mu}}{2\pi \alpha^{\prime}} \int d^2 \sigma \partial^{a}X^{\mu} \partial_{a}\rho \ \ (2)$

There is a factor of 2 from varying ${\partial X}$ that gives us a reduced denominator. We have also used the identity stated in Polchinski’s textbook, namely ${\delta X^{\mu}(\sigma) = \epsilon \rho(\sigma) a^{\mu}}$, where we can treat ${\epsilon}$ and ${a^{\mu}}$ as constants and therefore pull them in front of the integral.

Before we can move forward, there is something we have to remember. Recall the path integral formulation from our last discussion, where we found the variation to be proportional to the gradient. The result is written again below for convenience,

$\displaystyle [d\phi^{\prime}]e^{-S[\phi^{\prime}]} = [d\phi]e^{-S[\phi]}[1 + \frac{i\epsilon}{2\pi} \int d^2\sigma J^{a}\partial_{a}\rho + \mathcal{O}(\epsilon)^2] \ \ (3)$

If there is no contribution from the metric, then the measure in brackets becomes ${- \delta S}$. What this tells us is that the variation must be something like,

$\displaystyle \delta S = -\frac{i\epsilon}{2\pi} \int d^2\sigma J^{a}\partial_{a}\rho \ \ (4)$

Now notice, in computing both (2) and (4) we may establish the following interesting relation,

$\displaystyle \partial^{a}X_{a} \partial_{a} \rho \frac{\epsilon a_{\mu}}{2\pi \alpha^{\prime}} = -\frac{i\epsilon}{2\pi}J^{a} \partial_{a}\rho \ \ (5)$

The first step is to simplify. Immediately, we can see that we can cancel the ${\partial_{a}\rho}$ terms on both sides,

$\displaystyle \partial_{a}X^{\mu} \frac{\epsilon a_{\mu}}{2\pi \alpha^{\prime}} = -\frac{i\epsilon}{2\pi}J_{a} \ \ (6)$

This still leaves us with a bit of a mess. What we need to do is recall another useful fact. In the last section we studied the invariance of the path integral under change of variables, which, at the time, enabled us to obtain Noether’s theorem as an operator equation. Explicitly put, we had something of the general form ${ \frac{\epsilon}{2\pi i} \int d^{d} \sigma \sqrt{g} \rho(\sigma) \langle \nabla_{a}J^{a}(\sigma) ... \rangle}$. Notice that we have all of the ingredients. Given the Noether current is,

$\displaystyle J_{a} = a_{\mu}J_{a}^{\mu}$

We may substitute for ${J_{a}}$ in (6) and then work through the obvious cancellations that appear, including a cancellation of signs. Once this is done, we go on to obtain the following expression for the current,

$\displaystyle J_{a}^{\mu} = \frac{i}{\alpha^{\prime}} \partial_{a}X^{\mu} \ \ (7)$

Which is precisely what Polchinski gives in eqn. (2.3.13) on p.43 of his textbook. Automatically, we can see our currents are conserved. And, of course, we are free to switch to holomorphic and antiholomorphic indices and we can do so with relative ease,

$\displaystyle J_{a}^{\mu} = \frac{1}{\alpha^{\prime}}\partial X^{\mu}$

$\displaystyle \bar{J}_{a}^{\mu} = \frac{1}{\alpha^{\prime}}\bar{\partial} X^{\mu} \ \ (8)$

In the manner indicated above, we have successfully constructed the current following a spacetime translation. For this example the goal is to now use an operator to check the Ward identity and see if the overall logic is sound. What we require in the process are the appropriate residues, and to find these we will need to compute the OPEs. So to test some of the ideas from earlier discussions in the context of the given example.

Recall the formula for OPEs given the reverse of the sum of subtractions, namely the sum of contractions, as described in (11) of this post. In this formula recall that we have two operators which are normal ordered, $: \mathcal{F}:$ and $: \mathcal{G}:$. These are arbitrary functionals of X and typically the range of X is non-compact.

Now, in Polchinski’s first example we consider the case where ${\mathcal{F} = J_{a}^{\mu}}$ and ${\mathcal{G} = e^{ikX(z, \bar{z})}}$. In other words, we want to compute the product of the current and the exponential operator. As the product is normal ordered, there are no singularities and the classical equations of motion are satisfied. Instead, the singularities are produced from the contractions, or, in this case, the cross-contractions as ${z \rightarrow z_{0}}$. Furthermore, as a sort of empirical rule, it can be said that the most singular term in ${\frac{1}{z - z_{0}}}$ comes from the most cross-contractions. And we should recall that we compute the cross-contractions by hitting our operators with ${\delta / \delta X^{\mu}_{\mathcal{F}}}$ and ${\delta / \delta X^{\mu}_{\mathcal{G}}}$, respectively. Hence, from the master formula for cross-contractions,

$\displaystyle : \frac{i}{\alpha^{\prime}} :\partial X^\mu(z): :e^{ik X(z_0,{\bar z_{0}})}: = \exp [- \frac{\alpha^{\prime}}{2} \int d^2 z_1 d^2 z_2 \ln \mid z_{12}\mid^2 \frac{\delta}{\delta X_{\mathcal{F}}^{\mu}(z_1, \bar{z}_1)} \frac{\delta}{\delta X_{\mathcal{G} \mu}(z_2, \bar {z}_2)}] \ \ :\frac{i}{\alpha^{\prime}} \partial X^\mu(z) e^{i k X(z_{0},\bar{z}_{0})}:$

$\displaystyle = : \frac{i}{\alpha^{\prime}} \partial X^{\mu}(z) e^{i k X(z_{0},\bar{z}_{0})} : - \frac{i}{2} : \int d^2 z_1 d^2 z_2 \ln \mid z_{12} \mid^2 \frac{\delta(\partial X^{\mu}(z))}{\delta X_{\mathcal{F}}^{\mu}(z_1, \bar{z}_1)} \frac{\delta ( e^{i k X(z_0, \bar{z}_0)})}{\delta X_{\mathcal{G} \mu}(z_2, \bar{z}_2)} \ \ (9)$

Note that for ${\mathcal{G}}$, which, in this case is ${e^{ikX}}$, it is an eigenfunctional of ${\delta / \delta X_{\mathcal{G}} (z_{2}, \bar{z}_{2})}$. Likewise, for for ${\delta / \delta X_{\mathcal{F}}}$ we will end up with a delta function,

$\displaystyle = : \frac{i}{\alpha^{\prime}} \partial X^{\mu}(z) e^{i k X(z_{0},\bar{z}_{0})} : - \frac{i}{2} : \int d^2 z_1 d^2 z_2 \ln \mid z_{12} \mid^2 \partial (\delta^{\mu}_{\alpha} \delta^2(z_1, z)) i k^{\alpha} \delta^2(z_2, z_0) e^{i k X(z_{0}, \bar{z}_0)} : \ \ (10)$

Now, we can pull out the $ik^{\mu}$ which flips the sign,

$\displaystyle = : \frac{i}{\alpha^{\prime}} \partial X^{\mu}(z) e^{i k X(z_{0},\bar{z}_{0})}: + \frac{k^{\mu}}{2} : \partial (\int d^2 z_1 d^2 z_2 \ln \mid z_{12} \mid^2 \delta^2(z_1, z) \delta^2(z_2, z_{0}) e^{i k X(z_{0},\bar{z}_0)}) : \ \ (11)$

Notice that we have delta functions inside the integrand, so we are left with,

$\displaystyle : \frac{i}{\alpha^{\prime}} \partial X^{\mu}(z) e^{i k X(z_{0},\bar{z}_{0})} : + \frac{k^{\mu}}{2} : \partial ( \ln \mid z - z_{0} \mid^2 e^{i k X(z_0,\bar{z}_0)}) :$

$\displaystyle = : \frac{i}{\alpha^{\prime}} \partial X^{\mu}(z) e^{i k X(z_{0},\bar{z}_{0})} : + \frac{k^{\mu}}{2 (z - z_{0})} e^{i k X(z_0,\bar{z}_0)} \ \ (12)$

And so we obtain the following result,

$\displaystyle \frac{i}{\alpha^{\prime}} : \partial_{a}X^{\mu} : :e^{ikX}: \sim \frac{k^{\mu}}{2 (z - z_{0})} : e^{ikX}: \ \ (13)$

Where ${\sim}$ means the most singular pieces. We can also perform the same calculations for the antiholomorphic term,

$\displaystyle \frac{i}{\alpha^{\prime}} : \bar{\partial}X^{\mu}: : e^{ikX}: \sim \frac{k^{\mu}}{2(\bar{z} - \bar{z}_{0})} :e^{ikX}: \ \ (14)$

As Polchinski notes and as we see, the OPE is in agreement with the Ward identity. But we can still carry on a bit further. To conclude this example, recall explicitly to mind the Ward identity and our residues. Switching back to the holomorphic case and evaluating the LHS of (14) notice we find, picking out only the residues,

$\displaystyle \frac{i}{\alpha^{\prime}} : \partial X^{\mu} e^{ikX}: = (\frac{k^{\mu}}{2} + \frac{k^{\mu}}{2}) G = k^{\mu} G \ \ (15)$

So, we have that it must be equal to ${k^{\mu}}$ times the operator ${\mathcal{G}}$ from above. Now, given that ${\mathcal{G} = \mathcal{A}}$, the right-hand side of the Ward identity tells us that,

$\displaystyle k^{\mu} \mathcal{A} = \frac{1}{i \epsilon} \delta \mathcal{A} \ \ (16)$

And, again, from the Ward identity we can see in (15) that with a bit of algebra the variation of the operator must be,

$\displaystyle \delta \mathcal{A} = ik^{\mu}\epsilon \mathcal{A} \ \ (17)$

Where we are assuming the variation is only in one direction. Interestingly, as an aside, what is actually happening are the following transformation properties,

$\displaystyle \mathcal{A} = e^{ikX} \rightarrow e^{ikX + ik^{\mu}\epsilon}$

$\displaystyle X^{\mu} \rightarrow X^{\mu} + \epsilon \ \ (18)$

1.2. Example 2

In the first example we considered a spacetime translation. We can now look to the second example in Polchinski’s textbook, where we want to consider a worldsheet translation, particularly how the ${a}$ of the ${\sigma}$ coordinates transforms as ${\delta \sigma^{a} = \epsilon v^{a}}$. Here ${v^{a}}$ is a constant vector. It follows that from the action for free massless scalars is invariant under this transformation, with the above symmetry clearly understood given ${X}$ is a scalar and how ${\delta \sigma^{a}}$ does not change the measure of integration. And so, just as in the first example, what we want to do is investigate the construction of the conserved current as a result of this worldsheet symmetry transformation and then test the Ward identity.

The first step is to note that because we are dealing with a scalar theory we may write explicitly,

$\displaystyle \sigma^{a} \rightarrow \sigma^{\prime a} = \sigma + \epsilon v^{a} \ \ (19)$

Where, for any worldsheet symmetry transformation, the scalar fields simply transform as follows,

$\displaystyle X^{\prime \mu}(\sigma^{\prime}) = X^{\mu}(\sigma) \ \ (20)$

From which it also follows that,

$\displaystyle X^{\prime \mu}(\sigma + \delta \sigma) = X^{\mu}(\sigma) \implies X^{\prime \mu}(\sigma) = X^{\mu}(\sigma - \delta \sigma) \ \ (21)$

Where we should recognise that in brackets on the left-hand side of the first equality, ${\sigma + \delta \sigma = \sigma^{\prime}}$.

Of course, like the first example, we’re interested in how operators transform. And so we want to consider,

$\displaystyle \delta X^{\mu}(\sigma) = X^{\prime \mu}(\sigma) - X^{\mu}(\sigma) = X^{\prime \mu} (\sigma^{a} - \epsilon v^{a}) - X^{\mu}(\sigma) \ \ (22)$

Expanding and only keep the 1st terms, what we end up with is precisely an expression for how our operators transform,

$\displaystyle \delta X^{\mu}(\sigma) = -\epsilon(\sigma) v^{a}\partial_{a}X^{\mu} \ \ (23)$

Now, what we want to do is check with the Ward identity. So, like before, let’s start by varying the action and then build from there,

$\displaystyle \delta S = \delta [\frac{1}{4\pi \alpha^{\prime}}\int d^2\sigma \partial^{a}X^{\mu}\partial_{a}X_{\mu}]$

$\displaystyle = \frac{1}{2\pi \alpha^{\prime}} \int d^2\sigma \partial^{a}X^{\mu}\partial_{a}\delta X_{\mu} \ \ (24)$

Where ${\delta X_{\mu} = -\epsilon(\sigma)v^{a}\partial_{a}X_{\mu}}$. The implication is as follows. From (24) we can substitute for ${\delta X_{\mu}}$,

$\displaystyle \delta S = \frac{1}{2\pi \alpha^{\prime}} \int d^{2}\sigma \partial^{a}X^{\mu}\partial_{a}(-\epsilon(\sigma)v^{a}\partial_{a}X_{\mu})$

$\displaystyle = -\frac{\epsilon}{2\pi \alpha^{\prime}} \int d^2\sigma \partial^{a}X^{\mu}\partial_{a} v^{b}\partial_{b}X_{\mu} + \partial^{c}X^{\mu}\partial_{c}v^{d}\partial_{d}X_{\mu}$

$\displaystyle = -\frac{\epsilon}{2\pi \alpha^{\prime}} \int d^2\sigma \partial^{a}X^{\mu}\partial_{a} v^{b}\partial_{b}X_{\mu} \ [1] + \partial_{d}(\frac{1}{2}v^{d} \partial^{c}X^{\mu}\partial_{c}X^{\mu}) \ [2] \ \ (25)$

Where, for pedagogical purposes, the first and second integrands have been labelled [1] and [2] respectively. The reason is because it will be useful to recall these pieces separately in order to highlight some necessary computational logical and procedure. Before that, however, we should think of the conserved current. It follows, as we have already learned,

$\displaystyle -\delta S = \frac{i}{2\pi} \int d^2\sigma \sqrt{-g}J^{a}\partial_{a}\epsilon \ \ (26)$

Remember, looking at (2) in a previous entry, we can see clearly that ${J^{a}(\sigma)}$ is the coefficient of ${\partial_{a}\rho (\sigma)}$. In the first example we become more familiar with this fact. And what Polchinski is referencing in the single passing sentence that he provides prior to eqns. (2.3.15a) and (2.3.15b) is that we need to make contact with this formalism. It is convenient to now reassert the ${\rho(\sigma)}$ term,

$\displaystyle = -\frac{\epsilon}{2\pi \alpha^{\prime}} \int d^2\sigma [ (\partial^{a}X^{\mu}\partial_{b} v^{b}X_{\mu}) \partial_{b}(\rho(\sigma)) \ [1] + (\rho(\sigma)) \partial_{c}(\frac{1}{2}v^{c} \partial^{d}X^{\mu}\partial_{d}X^{\mu})] \ [2] \ \ (27)$

Now, let’s look at both pieces of (27). Piece [1] above looks fine and, on inspection, seems quite manageable. Piece [2], on the other hand, is not very nice. In taking one step forward, what we can do is integrate the second piece by parts. This has the benefit that we can eliminate the total derivative that arises and eliminate the surface terms. To save space, the result is given below,

$\displaystyle \delta S = \frac{\epsilon}{2 \pi \alpha'}\int d^2\sigma v^b\partial^a X^{\mu} \partial_b X_{\mu} \partial_a - \partial_b (\frac{1}{2}v^b \partial^a X^{\mu} \partial_a X_\mu)$

$\displaystyle =\frac{\epsilon}{2 \pi \alpha'}\int d^2\sigma [v^b(\partial^a X^\mu \partial_b X_{\mu} -\frac{1}{2}\delta^{a}_{b} \partial_b X^\mu \partial^b X_\mu) \partial_a] \ \ (28)$

How to interpret (28)? Notice something very interesting. We have the stress-energy tensor plus some additional terms outside the small brackets. If we make the appropriate substitution for the stress-energy tensor we therefore obtain,

$\displaystyle \delta S = -\dfrac{\epsilon}{2 \pi}\int d^2\sigma \, (v^c\,T_c^a)\partial_a \ \ (29)$

If we bring the constant epsilon back into the integrand, we have an integral over the worldsheet times a derivative of the parameter of an infinitesimal transformation. Whatever is left can be interpreted as a conserved current. Hence, then, if we go back and inspect (2) in this post we come to establish what Polchinski states in eqn. (2.3.15a). Our indices are slightly different up to this point, but this is merely superficial and when we rearrange things we find,

$\displaystyle J^{a} = iv^{b}T_{b}^{a}$

And then lowering the index on ${J}$,

$\displaystyle J_{a} = iv^{b}T_{ab} \ \ (30)$

This is our conserved current. In certain words, it is natural to anticipate a conserved current on the string worldsheet and also for this current to be related to the stress-energy tensor. Just thinking of the physical picture gives some idea as to why this is a natural expectation. But we are not quite done.

What we want to do, ultimately, is define the stress-energy tensor as an operator with full quantum corrections. But, as we are working in conformal field theory, there is an ambiguity about how we might define it related to normal ordering. Let’s explore this for a moment.

We should think of stringy CFTs by way of how we will define a set of basic operators, and then from this show what is the stress-energy tensor. Moreover, it is a property of the stress-energy tensor and the basic operators we utilise that will give definition to the CFT. In CFT language, it is given that the stress-energy tensor can be written as,

$\displaystyle T_{ab} = \frac{1}{\alpha^{\prime}} : \partial_{a}X^{\mu}\partial_{b}X_{\mu} - \frac{1}{2}\delta_{ab}(\partial X)^2 : \ \ (31)$

This is what Polchinski cites in eqn. (2.3.15b). We can still go a step further and discuss the topic of conformal invariance in relation to this definition. For instance, from the principles of conformal invariance, it remains the case that as discussed much earlier in these notes,

$\displaystyle T_{a}^{a} = 0$

Which is to say, as we should remember, the stress-energy tensor is traceless. This condition of tracelessness tells us how, if we were to go to holomorphic and antiholomorphic coordinates,

$\displaystyle T_{a}^{a} = 0 \rightarrow T_{z\bar{z}} = 0 = T_{\bar{z}z} \ \ (32)$

Where one may recall, also, the non-vanishing parts ${T_{zz}}$ and ${T_{\bar{z}\bar{z}}}$ from an earlier discussion in this collection of notes. It follows that if the stress-energy tensor is, in fact, traceless, we may invoke the conservation of the current such that,

$\displaystyle \nabla^{a}J_{a} = 0 = \nabla^{a} T_{ab} = 0 \ \ (33)$

Which is to say that we have full conservation for the full stress-energy tensor. We can write this in terms of holomorphic and antiholomorphic coordinates as expected,

$\displaystyle \bar{\partial}T_{zb} + \partial T_{\bar{z}b} = 0 \ \ (34)$

This gives us two choices:

$\displaystyle b = z \implies \partial T_{zz} = 0$

$\displaystyle b = \bar{z} \implies \partial T_{\bar{z}\bar{z}} = 0 \ \ (35)$

Where, as it was discussed some time ago, ${T_{zz} = T(z)}$ is a holomorphic function and ${T_{\bar{z}\bar{z}} = \bar{T}(\bar{z})}$ is an antiholomorphic function.

It is perhaps quite obvious at this point that we may also write,

$\displaystyle T(z) = -\frac{1}{\alpha^{\prime}} : \partial X^{\mu}\partial X_{\mu}:$

$\displaystyle \bar{T}(\bar{z}) = -\frac{1}{\alpha^{\prime}} : \bar{\partial} X^{\mu}\bar{\partial} X_{\mu}: \ \ (36)$

Now, returning to our current (30), we can be completely general in our study of the current,

$\displaystyle J_{z} = iv(z)T(z)$

$\displaystyle \bar{J}_{\bar{z}} = i\bar{v(z)}\bar{T}(\bar{z}) \ \ (37)$

If we have conservation of the current, then the above is the same as,

$\displaystyle \nabla_{a}J^{a} = \bar{\partial}J_{z} + \partial J_{\bar{z}} = 0 \ \ (38)$

Which is to say that the new currents are conserved provided ${v(z)}$, previously considered a constant vector, is holomorophic. Additionally, the current is of course associated with symmetries; but what are these symmetries? They are the conformal transformations.

If, in the bigger picture, what we want to do is find ${\delta X}$ due to symmetries ${J_{z} = ivT(z)}$, to proceed recall the Ward identity ${\mathcal{A} = X^{\mu}}$. It follows we need to compute an OPE for the stress-energy tensor with our scalar field (complete computation is given in the Appendix of this chapter, along with other important and useful OPEs),

$\displaystyle :T(z) : :X^{\mu}(z_{0}, \bar{z}_{0}): = -\frac{1}{\alpha^{\prime}} : \partial X\partial X: : X^{\mu}(z_{0}, \bar{z}_{0}) :$

$\displaystyle \sim -(\frac{2}{\alpha^{\prime}}) \cdot (-\frac{\alpha^{\prime}}{2}\partial_{z}\ln \mid z - z_{0}\mid^2) : \partial X(z_{0}) :$

$\displaystyle T X^{\mu} \sim \frac{1}{z - z_{0}}\partial X(z_{0})$

$\displaystyle \bar{T}X^{\mu} \sim \frac{1}{\bar{z} - \bar{z}_{0}}\bar{\partial}X(\bar{z}_{0}) \ \ (39)$

Which is what Polchinski states in eqn. (2.4.6). And now we can use the Ward identity and take the residue of the current with the coefficients of the OPE for the holomorphic and antiholomorphic pieces,

$\displaystyle iv(z_{0}) \partial X(z_{0}) + i \bar{v}(z_{0} \bar{\partial} X(\bar{z}_{0}) = \frac{1}{i\epsilon} \delta X \ \ (40)$

And so what we find is that, for the current we have constructed, we have a symmetry transformation of the following form,

$\displaystyle \delta X = -iv(z_{0})\partial X(z_{0}) - i\bar{v}(z_{0} \bar{\partial}X(\bar{z}_{0}) \ \ (41)$

For ${z_{0} \rightarrow z_{0} + \epsilon v(z_{0})}$. If we drop ${z_{0}}$ and generalise,

$\displaystyle \delta X^{\mu} = -\epsilon v(z)\partial X - \epsilon\bar{v}(z)\bar{\partial}X \ \ (42)$

For ${z \rightarrow z + \epsilon v(z)}$ which is an infinitesimal transformation, where the only constraint is that ${z \rightarrow z^{\prime} = f(z)}$ is holomorphic.

The reason the transformation is so simple,

$\displaystyle \delta X^{\mu} = X^{\prime \mu}(z^{\prime}, \bar{z}^{\prime}) - X^{\mu}(z, \bar{z}) = X^{\mu}(z - \epsilon v, z - \epsilon\bar{v}) - X^{\mu} \ \ (43)$

Where, after Taylor expansion,

$\displaystyle \delta X^{\mu} = - \epsilon v\partial X - \epsilon\bar{v}\bar{\partial}X \ \ (44)$

It is important to point out that ${f(z) = \xi(z)}$ represents a global rescaling (but can also represent a local rescaling). If ${\mid \xi \mid = 1}$ then we have a simple rotation, and in general no scaling.

To conclude, from the very outset of this chapter, we may also recall to mind that in the context of the conformal group we are working in 2-dimensions. When we ask, ‘what is the analogue of this symmetry in higher dimensions?’, the answer is that in higher dimensions we can construct scale invariance as well. Indeed, in D-dimensions, if you have ${y^{\mu} \rightarrow \lambda y^{\mu}}$ you have additional special conformal transformations.

References

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

Blumenhagen, R. and Plauschinn. (2009). ‘Introduction to Conformal Field Theory’.

Standard
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.

Standard
Stringy Things

# Notes on String Theory: Conformal Field Theory – Local Operators, the String Propagator, and Operator Product Expansions

1. Local Operators

In the last entry we introduced a theory of free massless scalars in flat 2-dimensions (i.e., a free X-CFT). From this we also introduced new terms and established notation relevant to our ongoing study of CFTs in string theory (Chapter 2 in Polchinski). What we now want to do is proceed with a review of a number of interrelated topics at the heart of stringy CFTs: namely, local operators, techniques with path integrals, string propagators, and finally operator product expansions. Each of these topics has a number of parts, and so we shall need to work piece by piece and then stitch everything together.

To begin, we note that in string perturbation theory, one of the main objects of interest is the expectation value of the path integral of a product of local operators (Polchinski, p.36). This interest is our entry point, and it represents a primary theme for much of the following discussion. So our first step should be to define what we mean by local operators. These objects may also be described as fields; however, in the context of CFTs, the notion of a field carries a different meaning than, for instance, the definition of a field in quantum field theory. In our case, a field may be viewed generally as a local expression, which may be the generic field ${\phi}$ that enters the path integral in QFT, or as a composite operator ${e^{i\phi}}$ or as a derivative ${\partial^{n}\phi}$ (Tong, p.69). These are all different types of fields or local operators in the CFT dictionary.

With a definition of local operators in mind, we opened the discussion by mentioning the expectation value of the path integral as a primary object of interest. Let us now consider some general expectation value. Consider, for instance, ${\mathcal{A}_{i}}$ that is some basis for a set of local operators. We may write the general expectation value as follows,

$\displaystyle \langle \mathcal{A}_{i_{1}}(z_{1}, \bar{z}_{1}) \mathcal{A}_{i_{2}}(z_{2}, \bar{z}_{2}) ... \mathcal{A}_{i_{n}}(z_{n}, \bar{z}_{n}) \rangle \ \ (1)$

If the basic idea, as mentioned, is to compute the expectation value of the path integral, a more technical or detailed description of our overarching interest is to understand the behaviour of this expectation value (1) in the limit of two operators taken to approach one another (Polchinski, p. 36). The tool that we use for such analysis is the operator product expansion (OPE). Understanding the definition of OPEs and how to compute them is one of the ultimate aims of studying stringy CFTs, important for more advanced topics that we will consider throughout the remainder of this paper. But before formally defining OPEs, it is useful to first build a deeper sense of intuition about their meaning. To do this, let as briefly review some more basics.

2. The Path Integral and Arbitrary Operator Insertions

What do we mean by path integral? And how do we understand this idea of local operator insertions? Additionally, how do we construct the important operator equations required to build a picture of OPEs? Polchinski offers several valuable contributions to a definition of the path integral, including a lengthy treatment in the Appendix of Volume 1. For our purposes, we might first emphasise the QFT view of the path integral as an integral over fields,

$\displaystyle Z = \int [dX]e^{-S} \ \ (2)$

We may describe this as a partition function. Now, if what we want to know is the expectation value given some operator, this implies that we want to employ the path integral representation to derive operator equations. For instance, as we read in Polchinski (p.34), given some operator we may compute,

$\displaystyle \left\langle \mathcal{F}[X] \right\rangle = \int [dX]e^{-S}\mathcal{F}[X] \ \ (3)$

Where ${\mathcal{F}[X]}$ is some functional of X, typically a product of two operators, and where $\langle \mathcal{F}[X] \rangle = \langle 0 \mid \mathcal{F} \mid 0 \rangle$ . For multiple entries in the form,

$\displaystyle \mathcal{F}_{1}[X(z_{1}, \bar{z}_{1})] \mathcal{F}[X(z, \bar{z})] \mathcal{F}_{2}[X(z_{2}, \bar{z}_{2})]$

We may write,

$\displaystyle \langle 0 \mid \mathcal{F}_{1} \mathcal{F} \mathcal{F}_{2} \mid 0 \rangle =\int [dX]e^{-S} \mathcal{F}_{1}[X(z_{1}, \bar{z}_{1})] \mathcal{F}[X(z, \bar{z})] \mathcal{F}_{2}[X(z_{2}, \bar{z}_{2})] \ \ (4)$

There is a notion of time-ordering present in (4), which we will discuss later. For now, we should note that the path integral of a total derivative is always zero. This fact will prove useful in just a moment and on many other occasions in the future. As Polchinski reflects, This is true for ordinary bosonic path integrals, which can be regarded as the limit of an infinite number of ordinary integrals, as well as for more formal path integrals as with Grassmann variables’ (Polchinski, pp. 34-35). Hence eq.(2.1.15) in Polchinski (p.35), where he considers the path integral with the inclusion of Grassmann variables,

$\displaystyle 0 = \int [dX] \frac{\delta}{\delta X_{\mu}(z, \bar{z})} \exp (-S)$

$\displaystyle = - \int [dX] \exp (-S) \frac{\delta S}{\delta X_{\mu} (z, \bar{z})}$

$\displaystyle = - \int \bigg \langle \frac{\delta S}{\delta X_{\mu} (z, \bar{z})} \bigg \rangle$

$\displaystyle = \frac{1}{\pi \alpha^{\prime}} \langle \partial \bar{\partial} X^{\mu}(z, \bar{z}) \rangle \ \ (5)$

There is something interesting with this result. If we recall the action for the free X-CFT in the last post, remember that we found the classical EoM to be ${\partial \bar{\partial} X^{\mu}(z, \bar{z}) = 0}$. Notice, then, that the result (5) is the analogue statement in the quantum theory for the classical equations of motion. What is this telling us? Let us dig a bit deeper.

First, consider how the same calculation in (5) holds if we have arbitrary additional insertions …’ in the path integral. We already considered what multiple entries in the path integral in (4). But there is a caveat: namely, these additional insertions cannot also be at ${z}$ (something we will elaborate below). Second, in the case of multiple entries in the path integral, which implies that we may write something of the form ${\int [dX] \frac{\delta}{\delta X^{\mu}(z, \bar{z})}[e^{-S}\mathcal{F}(z, \bar{z})]}$, one can think of some of the insertions as preparing a state in the theory. In other words, we should note that these insertions prepare arbitrary initial and final states in the theory (Polchinski, p.35). These arbitrary initial and final states perform a similar role should we instead consider boundary conditions, except with the offered convenience that we may now write the following path integral statement,

$\displaystyle \left\langle \partial\bar{\partial}X^{\mu}(z, \bar{z}) ... \right\rangle = 0 \ \ (6)$

Now, if (5) is the analogous statement in the quantum theory for the classical equations of motion, look at (6). Notice, as an operator statement, it is the same as in the Hilbert space formalism,

$\displaystyle \partial\bar{\partial}\hat{X}(z, \bar{z}) = 0 \ \ (7)$

Polchinski describes (7) as holding for all matrix elements of the operator ${\hat{X}(z, \bar{z})}$, with all relations that hold (6) being operator equations (Polchinski, p.34). These two points are important. It should also be noted that (7) is Ehrenfest’s theorem, which makes a lot of sense because it is telling us something that we already know or suspect: namely, the expectation values of the operators obey the classical equations of motion. But, again, this proves true only when the additional insertions …’ in the path integral are located away from ${z}$. So let us now look into this subtlety. If, for example, addition insertions cannot be coincident at ${z}$, then let us consider what happens when we do indeed have coincident points at ${z}$! It follows,

$\displaystyle 0 = \int [dX] \frac{\delta}{\delta X_{\mu}(z, \bar{z})}[exp(-S)X^{\nu}(z^{\prime}, \bar{z^{\prime}})]$

$\displaystyle = \int d[X] exp(-S) [\eta^{\mu \nu}\delta^{2}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime}) + \frac{1}{\pi\alpha^{\prime}}\partial_{z}\partial_{\bar{z}}X^{\mu}(z, \bar{z})X^{\nu}(z^{\prime}, \bar{z}^{\prime})]$

$\displaystyle = \eta^{\mu \nu} \langle \delta^{2}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime}) \rangle + \frac{1}{\pi\alpha^{\prime}}\partial_{z}\partial_{\bar{z}}\langle X^{\mu}(z, \bar{z})X^{\nu}(z^{\prime}, \bar{z}^{\prime})\rangle \ \ (8)$

Where the ${\delta^{2} (z^{\prime} - z, \bar{z}^{\prime} - \bar{z})}$ term comes from differentiating ${\frac{\delta X^{\mu}(z^{\prime}, \bar{z}^{\prime})}{\delta X^{\mu}(z, \bar{z})}}$ that appears in the computation. What we see in (8) is that at coincident points the classical equations of motion do not hold at the quantum level. This implies a few things. First, the good news is that we obtain our previous result that the EoM agrees as an operator statement of the ground state specifically under the conditions ${z \neq z^{\prime}}$. Second, the implication is clearly that with arbitrary additional insertions …’ in the path integral, so long that these are far away from ${z}$ and ${z^{\prime}}$, we may can rewrite (8) as,

$\displaystyle \frac{1}{\pi \alpha^{\prime}} \partial_{z}\partial_{\bar{z}} \langle X^{\mu}(z, \bar{z}) X^{\nu}(z^{\prime}, \bar{z}^{\prime}) \ ... \ \rangle = -\eta^{\mu \nu} \langle \delta^{2}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime}) \ ... \ \rangle \ \ (9)$

Where the ellipses are, again, the additional fields. Importantly, we may note that the following holds as an operator equation,

$\displaystyle \frac{1}{\pi \alpha^{\prime}} \partial_{z}\partial_{\bar{z}} X^{\mu}(z, \bar{z})X^{\nu}(z^{\prime},\bar{z}^{\prime}) = - \eta^{\mu \nu} \delta^{2}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime}) \ \ (10)$

We are going to want to solve this equation in the future, because the solution will prove useful when computing OPEs. In the meantime, what should be understood is that what we have accomplished here is that we’ve modified the EoM to take into account that there is a collision between points at ${z}$ and ${z^{\prime}}$. And we have also found that this behaviour can be derived as an operator statement. The purpose and greater logic for such an exercise will become increasingly clear. Meanwhile, notice that we now have a product of operators. Although it will not be proven here, it follows that in the Hilbert space formalism this product in the path integral becomes time-ordered (Polchinski, p.36). We also see that the delta function appears when the derivatives act on the time-ordering.

To summarise, these last results signal what has already been alluded (however vaguely) about the definition of OPEs in the final paragraph of Section 1. If, moreover, the general theme is so far one of path integrals and local operator insertions, the picture we are ultimately constructing is one of such insertions inside time-ordered correlation functions. These correlation functions can then be held as operator statements.

3. Time-ordered Correlation Functions, Normal Ordering, and the String Propagator

Before formally introducing and defining OPEs, we should spend a few more moments developing the picture and building intuition. For example, when it comes to the idea of time-ordered correlation functions, we will learn that solving the operator equation (10) gives us,

$\displaystyle \langle X^{\mu}(z, \bar{z})X^{\nu}(z^{\prime}, \bar{z}^{\prime}) \rangle = - \eta^{\mu \nu} \frac{\alpha^{\prime}}{2} \ln \mid z - z^{\prime} \mid^{2} \ \ (11)$

The computation required to arrive at this result may not yet have much meaning and may be too forward thinking. We will come to understand it soon. What can be understood at this juncture are some of the pieces of this equation. The most important note is that (11) is the propagator of the theory of massless scalars that we have been working with in our study of CFTs (i.e., the free X-CFT). Notice, on the left-hand side of the equality, we a two-point correlation function. As it has been stated, correlation functions are time-ordered. Let us focus on this notion of time-ordering. For instance, consider a Wick expansion for ${X^{\mu}(z, \bar{z})}$,

Where we have indicated the use of contraction notation that will be defined later. The first observation is that we have a two-point correlation function, and we have some term ${T}$. We also have colons on the right-hand side. For the ${T}$ term, it indicates that the expression is time-ordered in the same way one will find in basic QFT (Polchinski, p.36). Writing ${T}$ in full we find,

$\displaystyle T (X^{\mu} (z, \bar{z}), X^{\nu} (z^{\prime}, \bar{z}^{\prime}))$

$\displaystyle = X^{\mu} (z, \bar{z}) X^{\nu} (z^{\prime}, \bar{z}^{\prime}) \theta(z - z^{\prime}) + X^{\nu}(z^{\prime}, \bar{z}^{\prime})X^{\mu}(z, \bar{z})\theta(z^{\prime} - z) \ \ (13)$

Now, looking again at (13), it is worth pointing out a few other things. Firstly, what we will learn in the future, particularly as we advance our discussion on CFTs, is that this time-ordering will prove very useful. Eventually we are going to want to make conformal transformations from an infinite cylinder to the complex plane, and we will learn that time-ordering on the cylinder corresponds to radially ordering on the complex plane. Reversely, we will see that radial ordering on the complex plane corresponds with time-ordering in the path integral. This is a featured point of study in Section 2.6 of Polchinski and it is something we will discuss later. Secondly, for the colons on the right-hand side, they indicate normal ordering. We saw normal ordering in the past discussion on the free string string spectrum using light-cone gauge quantisation. Notice, then, that on the far right-hand side we have a normal ordered product. The definition of normal ordered operators follows as (Polchinski, p.36),

$\displaystyle :X^{\mu}(z, \bar{z}): = X^{\mu}(z, \bar{z}) \ \ (14)$

And for the normal ordered product we have,

$\displaystyle :X^{\mu}(z_{1}, \bar{z}_{1}), X^{\nu}(z_{2}, \bar{z}_{2}): = X^{\mu}(z_{1}, \bar{z}_{1}) X^{\nu}(z_{2}, \bar{z}_{2}) + \frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \ln \mid z_{12} \mid^{2} \ \ (15)$

Where ${z_{ij} = z_{i} - z_{j}}$. Furthermore, for arbitrary numbers of fields, the normal ordered product may be written as,

$\displaystyle :X^{\mu_{1}} (z_{1}, \bar{z}_{1}) ... X^{\mu_{n}}(z_{n}, \bar{z}_{n}): = X^{\mu_{1}}(z_{1}, \bar{z}_{1}) ... X^{\mu_{n}}(z_{n}, \bar{z}_{n}) + \sum \text{subtractions} \ \ (16)$

Where, for the subtractions, we sum the pairs of fields from the product and then replace each pair with its expectation value ${\frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \ln \mid z_{ij} \mid^{2}}$. We will elaborate more on (16) later. Meanwhile, consider again the operator equation (10). If what we want to do is define a product of operators that would satisfy the classical EoM, then from (16) and using (10) we can compute,

$\displaystyle \partial_{z} \partial_{\bar{z}} :X^{\mu}(z_{1}, \bar{z}_{1}) X^{\nu}(z_{2}, \bar{z}_{2}): = \partial_{z} \partial_{\bar{z}} X^{\mu}(z_{1}, \bar{z}_{1}) X^{\nu}(z_{2}, \bar{z}_{2}) + \frac{\alpha^{\prime}}{2}\eta^{\mu \nu} \partial_{z} \partial_{\bar{z}} \ln \mid z_{12} \mid^{2}$

$\displaystyle = - \pi \alpha^{\prime} \eta^{\mu \nu} \delta^{2}(z_{1} - z_{2}, \bar{z}_{1} - \bar{z}_{2}) + \frac{\alpha^{\prime}}{2} \eta^{\mu \nu} \partial_{z}\partial_{\bar{z}} \ln \mid z_{12} \mid^2$

$\displaystyle = - \pi \alpha^{\prime} \eta^{\mu \nu} \delta^{2}(z_{1} - z_{2}, \bar{z}_{1} - \bar{z}_{2}) + \pi \alpha^{\prime} \eta^{\mu \nu} \delta^{2} (z_{1} - z_{2}, \bar{z}_{1} - \bar{z}_{2}) = 0 \ \ (17)$

Where, for the last line in the computation, we used the standard result,

$\displaystyle \partial \bar{\partial} \ln \mid z \mid^{2} = 2\pi \delta^{2}(z, \bar{z}) \ \ (18)$

Which is derived from an application of Stokes’ theorem.

Importantly, (17) is precisely the property that Polchinski highlights in equation (2.1.23) on p.36 of his textbook. What (17) is telling us is that, on the last line, ${- \pi \alpha^{\prime} \eta^{\mu \nu} \delta^{2}(z_{1} - z_{2}, \bar{z}_{1} - \bar{z}_{2})}$ are the quantum corrections to the classical EoM. It is also telling us that, as we want to define a product of operators that satisfy the classical EoM, we must necessarily induce normal ordering.

So what does this all mean? In order to further extend the picture being developed here, we are lead directly to a definition of OPEs.

4. Operator Product Expansions

We may now define operator product expansions. The definition follows (pp. 37-38) directly from the intuition and logic that we have so far established, notably that OPEs may be considered a direct statement about the behaviour of local operators as they approach one another. The formula for OPEs is as follows,

$\displaystyle \langle \mathcal{O}_{i}(z, \bar{z})\mathcal{O}_{j}(z^{\prime}, \bar{z}^{\prime}) \rangle = \sum_{k} C_{ij}^{k}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime})\langle \mathcal{O}_{k}(z^{\prime}, \bar{z}^{\prime}) \rangle \ \ (19)$

Which is, again, an operator statement. This means that it also holds inside a general expectation value. Saving the general formula for OPEs until later, note that in (19) the ${C_{ij}^{k}(z - z^{\prime}, \bar{z} - \bar{z}^{\prime})}$ should be considered as a set of functions that depend only on the separation between the two operators (i.e., there is translational invariance).

To summarise, if OPEs describe what happens when local operators approach one another other, we have already developed a sense of technical intuition for why the key idea is one of having two local operators inserted in such a way that they are situated close to one another but not at coincident points. As we have already discussed, upon insertion of local operators at ${z_{1}}$ and ${z_{2}}$ for example, we obtain some normal ordered product. Then, what we can do is compute their approximation by way of a string of operators at only one of the insertion points (Tong, p.69). There can be any number of operator insertions, which is of course why we have included `…’ in the general formula for the OPE (19); it denotes insertions that are not coincident at ${z}$. (From this point forward, the ellipse will be removed and the following statement will be implied). This leads us directly to an illustration of OPEs as provided in Polchinski’s textbook.

In figure 4.1, we see that we have a number of local operator insertions, ${z_{1}}$ to ${z_{4}}$, hence what we would be computing is the expectation of 4 local operators. Given that the OPE describes the limiting behaviour of ${z_{1} \rightarrow z_{2}}$ as a series, where the pair of operators are replaced by a single operator at ${z_{2}}$, one way to think about this is analogous to the Taylor series in calculus (i.e., the OPE plays a similar role in quantum field theory). In fact, the analogue of computing a Taylor series is apt, as we will see when we start computing OPEs.

Another thing to note is that the circle in the picture illustrates the radius of convergence, such that this radius is computed as the distance to the nearest other operator positioned on the circle. In CFTs, OPEs have a finite radius of convergence.

Now, from our previous discussions, and from the formal definition of OPEs, we can see quite clearly why they are always to be understood as statements which hold as operator insertions inside time-ordered correlation functions. Should one ask, ‘what are the observables in string theory?’, the answer is that we compute a set of correlation functions of local/composite operators at their insertion points. So, should we take for example the Polyakov action, ${S_{P}}$, and compute the correlation functions for the CFT, one motivation is to show the correlation function to be related to the scattering amplitude in 26-dimensional spacetime (in the case of the bosonic string). So, in perturbative string theory, we look at the critical theory – that is, the critical coefficients and components of the correlation function,

$\displaystyle \langle A_{ij}(z_{i}\bar{z}_{j}) ... A_{ij}(z_{n}, \bar{z}_{n})) \rangle \ \ (20)$

Where we are interested in the singular behaviour. Moreover, recall the definition of the normal ordered product (15). Notice that we have very interesting log behaviour. If what we want to know exactly is what will happen with the product of the two operators as ${z_{1} \rightarrow z_{2}}$, this implies that we have an operator singularity. As we start computing OPEs and moving forward in our study of string CFTs, it will become very clear why this singular behaviour is actually the only thing we care about.

In the next post, we will extend our discussion of OPEs. Following that, we will look to derive the Ward Identities and then turn our attention to the Virasoro algebra among other important topics in Chapter 2 of Polchinski’s textbook.

References

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

David Tong. (2009). ‘String Theory’ [lecture notes].

Standard