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According to Cardy's lectures on Conformal Field theory, the general form of an operator product expansion is $$ \phi_{i}(r_{i})\cdot\phi_{j}(r_{j})=\sum_{k}C_{ijk}(r_{i}-r_{j})\phi_{k}((r_{i}+r_{j})/2)), $$ see page 5 Eq.(4).

I don't understand why the fields on the right hand side of this equation necessarily only depend on $(r_{i}+r_{j})/2$. In principle it seems to me that they could depend on both $r_{i}$ and $r_{j}$ or equivalently on both $(r_{i}+r_{j})/2$ and $r_{i}-r_{j}$.

Take the simple example of vertex operators which fulfills $$ :e^{i\alpha\phi(z)}::e^{i\beta\phi(w)}: = e^{-\alpha\beta\langle\phi(z)\phi(w)\rangle} :e^{i\alpha\phi(z)+i\beta\phi(w)}: $$ Typically we have $e^{-\alpha\beta\langle\phi(z)\phi(w)\rangle}\propto\ln(z-w)$ from which I can understand why the $C_{ijk}$ only depend on $z-w$. But it is not clear to me, why the field $e^{i\alpha\phi(z)+i\beta\phi(w)}$ only depends on $(z+w)/2$.

I would be very happy if someone could explain this point to me.

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  1. The most general OPE is of the form $$\phi_i(x)\phi_j(y)~=~\sum_k C_{ij}^k(x-y, y-z,z-x) ~\phi_k(z),\tag{1}$$ where $z$ is a third point. Eq. (1) is a shorthand for $$\langle \ldots \phi_i(x)\phi_j(y)\ldots\rangle~=~\sum_k C_{ij}^k(x-y, y-z,z-x) ~\langle \ldots\phi_k(z)\ldots\rangle,\tag{2}$$ where $\ldots$ denotes other operator insertions. In Euclidean signature, for a convergent$^1$ sum, the points $x$ and $y$ should be closer to $z$ than other insertion points.

  2. The OPE is typically assumed to be of the form $$\phi_i(x)\phi_j(y)~=~\sum_k C_{ij}^k(x-y) ~\phi_k(z_{\lambda}),\tag{3}$$ where $z_{\lambda}$ is taken to lie on the line$^2$ $$ z_{\lambda} ~=~ \lambda x + (1-\lambda)y \tag{4} $$ that goes through the points $x$ and $y$. Here $\lambda\in\mathbb{R}$ is a fixed conventional constant, typically chosen to be, say, $0$, $\frac{1}{2}$, or $1$. Cardy picks $\lambda=\frac{1}{2}$. Many string theory textbooks on CFT choose $\lambda=0$.

  3. It is implicitly assumed that the $k$-sum on the right-hand side of the OPE (3) runs over a complete set of local operators $\phi_k$ in the theory. This in particular means that for each $k$, the derivatives of $\phi_k$ wrt. $z_{\lambda}$ should again be some linear combination of the $\phi_{\ell}$'s.

  4. Note that if we pick another conventional constant $\mu\in\mathbb{R}$, then the difference $$ z_{\lambda}-z_{\mu} ~=~(\lambda-\mu)(x-y) \tag{5} $$ is proportional to the difference $x-y$.

  5. In particular, the differences $x-z_{\lambda}$ and $y-z_{\lambda}$ are again proportional to the difference $x-y$. Naively speaking, if it made sense to Taylor expand the left-hand side of the OPE (3) around the point $z_{\lambda}$ instead of the two points $x$ and $y$, we would get an operator that depends on the two points $z_{\lambda}$ and $x-y$. Such heuristic arguments make the right-hand side of the OPE (3) a plausible Ansatz.

  6. Well, that's all we want to say about the existence of the OPE (3). Concerning uniqueness, if we Taylor expand $\phi_k(z_{\lambda})$ in the OPE (3) around the point $z_{\mu}$ instead of $z_{\lambda}$, then the OPE would still be of the form $$\phi_i(x)\phi_j(y)~=~\sum_k \tilde{C}_{ij}^k(x-y) ~\phi_k(z_{\mu}).\tag{4}$$ The coefficients $\tilde{C}_{ij}^k(x-y)$ might be different, but the crucial point is that they would still only depend on the difference $x-y$; not on $x$ and $y$ individually.

References:

  1. J. Cardy, Conformal Field Theory and Statistical Mechanics, arXiv:0807.3472; eqs. (3) & (4).

  2. P. Ginsparg, Applied Conformal Field Theory, arXiv:hep-th/9108028; eq. (2.11).

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$^1$ For a generic non-CFT the rhs. of eq. (2) is only an asymptotic series.

$^2$ Strictly speaking the line should be an affinely parametrized geodesic.

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