# Operator product expansion (OPE) The **operator product expansion** tells us about the behaviour when two local operators in conformal field theories approach each other. It takes the form $ \mathcal{O}_i(z, \bar{z}) \mathcal{O}_j(w, \bar{w})=\sum_k C_{i j}^k(z-w, \bar{z}-\bar{w}) \mathcal{O}_k(w, \bar{w}) $where $\mathcal{O}_k$ is a local operator, not necessarily [[0029 Primary operator|primary]]. The statement should be understood to hold only when inserted into some time-ordered correlator. ### Refs - [[Rsc0002 Alday Notes on CFT]] - [[Rsc0006 Tong String theory]] - [[Rsc0007 BlumenhagenPlauschinn Book on CFT]] ## Proof Consider inserting two points in a region of certain radius near the origin. This creates a state $|\psi\rangle=\phi_{1}(x) \phi_{2}(0)|0\rangle$. Because any state can be expanded in eigenstates of the dilatation operator, we can write $\phi_{1}(x) \phi_{2}(0)|0\rangle=\sum_{\text {primaries } \phi} C_{\Delta}(x, \partial) \phi_{\Delta}(0)|0\rangle$. ## Alternative form - $\phi_{1}(x) \phi_{2}(0)=\sum_{\text {primaries } \phi} C_{\Delta}(x, \partial) \phi_{\Delta}(0)$ - i.e. not acting on the vacuum state - true as along as there exists a sphere including only these two operators. ## Computing OPE coefficients 1. consider a 3-pt function 2. use OPE so it becomes $C_{\Delta(x,\partial)}$ times 2-pt fn. 3. we know 3-pt and 2-pt function for a CFT, so can solve for $C_{\Delta(x,\partial)}$ by expansion near some point ## Constraints from [[0028 Conformal symmetry|conformal symmetry]] - leading term multiplying the primary operator not determined - i.e. $C_{ij}^k$ not known - but all coefficient of descendants are fixed by conformal symmetry once $C_{ij}^k$ are known ## Applicability - general QFTs do not have a convergent OPE ## Relation to commutation relations and symmetry transformations - see e.g. [[Rsc0006 Tong String theory]] - the singular term in OPE contain the same information as commutation relations as well as telling us how operators transform under symmetries - knowing OPE between conformal current and an operator => knowing how a field transforms under conformal symmetry; - knowing how an operator transforms => know something about the OPE