Correlators, OPE, and Conformal Blocks
Conformal symmetry fixes much of a correlation function, leaving dimensions, representations, and operator-product coefficients as dynamical data. This chapter shows how those data build correlators through conformal blocks and how agreement between different expansions produces crossing equations. Positivity requires an additional unitary structure.
Enter this chapter
Section titled “Enter this chapter”Begin with Scalar Two- and Three-Point Functions to see what symmetry fixes and which normalization choices remain.
Helpful background. Free-Field OPE Preview separates the general local Wilson expansion from its conformal specialization. The State–Operator Correspondence supplies radial states and cylinder energies. Tempered Distributions and Fourier Calculus supplies the distributional meaning of momentum-space correlators. Multiplets, Invariants, and Selection Rules supplies internal-symmetry sectors.
Topics
Section titled “Topics”- Scalar Two- and Three-Point Functions — Determines scalar correlators at separated points and identifies their independent constants.
- Spinning Correlators and Tensor Structures — Constructs independent tensor structures and their conservation and permutation constraints.
- From the Local OPE to Conformal Data — Separates primary data from descendant coefficients fixed by symmetry.
- OPE Convergence, Associativity, and Domain Control — Explains when channel expansions converge and may be compared.
- Cross Ratios and Four-Point Kinematics — Defines cross ratios together with their ordering and analytic-sheet information.
- Conformal Blocks and Casimir Equations — Selects a conformal family’s contribution using a Casimir equation and its OPE asymptotic.
- Conformal Partial Waves and the Shadow Formalism — Distinguishes a physical block from the block-plus-shadow combination in a partial wave.
- Crossing Equations and Positivity — Combines equality of channel expansions with the appropriate positive pairing.
- Mixed Correlators and Global-Symmetry Sectors — Organizes coupled correlators into symmetry sectors with matrix positivity.
- Momentum-Space Correlators and Conformal Ward Identities — Retains contact terms and anomalies when Fourier transforming conformal Ward identities.
How the ideas fit together
Section titled “How the ideas fit together”The operator product expansion (OPE) replaces a nearby pair of operators by a sum of local operators. Conformal symmetry groups each primary and its descendants into one block, so changing the pair being fused gives a different decomposition of the same correlator. On a common convergence domain, equality of these decompositions is crossing. Reflection positivity then constrains the coefficients: squared real coefficients occur in suitable identical-scalar problems, while mixed systems require positive-semidefinite matrices. Simmons-Duffin 2017, §§ 8–10.2.
A Casimir eigenvalue alone does not select the physical block: the shadow solution shares it. The OPE asymptotic selects the branch, and Lorentzian continuation still requires an operator ordering and a path around singularities.
Review the chapter
Section titled “Review the chapter”Can crossing symmetry alone justify treating every coefficient in a mixed-correlator problem as a nonnegative number?
Solution
No. Crossing equates channel decompositions. Nonnegativity comes from reflection positivity and a specified pairing. In mixed systems several OPE coefficients for one exchanged operator form a vector; its outer product is positive semidefinite even when individual off-diagonal entries are negative.
Where to go next
Section titled “Where to go next”Continue to Conformal Field Theory in One Dimension for an explicit block-and-crossing problem. For computation, Numerical Bootstrap develops functional exclusions.