Abstract
Caron-Huot has recently given an interesting formula that determines OPE data in a conformal field theory in terms of a weighted integral of the four-point function over a Lorentzian region of cross-ratio space. We give a new derivation of this formula based on Wick rotation in spacetime rather than cross-ratio space. The derivation is simple in two dimensions but more involved in higher dimensions. We also derive a Lorentzian inversion formula in one dimension that sheds light on previous observations about the chaos regime in the SYK model.
Highlights
The operator product expansion in a conformal field theory implies that one can write a four-point correlation function as a discrete sum of conformal blocks corresponding to the physical operators of the theory: O1(x1) · · · O4(x4) = p∆,J G∆∆i,J (xi). ∆,J (1.1)The conformal block G∆∆i,J (xi) gives the total contribution to their four-point function coming from operators in a multiplet with a primary of dimension ∆ and spin J
The Euclidean inversion formula (1.6) is an inner product between our four-point function and the partial wave Ψ∆i where we replace all operators by their shadows ∆ =
CFT four-point functions are bounded in the Regge limit [18]
Summary
A much more interesting formula for I∆,J has been presented by Caron-Huot [9] This involves an integral over two Lorentzian regions, with an integrand given by a special type of conformal block multiplied by a double commutator, either [O1, O3][O2, O4] or [O1, O4][O2, O3] , depending on the region. This formula has several advantages, such as the fact that it can be analytically continued in the spin J, and that for real dimension and spin the integrand satisfies positivity conditions. We present a separate derivation for the interesting case of dimension one, where lightcone coordinates are not available but Caron-Huot’s formula does have a nontrivial analog, which played a role in [8]
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