Abstract
I derive a formula for the coupling-constant derivative of the coefficients of the operator product expansion (Wilson OPE coefficients) in an arbitrary curved space, as the natural extension of the quantum action principle. Expanding the coefficients themselves in powers of the coupling constants, this formula allows to compute them recursively to arbitrary order. As input, only the OPE coefficients in the free theory are needed, which are easily obtained using Wick’s theorem. I illustrate the method by computing the OPE of two scalars ϕ in hyperbolic space (Euclidean Anti-de Sitter space) up to terms vanishing faster than the square of their separation to first order in the quartic interaction gϕ4, as well as the OPE coefficient at second order in g.
Highlights
Right-hand side of (1.1) all operators OB up to a fixed dimension [OB] ≤ ∆, the difference between left- and right-hand side vanishes like d∆−, ki=1[OAi] where d is the largest distance between y and any of the xi
I derive a formula for the coupling-constant derivative of the coefficients of the operator product expansion (Wilson OPE coefficients) in an arbitrary curved space, as the natural extension of the quantum action principle
It seems possible to define a quantum field theory not by its correlation functions, but instead by the OPE coefficients which encode the algebraic properties of the theory and the one-point expectation values OB(y) Ω, which determine the quantum state Ω
Summary
Given the geodesic from x to y and the tangent vector ξ = ξ(x, y) along this geodesic at the point x, we have expx(ξ) = y This is well-defined in a sufficiently small neighbourhood of x, called normal geodesic neighbourhood, where a unique geodesic between x and y exists, and the length of the tangent vector |ξ| ≡ gμν(x)ξμξν is equal to the geodesic length d(x, y). If M is complete and has non-positive sectional curvature (e.g., flat Euclidean space Rn or hyperbolic space Hn), the exponential map is even globally well-defined and the normal geodesic neighbourhood of any point is equal to the whole M , or if M is not connected to its universal cover [45, Ch. 1].
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