Abstract
We describe in more detail the general relation uncovered in our previous work between boundary correlators in de Sitter (dS) and in Euclidean anti-de Sitter (EAdS) space, at any order in perturbation theory. Assuming the Bunch-Davies vacuum at early times, any given diagram contributing to a boundary correlator in dS can be expressed as a linear combination of Witten diagrams for the corresponding process in EAdS, where the relative coefficients are fixed by consistent on-shell factorisation in dS. These coefficients are given by certain sinusoidal factors which account for the change in coefficient of the contact sub-diagrams from EAdS to dS, which we argue encode (perturbative) unitary time evolution in dS. dS boundary correlators with Bunch-Davies initial conditions thus perturbatively have the same singularity structure as their Euclidean AdS counterparts and the identities between them allow to directly import the wealth of techniques, results and understanding from AdS to dS. This includes the Conformal Partial Wave expansion and, by going from single-valued Witten diagrams in EAdS to Lorentzian AdS, the Froissart-Gribov inversion formula. We give a few (among the many possible) applications both at tree and loop level. Such identities between boundary correlators in dS and EAdS are made manifest by the Mellin-Barnes representation of boundary correlators, which we point out is a useful tool in its own right as the analogue of the Fourier transform for the dilatation group. The Mellin-Barnes representation in particular makes manifest factorisation and dispersion formulas for bulk-to-bulk propagators in (EA)dS, which imply Cutkosky cutting rules and dispersion formulas for boundary correlators in (EA)dS. Our results are completely general and in particular apply to any interaction of (integer) spinning fields.
Highlights
We have seen that while conformal symmetry alone does not distinguish between boundary correlators in Euclidean anti-de Sitter (EAdS) and the Bunch-Davies vacuum of de Sitter (dS), the requirement of unitarity differentiates between them by imposing different constraints on the spectrum of scaling dimensions of states and the operator product expansion (OPE) coefficients
While in EAdS unitarity requires the OPE coefficients to be real [114], in dS this is not a requirement: at linear order in the coupling they are proportional to the sine factor (3.72), which is a function of the scaling dimensions and the spins of the particles
While any diagram contributing to a boundary correlator in dS can be written as a linear combination of EAdS Witten diagrams [36], since the unitary values of the scaling dimensions in EAdS and dS do not coincide and the couplings in EAdS and dS differ by sinusoidal factors (3.72), it should be emphasised that such Witten diagrams are generally not generated by a unitary theory in anti-de Sitter space
Summary
Internal legs of Witten diagrams, which connect two bulk points, are associated bulkto-bulk propagators which satisfy the wave equation with a Dirac delta function source:. Analogous to the exponential plane wave expansion in flat space, it is useful to decompose bulk-to-bulk propagators in a basis of bi-local Harmonic functions ΩAν,dJS. These are trace- and divergence-free, and provide a complete basis of orthogonal solutions to the homogeneous wave equation (see appendix D of [72]),. For the Dirichlet boundary condition, which is normalisable, the decomposition of the bulk-to-bulk propagator for a spin-J field in terms of harmonic functions ΩAν,dJS is given by the spectral integral [72]: ΠA∆d+S,J (x; x) =. In recent years it has been shown [36,37,38] that the above harmonic analysis on AdS space can be extended to de Sitter space via analytic continuation, which we review
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