Equivalence between Euclidean and in-in formalisms in de Sitter QFT

We study the relation between two sets of correlators in interacting quantum field theory on de Sitter space. The first are correlators computed using in-in perturbation theory in the expanding cosmological patch of de Sitter space (also known as the conformal patch, or the Poincaré patch), and for which the free propagators are taken to be those of the free Euclidean vacuum. The second are correlators obtained by analytic continuation from Euclidean de Sitter; i.e., they are correlators in the fully interacting Hartle-Hawking state. We give an analytic argument that these correlators coincide for interacting massive scalar fields with any $m^2>0$. We also verify this result via direct calculation in simple examples. The correspondence holds diagram by diagram, and at any finite value of an appropriate Pauli-Villars regulator mass $M$. Along the way, we note interesting connections between various prescriptions for perturbation theory in general static spacetimes with bifurcate Killing horizons.

Figure 1

Standard Carter-Penrose diagram of de Sitter space. Region I is the static patch, and the Poincaré patch consists of regions I and II. The causal pasts of points $X_1$ and $X_2$ are the shaded regions. See Sec. 2a for details.

Figure 2

Carter-Penrose diagram of de Sitter space with $\theta =\mathrm{const}$ surfaces (schematically) indicated by solid lines and $t=\mathrm{const}$ surfaces by dashed lines.

Figure 3

Carter-Penrose diagram of de Sitter space with timelike geodesics from point $O$ drawn with solid lines and spacelike geodesics from it drawn with dashed lines.

Figure 4

A deformation of the Euclidean contour.

Figure 5

Singularities and branch cuts of propagators.

Figure 6

Deformed contour for external points at $t_i$ with finite imaginary parts.

Figure 7

$t$-integration contour for the in-in formalism in the static patch. The open circles denote external points at times $t_i$.

Figure 8

Typical $\lambda$ contour for the in-in formalism in the Poincaré patch. All external points have $\lambda >{\lambda }_f$.

Figure 9

Deformed contour for external points with finite imaginary parts in the Poincaré patch.

Figure 10

Singularities in the complex ${\lambda }_1$ plane. Two external points lie at ${\tilde{\lambda }}_1$, ${\tilde{\lambda }}_3$ and two internal points lie at ${\lambda }_2$, ${\lambda }_3$ as indicated by the open circles. Filled circles are singularities and dashed lines are branch cuts.

Figure 11

The ${\lambda }_1$ contour in Fig. 10 deformed as much to the right as possible for the given values of the spatial coordinates as described in the text.

Figure 12

A Feynman diagram in which arrows indicate past relations as described in the text, determined by some particular set of spatial coordinates.

Figure 13

The 1-loop corrections to the propagator.