Propagators and Gaussian effective actions in various patches of de Sitter space

Abstract

We consider time-ordered (or Feynman) propagators between two different α−states of a linear de Sitter Quantum Field in the global de Sitter manifold and in the Poincaré patch. We separately examine α − β, In-In and In-Out propagators and find the imaginary contribution to the effective actions. The In-In propagators are real in both the Poincaré patch and in the global de Sitter manifold. On the other side the In-Out propagators at coincident points contain finite imaginary contributions in both patches in even dimensions, but they are not equivalent. In odd dimensions in both patches the imaginary contributions are zero. For completeness, we also consider the Static patch and identify in our construction the state that is equivalent to the Bunch-Davies one in the Poincaré patch.

Quantum fields in the static de Sitter universe

“…There is still the freedom to add to the two-point function a symmetric part that, as such, does not contribute to the commutator. The so-called α-vacua [4,12,16,17,32] belong to this second class of states. Let us briefly sketch their construction in the static patch coordinates.…”

“…When the temperature is an integer multiple of the Hawking-Gibbons temperature, i.e., when β ¼ 2π=N, we may use Eq. (3.12) to derive another representation of the two-point function as a finite sum of Legendre functions [as opposed to the infinite Matsubara-type series (7.9)]; this is obtained by translating the Bunch-Davies maximal analytic two-point function in the imaginary time variable within the analyticity strip ð−2π < Imt < 0Þ (see also [4,33]),…”

“…The nonstationarity of the de Sitter metric indicates that to compute loop quantum corrections one should make use of the Schwinger-Keldysh rather than the Feynman diagrammatic technique; in the first step, initial data have to be imposed on a Cauchy surface at some time t 0 to define the correlation functions. No matter what state is chosen, secular growing infrared contributions appear in the loops (see e.g., [1] for a review); these effects are global and sensitive to the initial conditions [1][2][3][4]. Summarizing, when quantizing fields in curved space-times, the field dynamics may and in general does depend on the choice of coordinates through the choice of the initial data and this is crucial, in particular, for understanding the properties of de Sitter quantum physics.…”

Scalar and spinor effective actions in global de Sitter

“…Just like antide Sitter (AdS) space, the dS space also has a maximal isometry group. As a result, the canonical quantization of free fields is possible in dS space and the effective action can be computed analytically [1][2][3][4][5][6][7][8]. It has been known for a long time that dS space has a one-parameter family of vacua invariant under the dS isometry group [3][4][5].…”

“…Although the effective action of a scalar field in the global patch of dS d space in the in-/out-state formalism has been computed in [8,10], the authors have been focusing on the imaginary part of the effective action. Besides, as far as we are aware there has been no attempts in computing such effective action of a Dirac spinor field in dS space of arbitrary dimension.…”