De Sitter Green's Functions in Quantum Field Theory

Updated 21 August 2025

De Sitter Green's functions are two-point correlators that solve wave equations on a maximally symmetric de Sitter background, essential for understanding quantum fields in curved spacetime.
They incorporate diverse regularization schemes and embed techniques to manage UV/IR divergences and reveal hidden SL(2,ℝ)×SL(2,ℝ) symmetries underlying quasinormal mode spectra.
Applications range from precise predictions in cosmological perturbations and quantum gravity to innovative approaches in holography and entanglement entropy analysis.

De Sitter Green's functions are fundamental two-point correlation functions that solve the relevant wave or propagation equations on de Sitter spacetime. They encode the structural, causal, and quantum properties of fields—scalars, vectors, and tensors—on this maximally symmetric, constant curvature background, and underlie both the dynamics of cosmological perturbations and the structure of quantum states in the inflationary universe and related settings. The construction, interpretation, and regularization of these Green's functions is an essential component of quantum field theory in curved backgrounds, with far-reaching implications in cosmology, gravitational theory, and holography.

1. Regularization and Renormalization Schemes

A central challenge in quantum field theory on de Sitter space is the treatment of ultraviolet (UV) and infrared (IR) divergences in Green's functions arising from loop corrections. Three regularization schemes are shown to produce equivalent physical answers for the one-loop corrected scalar propagator when UV and IR cutoffs are implemented carefully (Xue et al., 2011):

Brute-force cutoff: Introduces explicit comoving IR and physical UV momentum cutoffs. The UV cutoff corresponds to a fixed physical energy, Λ₍UV₎, implemented via the comoving value Λ_comoving = a(η) Λ₍UV₎ (a is the scale factor), while the IR cutoff, Λ₍IR₎, is fixed in comoving coordinates.
Dimensional regularization: Extends spacetime dimension to d = 4 + δ, computes the loop integrals, and extracts divergences in the δ → 0 limit. Scalar mode functions acquire δ-dependent logarithmic corrections, shifting the logarithms in the result.
Pauli-Villars regularization: Introduces massive regulator fields with coefficients chosen to cancel divergences. The regulator fields separate momentum integrals into UV and IR regions.

For a massless, minimally coupled scalar, the one-loop diagram correction to the Green's function in all three schemes takes the form

$G_T(n=0) \sim \frac{H^2}{p^3}\bigg[2\log\left(\frac{p}{\mu_{IR}}\right) + 2\log\left(\frac{H}{\mu_{UV}}\right)\bigg]$

where $H$ is the Hubble constant and $p$ is the external momentum (Xue et al., 2011). The UV cutoff is fixed in physical coordinates while the IR cutoff is in comoving units, reflecting the role of expansion in stretching wavelengths.

Implementing these regularizations consistently is critical for robust predictions of the power spectrum and higher-order correlators of cosmological perturbations.

2. Symmetry Structures and Conformal Mechanics

The static patch Green's functions for scalars and gravitons are controlled by hidden $SL(2,\mathbb{R}) \times SL(2,\mathbb{R})$ symmetry algebras generated by conformal Killing vectors of the de Sitter static patch (Anninos et al., 2011). This algebraic structure organizes both the analytic structure and spectral features of the Green's function:

The retarded Green's function is determined by the ratio of coefficients in the solution to the wave equation with appropriate (ingoing) horizon boundary conditions.
Poles in the frequency domain correspond to quasinormal modes, which are organized by the algebra's ladder operators acting on primary states.
In the special case of conformally coupled scalars and four-dimensional gravitons, the symmetry algebra is enhanced and admits a supersymmetric structure, resulting in an evenly spaced (degenerate) quasinormal mode spectrum.

Moreover, the Green's functions can be reproduced from a worldline conformal quantum mechanics model by "level-matching" left- and right-moving sectors. This connection points to the existence of an alternative (one-dimensional) dual description of static patch de Sitter physics.

3. Embedding Techniques and Causal Structure

A powerful method for constructing Green's functions on de Sitter spacetime is the embedding formalism, which exploits the maximal symmetry by realizing de Sitter as a hyperboloid in a higher-dimensional flat space (Chu, 2013, Chu, 2013). For a minimally coupled scalar field, the Green’s function $G_d[x,x']$ is obtained by integrating the flat ambient (d+1)-dimensional Green’s function $\bar{G}_{d+1}$ against a line source emanating orthogonally from the de Sitter hypersurface: $G_d[x,x'] = \int_0^\infty d\rho' \, (H\rho')^{d-2} \bar{G}_{d+1}(X[1/H, x] - X'[{\rho'}, x'])$ where $X$ denotes the embedding coordinates (Chu, 2013).

For massive fields, the integral kernel is weighted by a charge density determined by the extra-dimensional zero mode equation, typically yielding a solution in terms of Bessel functions (Chu, 2013). The embedding approach makes the separation between null-cone and tail (inside light-cone) terms manifest: the ambient line source naturally generates both the direct part and the tail contribution of the retarded Green's function.

This reveals how de Sitter causality is inherited from the causal structure of flat space and enables direct calculations without solving the wave equation in de Sitter coordinates.

4. Regularization of Green’s Functions and Trace Anomaly

The Green’s function in position space, when computed naïvely, often exhibits logarithmic infrared divergences, especially for massless or minimally coupled fields (Zhang et al., 2019). The adiabatic regularization method addresses this by subtracting the high-frequency, WKB-approximated divergent pieces from the exact mode expansion at the appropriate adiabatic order. The minimal necessary order depends on the field's coupling:

Second order is required for minimal coupling ( $\xi = 0$ )
Zeroth order suffices for conformal coupling ( $\xi = 1/6$ ).

Fourier transforming the regularized power spectrum yields UV- and IR-finite Green's functions even for general mass and coupling parameters. Importantly, in the massless, conformal limit, the regularized Green’s function and stress tensor vanish, demonstrating the absence of a trace anomaly if the minimal subtraction rule is correctly implemented (Zhang et al., 2019).

5. Gravitational and Gauge Field Green’s Functions

Green’s functions for vector and tensor fields on de Sitter space exhibit additional features due to gauge invariance and residual symmetry:

Vector field propagators split into transverse (gauge-invariant) and longitudinal (gauge-dependent) parts. The transverse part has a smooth massless limit, agreeing with known flat-space results and vanishing at antipodal points in the Ricci-flat limit (Narain, 2014).
On the gravitational side, Green’s functions for the linearized (physical) graviton can be explicitly constructed via mode expansions on the Euclidean four-sphere, using Wick rotation from Lorentzian signature. The resulting two-point functions may not be positive-definite on the full physical phase space due to contributions from the finite-dimensional space of Killing forms. Positivity can be restored by projecting out problematic low-energy modes, yielding a Hadamard state that is fully gauge-invariant and positive but invariant only under the O(4) subgroup of the isometry group (Gérard et al., 1 May 2024).
For the Maxwell field, the conformal invariance allows direct use of the scalar Minkowski Green’s function in conformally flat charts, reducing complexity and facilitating practical calculations (Faci et al., 2011).

6. Holography, Replica Trick, and Entanglement

De Sitter Green’s functions play a central role in formulations of holography in de Sitter space:

In static patch holography, the proposal for holographic entanglement entropy is built from two-point functions of twist operators, which can be written as (combinations of) de Sitter scalar Green’s functions (Ruan et al., 20 Aug 2025). The entropy functional

$S_{EE} = -C \log G_{dS}(x, x')$

with general $G_{dS}$ a linear combination of global and antipodal contributions, recovers the geodesic limit for large operator dimension and overcomes the failure of the naive Ryu–Takayanagi surface prescription.

Strong subadditivity of the entropy imposes sharp constraints on the allowed linear combinations of Green's functions and the location of the stretched horizon. Only a restricted subset of parameter space yields a function with concavity properties required for von Neumann entropy.
In dS₃/CFT₂ holography, bulk Green's functions associated with local excitations are not simply obtainable by Wick rotating the AdS correlators. The conjugation operation in the dual CFT must be modified to relate global Ishibashi states built from primaries of different conformal dimensions, combining antipodal symmetries to produce the correct dS₃ two-point function (Doi et al., 23 May 2024).

7. Applications and Theoretical Significance

De Sitter Green’s functions are not only technical tools for calculations but encode profound structural information:

They determine how loop corrections affect primordial spectra, modifying predictions for non-Gaussianity in the cosmic microwave background (Xue et al., 2011).
The SL(2,ℝ) × SL(2,ℝ) symmetry structure of static patch Green’s functions underpins the algebraic origin of quasinormal spectra and supports the existence of putative dual worldline conformal quantum mechanics models (Anninos et al., 2011).
Green’s functions built from the embedding formalism yield explicit, causal, geometrically intuitive expressions, facilitating calculations in quantum gravitational and semiclassical settings (Chu, 2013, Chu, 2013).
In quantum gravity and holography, the properties and definitions of Green’s functions are central to consistent formulations of entanglement entropy and the physical interpretation of de Sitter holography, with strong subadditivity providing a stringent test of candidate frameworks (Ruan et al., 20 Aug 2025).

The paper and construction of de Sitter Green’s functions thus integrate representation theory, microlocal analysis, embedding geometry, and quantum information principles, forming a critical foundation for both fundamental and phenomenological investigations in modern cosmology and gravitational theory.