What is a photon in de Sitter spacetime?

TL;DR

This work shows that in four-dimensional de Sitter space, photon states naturally populate the Hilbert space of generic QFTs as part of the photon UIR of SO(1,4) with $\Delta=2$ and spin 1, even in the absence of a gauge field. By deriving a Källén-Lehmann representation for antisymmetric two-point functions and analyzing a wide class of composite operators built from massive fields, the authors demonstrate that photon states can be interpolated by non-gauge operators and that some of these operators exhibit late-time decay slower than the Maxwell field strength, challenging the notion that photons dominate the infrared. The paper develops both flat-space and de Sitter KL formalisms, provides inversion formulas to extract spectral densities, and establishes non-perturbative bounds on electric and magnetic field correlators in de Sitter, with potential relevance to primordial magnetogenesis. One-loop computations show that the creation of photon states and the enhanced late-time behavior persist in weakly interacting theories, indicating a robust representation-theoretic structure for QFT in curved spacetime with cosmological implications. These results highlight a powerful, symmetry-based approach to QFT in de Sitter that can illuminate the infrared behavior of gauge-like excitations and possibly guide cosmological magnetogenesis scenarios.

Abstract

The states of a single photon in four-dimensional de Sitter (dS) spacetime form a Unitary Irreducible Representation (UIR) of SO(1,4), which we call the photon UIR. While in flat spacetime photons are intimately tied to gauge symmetry, we demonstrate that in de Sitter, photon states emerge generically in any quantum field theory, even without an underlying U(1) gauge field. We derive a Källén-Lehmann representation for antisymmetric tensor two-point functions and show that numerous composite operators constructed from massive free fields can create states in the photon UIR. Remarkably, we find that some of these operators exhibit two-point functions with slower late-time and large-distance decay than the electromagnetic field strength itself, challenging the conventional notion that photons dominate the infrared regime. Using our spectral representation, we establish non-perturbative bounds on the late-time behavior of electric and magnetic fields in de Sitter, with potential implications for primordial magnetogenesis. Through one-loop calculations, we demonstrate that both the creation of photon states and the enhanced late-time large-distance behavior persist in weakly interacting theories.

Figures (10)

• Figure 1: A representation of the salient features of the Källén-Lehmann decomposition. The blue vertical line is the contour of integration over the principal series, the crosses are the poles of the principal series spectral density, the empty circles are complementary series contributions, which come as a discrete sum in all known examples, and the full orange circles represent the photon UIR. In eq. (\ref{['eq:latetimeB']}) we indicate the set of scaling dimensions corresponding to the red circles as $\{\Delta_C\}$ and the blue crosses as $\{\Delta_P\}$. $\Delta_c$ and $\Delta_p$ are the scaling dimensions with lowest real part of the sets $\{\Delta_C\}$ and $\{\Delta_P\}$ respectively. In the particular configuration shown in this picture, the dominant late time power law is set by $\Delta_c$. This figure could apply to both the parity even $(+)$ and odd $(-)$ contributions.
• Figure 2: A diagrammatic representation of eq. (\ref{['eq:phi2']}). Because of the orthogonality of Gegenbauer polynomials on the sphere, diagrams on the sphere factorize.
• Figure 3: The spectral density (\ref{['eq:rhophi']}) for $\phi$ starting in the principal series for various values of the coupling $g$. We use the parametrization $\Delta=\frac{3}{2}+i\mu$ and fix $\mu_\chi=\frac{1}{2}$, $m_\phi R=2$ and $c_m=0$. Taking $g\to0$ the spectral density approaches a Dirac delta function around $\mu=\mu_\phi$. For finite coupling $g$, the spectral density is instead peaked around the shifted value $\mu=\text{Im}[\Delta_1]$ (eq. (\ref{['eq:imdelta']})). The position of the peak depends on the renormalization constant $c_m$, while the width is set by Re$[\Delta_1]-\frac{3}{2}$ and is scheme-independent (see eq. (\ref{['eq:redelta']})).
• Figure 4: The poles in the spectral density (\ref{['eq:rhophi']}) which have $O(1)$ residue. The dots represent the poles as $g\to0$, the free theory. In the strict $g=0$ case, they pinch the contour over the principal series and effectively become two delta functions. The crosses represent the position of the poles at order $g^2$, $\Delta_1$ and $\Delta_2$ (defined in eq. (\ref{['eq:d1d2def']})) and their shadows. The blue line is the principal series, while the red line is the complementary series. For this figure we chose $\chi$ and $\phi$ to start off in the principal series in the free theory.
• Figure 5: The spectral density (\ref{['eq:rhophi']}) for $\phi$ starting in the complementary series. We use the parametrization $\Delta=\frac{3}{2}+i\mu$ and fix $\mu_\chi=\frac{1}{2}$, $m_\phi R=\frac{1}{2}$ and the renormalization constant $c_m=0$. Changing $g$ leads to indistinguishable differences in this plot, if we stay in the perturbative regime.