Axions on de Sitter space

Technical Summary: Axions on de Sitter Space

Problem Statement
The paper investigates the quantum field theory (QFT) of a massless, minimally coupled compact scalar field (an axion) on global $D$-dimensional de Sitter space ($dS_D$). While the non-compact version of this theory is well-studied, the compact case—where the field takes values on a circle $S^1$ rather than the real line $\mathbb{R}$—has received limited attention despite its relevance to cosmology (e.g., inflationary models, axion-like particles) and its emergence in string compactifications. The central problem is to construct the full Hilbert space of this theory, determine its symmetry properties under the de Sitter group $SO(D, 1)$, and understand how the compactness of the field alters the vacuum structure and particle content compared to the non-compact case.

Methodology
The authors employ a dual approach, analyzing the theory from both Lorentzian and Euclidean perspectives:

1. Canonical Quantization (Lorentzian):

   • The field is decomposed into a zero-mode sector (spatially constant) and an oscillator sector (spatially varying modes).
   • The oscillator modes are identified with the standard Bunch–Davies modes, which form a Unitary Irreducible Representation (UIR) of $SO(D, 1)$.
   • The zero-mode sector is treated as a quantum mechanical particle on a circle, $L^2(S^1)$, due to the periodicity $\phi \sim \phi + 2\pi$.
   • The authors explicitly construct the conserved charges associated with the $SO(D, 1)$ isometries (rotations and boosts) and analyze their action on the full Hilbert space.
   • Field-strength Wightman two-point functions are computed in various "zero-particle" states to test for Hadamard conditions and symmetry invariance.

2. Path Integral Formulation (Euclidean):

   • The theory is Wick-rotated to the Euclidean $D$-sphere ($S^D$).
   • The authors express the partition function in terms of Harish-Chandra characters of $SO(D, 1)$.
   • They demonstrate that the standard sphere path integral only captures the neutral sector (oscillator Fock space). To access charged sectors, they introduce vertex operator insertions at the poles of the sphere.
   • By summing over these charged sectors, they derive a "decorated" path integral that factorizes into the standard sphere partition function and a quantum mechanical partition function for a rotor.

3. Duality Check:

   • In $D=3$, the authors utilize the duality between a compact scalar and electromagnetism (a 1-form gauge field) to map their results to the Hilbert space of a photon on $dS_3$.

Key Contributions and Results

• Structure of the Hilbert Space:
  The full Hilbert space is not a standard Fock space over a single vacuum. Instead, it is a tensor product:
  $$\mathcal{H} = L^2(S^1) \otimes \mathcal{F}_{\text{osc}}$$
  where $\mathcal{F}_{\text{osc}}$ is the oscillator Fock space and $L^2(S^1)$ describes the quantum mechanics of the zero mode. The states in $L^2(S^1)$ are labeled by an integer momentum $n \in \mathbb{Z}$, corresponding to the conserved $U(1)$ shift charge.

• Vacuum Structure and Normalizability:
  Unlike the non-compact scalar case (where the unique de Sitter-invariant vacuum is non-normalizable), the compact scalar possesses a family of normalizable "zero-particle" states $|\Omega_n\rangle = |n\rangle_{\text{QM}} \otimes |0\rangle_{\text{osc}}$.

  • Only the neutral state $|\Omega_0\rangle$ (where $n=0$) is invariant under the full $SO(D, 1)$ group.
  • Charged states $|\Omega_n\rangle$ ($n \neq 0$) are invariant only under spatial rotations $SO(D)$ but not under de Sitter boosts.

• Observer-Dependent Particle Number:
  A significant finding is that the notion of "particle number" is observer-dependent in charged sectors. While a geodesic observer in the state $|\Omega_n\rangle$ measures zero oscillator quanta, a second geodesic observer related by a de Sitter boost will detect a non-zero number of particles. This occurs because the boost generators mix the zero-mode charge with the oscillator sector, effectively "creating particles" when acting on charged zero-particle states. This effect is absent in the neutral sector and in massive scalar theories.

• Correlation Functions:
  The field-strength two-point functions computed in charged sectors $|\Omega_n\rangle$ satisfy the Hadamard condition (ensuring physical consistency). However, they are not de Sitter invariant at finite times for $n \neq 0$. De Sitter invariance is only recovered asymptotically at early and late times ($t \to \pm \infty$), where the zero-mode contribution decays exponentially.

• Euclidean Path Integral and Thermal Behavior:
  The authors show that the standard Euclidean sphere path integral $Z_{S^D}$ corresponds to the neutral sector. To capture the full Hilbert space, one must sum over vertex operator insertions. This sum yields a decorated partition function:
  $$\tilde{Z}_{S^D} = Z_{\text{QM}} Z_{S^D}$$
  Here, $Z_{\text{QM}}$ is the partition function of a quantum rotor at an effective, dimension-dependent temperature. This provides a Euclidean avatar of the Lorentzian Hilbert space structure.

• Duality in $dS_3$:
  In three dimensions, the compact scalar is dual to a photon. The oscillator sector maps to the photon's UIR, while the scalar's zero-mode sector maps to magnetic monopole sectors in Maxwell theory. The existence of these monopole sectors is topologically permitted because the spatial slice of global $dS_3$ is a 2-sphere ($S^2$), which supports non-trivial magnetic flux.

Significance and Claims
The paper claims to provide the first comprehensive account of the quantum theory of a massless compact scalar on global de Sitter space. Its primary significance lies in:

1. Correcting the Hilbert Space: Demonstrating that the standard Fock space construction is insufficient for compact scalars on $dS$ and that the full Hilbert space includes a discrete tower of charged zero-particle states.
2. Symmetry Breaking of Particle Number: Revealing that de Sitter boosts do not preserve particle number in charged sectors, a phenomenon with no analogue in standard massive QFTs or the non-compact scalar case.
3. Euclidean-Lorentzian Bridge: Resolving the apparent tension between the topology of the Euclidean sphere (which lacks cycles) and the Lorentzian charged sectors by showing how vertex operator insertions in the Euclidean path integral recover the full Lorentzian Hilbert space.
4. Topological Realization: Explicitly linking the scalar zero-mode to magnetic monopoles in the dual $dS_3$ photon theory, grounding the abstract scalar results in a concrete gauge-theoretic context.

The authors maintain a modest tone, noting that while they have established the free theory, the behavior of these charged sectors in interacting theories (e.g., with a potential) and in the presence of dynamical gravity remains an open area for future investigation.