Anti-de Sitter, plane waves and quantum field theory (2509.09257v1)

Abstract: We present a new plane-wave expansion of the Wightman functions of anti de Sitter scalar fields and showcase its conceptual and technical importance in AdS quantum field theory. We deduce from it a new integral representation of the Feynman propagator which helps in clarifying the relation between AdS Euclidean and Lorentzian Feynman diagrams in concrete examples. The plane-wave expansion makes it possible also to demonstrate numerous new nontrivial formulas for Bessel and Legendre functions, and we provide two examples.

• The paper provides a manifestly covariant plane-wave expansion for AdS scalar fields that unifies analytic structures and clarifies boundary conditions.
• It introduces novel integral representations and special function identities, bridging Euclidean and Lorentzian frameworks for Feynman diagrams.
• The results facilitate practical computations in the Poincaré patch, offering insights for applications in the AdS/CFT correspondence and bulk reconstruction.

Plane-Wave Expansions and Analytic Structure in Anti-de Sitter Quantum Field Theory

Introduction

This work addresses a longstanding gap in the formalism of quantum field theory (QFT) on anti-de Sitter (AdS) space: the absence of a manifestly covariant plane-wave expansion for Wightman functions of scalar fields. The author constructs such an expansion, paralleling the well-established de Sitter (dS) case, and demonstrates its utility for both conceptual understanding and technical computations in AdS QFT. The new representation clarifies the analytic structure of AdS correlation functions, provides a bridge between Euclidean and Lorentzian Feynman diagrams, and yields novel integral identities for special functions.

Analytic Structure and Plane-Wave Expansions

The paper begins by reviewing the role of plane-wave expansions in Minkowski and de Sitter QFTs. In Minkowski space, the two-point Wightman function is a superposition of plane waves, with analyticity in complexified spacetime domains reflecting the spectrum condition. In dS space, analogous expansions exist, with plane waves constructed as $(\xi \cdot z)^\lambda$ for $\xi$ on the lightcone and $z$ in complexified dS. The resulting two-point function is a holomorphic integral over the cone, manifestly dS-invariant, and expressible in terms of Legendre functions.

For AdS, the situation is more subtle due to the global structure of the manifold, the presence of closed timelike curves, and the necessity of working on the universal covering space. The author constructs AdS plane waves as $(z \cdot \zeta)^\lambda$, with $z$ in complexified AdS and $\zeta$ on a complexified null cone. The key technical advance is the identification of appropriate integration cycles in relative homology, ensuring invariance and analyticity for generic $\lambda$.

Explicit Formulas and Special Function Identities

The main result is the explicit plane-wave expansion for the AdS Wightman function: $$W^{AdS_d}_{\lambda}(z_1, z_2) = C_d(\lambda) \int_{\gamma(z_1)} (z_1 \cdot \zeta)^\lambda (z_2 \cdot \zeta)^{1-d-\lambda} d\mu_\gamma(\zeta)$$ where $C_d(\lambda)$ is a normalization constant, and $\gamma(z_1)$ is a cycle in the complexified null cone avoiding the singularity at $z_1 \cdot \zeta = 0$. This representation is valid in the domain of holomorphy corresponding to positive energy and is manifestly AdS-invariant.

The expansion leads to new integral representations for Legendre functions of the second kind and, in particular, to a nontrivial multiplication theorem for $Q_\lambda(x \cosh u)$ in $d=2$. The formalism generalizes to higher dimensions and to tensorial and spinorial fields via differential operators.

Poincaré Patch, Källén-Lehmann Representation, and Feynman Propagators

The author analyzes the Poincaré foliation of AdS, which is crucial for practical computations and for connecting with the AdS/CFT correspondence. In these coordinates, the two-point function admits a diagonal (Källén-Lehmann-type) representation: $$W^{AdS_d}_\lambda(z_1, z_2) = \int_0^\infty W^{M_{d-1}}_m(x_1, x_2) J_\nu(m u) J_\nu(m u') dm^2$$ where $W^{M_{d-1}}_m$ is the Minkowski Wightman function in $d-1$ dimensions, and $J_\nu$ are Bessel functions. This leads directly to a new integral representation for the AdS Feynman propagator in the Poincaré patch: $$G^{AdS_d}_\lambda(x_1, x_2) = \int_0^\infty G^{M_{d-1}}_m(x_1, x_2) J_\nu(m u) J_\nu(m u') dm^2$$ This representation is particularly useful for relating Euclidean and Lorentzian computations and for explicit evaluation of Feynman diagrams.

Implications for AdS Feynman Diagrams and Wick Rotation

A significant application of the new formalism is the clarification of the relationship between Euclidean and Lorentzian AdS Feynman diagrams. The integral representation of the propagator allows for a direct Wick rotation, showing that certain diagrams (e.g., "banana" diagrams) computed in Euclidean AdS coincide with their Lorentzian counterparts in the Poincaré patch. The author provides explicit calculations for one- and two-line diagrams, demonstrating AdS invariance and agreement with Euclidean results.

The formalism also suggests a prescription for computing Witten diagrams in real AdS space, with external legs given by boundary values of the constructed plane waves. The question of whether all such diagrams integrated over the Poincaré patch yield AdS-invariant results is left open, but the evidence from explicit examples is positive.

Theoretical and Practical Implications

The construction of a manifestly covariant plane-wave expansion for AdS scalar fields fills a foundational gap in the analytic approach to AdS QFT. It provides a unified framework for understanding the analytic structure of correlation functions, the role of boundary conditions, and the connection between Euclidean and Lorentzian formulations. The explicit integral representations facilitate practical computations of Feynman diagrams and may have implications for the AdS/CFT correspondence, particularly in the context of bulk reconstruction and the paper of singularities in CFT correlators.

The new identities for special functions arising from the formalism may also find applications in mathematical physics, particularly in the paper of integral transforms and spectral theory on symmetric spaces.

Conclusion

This work establishes a comprehensive plane-wave formalism for AdS QFT, analogous to the well-developed dS case, and demonstrates its utility for both conceptual and technical aspects of the theory. The results clarify the analytic structure of AdS correlation functions, provide new computational tools for Feynman diagrams, and yield novel mathematical identities. Future directions include the extension to interacting fields, higher-spin and tensorial cases, and a systematic paper of the invariance properties of more general diagrams in the Poincaré patch. The formalism is expected to have significant impact on both the mathematical foundations and practical computations in AdS QFT and related areas.