We study two-point functions of symmetric traceless local operators in the
bulk of de Sitter spacetime. We derive the K”all’en-Lehmann spectral
decomposition for any spin and show that unitarity implies its spectral
densities are nonnegative. In addition, we recover the K”all’en-Lehmann
decomposition in Minkowski space by taking the flat space limit. Using harmonic
analysis and the Wick rotation to Euclidean Anti de Sitter, we derive an
inversion formula to compute the spectral densities. Using the inversion
formula, we relate the analytic structure of the spectral densities to the
late-time boundary operator content. We apply our technical tools to study
two-point functions of composite operators in free and weakly coupled theories.
In the weakly coupled case, we show how the K”all’en-Lehmann decomposition is
useful to find the anomalous dimensions of the late-time boundary operators. We
also derive the K”all’en-Lehmann representation of two-point functions of
spinning primary operators of a Conformal Field Theory on de Sitter.
Examining the Conclusions and Outlining a Future Roadmap
The conclusions of the text are as follows:
- K”all’en-Lehmann spectral decomposition holds for any spin in the bulk of de Sitter spacetime.
- Unitarity implies that the spectral densities of the K”all’en-Lehmann decomposition are nonnegative.
- By taking the flat space limit, the K”all’en-Lehmann decomposition in Minkowski space can be recovered.
- Using harmonic analysis and the Wick rotation to Euclidean Anti de Sitter, an inversion formula can be derived to compute the spectral densities.
- The analytic structure of the spectral densities is related to the late-time boundary operator content.
- The K”all’en-Lehmann decomposition is useful for finding the anomalous dimensions of late-time boundary operators in weakly coupled theories.
- The K”all’en-Lehmann representation of two-point functions of spinning primary operators of a Conformal Field Theory on de Sitter can be derived.
Future Roadmap
Potential Challenges
- Further exploration and understanding of the K”all’en-Lehmann spectral decomposition in different physical systems and scenarios may present challenges.
- The derivation and application of the inversion formula to compute spectral densities may require advanced mathematical techniques and analysis.
- Investigating the analytic structure of the spectral densities and its connection to the late-time boundary operator content could involve complex calculations and modeling.
- Determining the anomalous dimensions of late-time boundary operators in weakly coupled theories may require extensive computations and numerical methods.
- Deriving the K”all’en-Lehmann representation for two-point functions of spinning primary operators in a Conformal Field Theory on de Sitter may involve intricate mathematical formalism and considerations.
Potential Opportunities
- Further understanding and application of the K”all’en-Lehmann spectral decomposition could provide valuable insights into the behavior of various physical systems and theories.
- The nonnegativity of spectral densities implied by unitarity opens up possibilities for exploring new properties and constraints in different contexts.
- Investigating the flat space limit of the K”all’en-Lehmann decomposition and its relation to Minkowski space can lead to a deeper understanding of the connections between different spacetimes.
- The derivation and use of the inversion formula to compute spectral densities provides a powerful tool for analyzing and modeling various systems.
- Exploring the relationship between the analytic structure of spectral densities and the late-time boundary operator content can offer insights into the underlying dynamics and symmetries of the system.
Roadmap Summary
- Further investigate the K”all’en-Lehmann spectral decomposition in different physical systems. Consider its implications and applications in diverse scenarios.
- Explore and understand the nonnegative nature of the spectral densities implied by unitarity. Investigate its consequences and potential constraints in different contexts.
- Investigate the relationship between the K”all’en-Lehmann decomposition in de Sitter spacetime and its counterpart in Minkowski space through the flat space limit. Understand the connections between different spacetimes.
- Derive and utilize the inversion formula to compute spectral densities. Apply it to analyze and model a range of systems.
- Study the relationship between the analytic structure of spectral densities and the late-time boundary operator content. Use this understanding to gain insights into the system’s dynamics and symmetries.
- Apply the K”all’en-Lehmann decomposition in weakly coupled theories to find the anomalous dimensions of late-time boundary operators. Explore its implications and applications in these scenarios.
- Derive the K”all’en-Lehmann representation for two-point functions of spinning primary operators in a Conformal Field Theory on de Sitter. Understand its implications and connections to other field theories.