Causality Constraints in Conformal Field Theory (1509.00014v2)

Abstract: Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In d-dimensional conformal field theory, we show how such constraints are encoded in crossing symmetry of Euclidean correlators, and derive analogous constraints directly from the conformal bootstrap (analytically). The bootstrap setup is a Lorentzian four-point function corresponding to propagation through a shockwave. Crossing symmetry fixes the signs of certain log terms that appear in the conformal block expansion, which constrains the interactions of low-lying operators. As an application, we use the bootstrap to rederive the well known sign constraint on the $(\partial\phi)^4$ coupling in effective field theory, from a dual CFT. We also find constraints on theories with higher spin conserved currents. Our analysis is restricted to scalar correlators, but we argue that similar methods should also impose nontrivial constraints on the interactions of spinning operators.

• The paper reveals that crossing symmetry enforces specific sign constraints on logarithmic terms in Euclidean correlators, linking causality with operator dynamics.
• It employs a Lorentzian shockwave four-point function to analytically derive bounds on low-lying operators using the conformal bootstrap methodology.
• The study rederives the classic positive constraint on the φ⁴ coupling and extends implications to theories with conserved higher-spin currents, informing holographic duality.

Causality Constraints in Conformal Field Theory

The paper "Causality Constraints in Conformal Field Theory" by Thomas Hartman, Sachin Jain, and Sandipan Kundu investigates how causality constraints in quantum field theory (QFT) can inform our understanding of operator interactions in conformal field theories (CFTs). Causality, a fundamental principle in QFT, dictates that effects cannot precede their causes, and this property manifests in the analytic structure of correlators in both Lorentzian and Euclidean signatures. This work explores these causality constraints using the conformal bootstrap approach, a technique that exploits crossing symmetry to derive bounds on CFT data.

Key Results

The authors demonstrate that, in a CFT, crossing symmetry can impose nontrivial restrictions on the signs of certain logarithmic terms appearing in the conformal block expansion of Euclidean correlators. These constraints, which are analogous to those imposed by causality in Lorentz-invariant QFT, can be derived analytically from the conformal bootstrap equations. The primary focus is on a particular bootstrap setup involving a Lorentzian four-point function describing the propagation through a shockwave, a methodological choice that enables the authors to translate causality into specific constraints on the interactions of low-lying operators.

A significant numerical result is the rederivation of the classic causality-imposed constraint on the $(\phi)^4$ coupling in effective field theory, showing its positive nature. This paper goes further by identifying constraints on theories with conserved higher-spin currents, suggesting avenues for exploring theories with such symmetries.

Implications

The research carries implications for our understanding of both the theoretical foundations and the practical applications of conformal field theories, particularly in light of their role in the AdS/CFT correspondence. The causality constraints provide insights into the connections between low-energy effective actions and their high-energy completions within a CFT framework. Furthermore, these results clarify the role of crossing symmetry in maintaining causal consistency and offer tools to probe the dynamics of operators across various kinematic limits.

The exploration of shockwave states finds its relevance in studying the dynamics similar to that encountered in the quantum gravity context, and this paper's approach could inform future investigations into the holographic duality between gravitational theories and boundary CFTs.

Future Developments

The constraints proposed here open new research directions concerning emergent geometry in large-N CFTs, with the potential to derive more rigorous bounds involving stress tensor correlation functions. An important future endeavor is to extend the methodology to incorporate external operators with spin, potentially deriving new constraints on higher-spin interactions.

Additionally, the connection to large-N limits hints at possible applications within the broader context of quantum gravity, where the interplay between causality, analyticity, and crossing symmetry could reveal more about the emergence of spacetime geometry from quantum degrees of freedom.

Overall, the paper successfully bridges causality constraints in QFT with CFT dynamics by leveraging the conformal bootstrap methodology, providing a platform for deeper explorations into both theoretical constructs and their implications across various domains of high-energy physics.