Averaged Null Energy Condition from Causality (1610.05308v1)

Abstract: Unitary, Lorentz-invariant quantum field theories in flat spacetime obey microcausality: commutators vanish at spacelike separation. For interacting theories in more than two dimensions, we show that this implies that the averaged null energy, $\int du T_{uu}$, must be positive. This non-local operator appears in the operator product expansion of local operators in the lightcone limit, and therefore contributes to $n$-point functions. We derive a sum rule that isolates this contribution and is manifestly positive. The argument also applies to certain higher spin operators other than the stress tensor, generating an infinite family of new constraints of the form $\int du X_{uuu\cdots u} \geq 0$. These lead to new inequalities for the coupling constants of spinning operators in conformal field theory, which include as special cases (but are generally stronger than) the existing constraints from the lightcone bootstrap, deep inelastic scattering, conformal collider methods, and relative entropy. We also comment on the relation to the recent derivation of the averaged null energy condition from relative entropy, and suggest a more general connection between causality and information-theoretic inequalities in QFT.

• The paper demonstrates that microcausality in Lorentz-invariant QFTs implies a positive averaged null energy condition (ANEC).
• It utilizes the lightcone operator product expansion to derive a sum rule that isolates the inherently positive contribution of the null energy operator.
• The study extends the analysis to higher-spin operators, yielding stringent constraints on 3-point function couplings in conformal field theories.

Summary of "Averaged Null Energy Condition from Causality"

The paper by Hartman, Kundu, and Tajdini explores the implications of causality in quantum field theories (QFTs) on the averaged null energy condition (ANEC). This paper is anchored in the framework of unitary, Lorentz-invariant QFTs in flat spacetime, where microcausality is obeyed. Specifically, they demonstrate that for interacting theories in more than two dimensions, microcausality implies a positive averaged null energy: $\int du T_{uu} \geq 0$.

The central premise is that non-local operators such as the averaged null energy emerge in operator product expansions (OPEs) of local operators in the lightcone limit and contribute significantly to $n$-point functions. The paper derives a sum rule that isolates this contribution, which is intrinsically positive. Additionally, the argument is extended to higher-spin operators other than the stress tensor, leading to an infinite series of constraints of the form $\int du X_{uuu\cdots u} \geq 0$. These constraints impose novel inequalities on the coupling constants of spinning operators in conformal field theories (CFTs).

By leveraging the connection between causality and information-theoretic inequalities in QFT, the authors further establish a relationship with recent derivations of the ANEC. They conjecture that causality in QFT has broader links to entanglement and information-theoretic constraints, suggesting that these bounds might be viewed as complementary perspectives on quantum field theory behavior.

Key Numerical Results and Claims

1. Lightcone OPE and ANEC: The paper provides an operator expression for the lightcone OPE, showing that the averaged null energy operator appears as a leading contribution. The condition $\int du T_{uu} \geq 0$ in states follows from the positivity of certain 4-point functions derived via causality arguments.
2. Causality and Sum Rule: The authors derive a sum rule that expresses the average null energy expectation value as an integral, which is inherently positive due to the causality constraints imposed by microcausality in QFTs.
3. Higher Spin Constraints: The generalization to higher-spin operators indicates that operators of even spin greater than two must satisfy positivity constraints, producing stronger bounds on 3-point function couplings than methods like deep inelastic scattering, the lightcone bootstrap, and previous conformal collider bounds.

Implications and Future Directions

This work has substantial implications for theoretical physics, particularly in understanding QFTs through geometric and information-theoretical lenses. The insistence on the positivity of the averaged null energy condition constrains the types of interactions and fields that can exist in a consistent QFT or CFT. This provides a robust check on potential theories and could guide the search for new, physically meaningful models.

On a more speculative note, the connection between causality and quantum information posited here hints at possible deeper insights into the structure of spacetime. Understanding these relationships could influence the ongoing exploration of quantum gravity, particularly through frameworks like AdS/CFT correspondence.

For future developments, the paper suggests examining other regimes and setups within higher dimensional and non-conformal theories. It also proposes further exploration of potential connections with entropic principles in holographic settings, which could enrich both our conceptual framework and technical toolkit in theoretical physics. Overall, while shedding significant light on causality’s role in enforcing energy conditions, this work opens several avenues for advanced research in the field of quantum field theories and gravity.