Negative Average Null Energy in QFT

• Negative average null energy is a quantum phenomenon where the integrated null-projected energy density becomes negative, defying classical energy conditions.
• Smeared and double-smeared formulations impose rigorous bounds on negative energy in QFT, balancing quantum fluctuations with gravitational constraints.
• Studies in interacting field theories and holographic models show that negative energy is capped by parameters like the central charge, preserving causal structure.

Negative average null energy refers to the phenomenon in quantum and semiclassical field theories whereby the integral of the null-projected energy density—often along a null geodesic or “smeared” over some domain—can be negative, contrary to the classical expectation imposed by the null energy condition (NEC). While the NEC is a cornerstone of classical general relativity and a critical assumption for singularity theorems, its averaged and quantum generalizations (ANEC, SNEC, DSNEC) represent both a mechanism for understanding negative energy densities and a set of rigorous constraints on how negative null energy can be in both free and interacting quantum field theories, especially within the context of gravity and holography.

1. Classical and Quantum Formulations of the Averaged Null Energy Condition

The classical Averaged Null Energy Condition (ANEC) asserts that the null energy integrated over a complete achronal null geodesic is nonnegative: $$\int_{-\infty}^{+\infty}\langle T_{\mu\nu}(x(\lambda))\,k^\mu k^\nu\rangle\,d\lambda \ge 0.$$ Here, $k^\mu$ is the affinely parameterized tangent vector to a complete null geodesic $\gamma$. In classical general relativity, the ANEC underpins no-go theorems for exotic spacetimes, including traversable wormholes and closed timelike curves (0705.3193).

In quantum field theory (QFT), the expectation value $\langle T_{\mu\nu} k^\mu k^\nu \rangle$ can be locally negative due to phenomena such as Casimir effects and vacuum fluctuations. Explicit violations of the classical ANEC exist for quantum fields in curved backgrounds, e.g., Minkowski spacetime compactified on $S^1$ (negative Casimir stress) and Schwarzschild spacetime in the Boulware vacuum (0705.3193).

2. Smeared and Double-Smeared Null Energy Conditions

To address the unboundedness of negative averaged null energy in QFT, quantum “smeared” versions of the NEC have been formulated.

• Smeared Null Energy Condition (SNEC): For any smooth, nonnegative, normalized sampling function $f(\lambda)$ along a null geodesic, SNEC bounds the negative null energy by

$$\int d\lambda\,f(\lambda)\,\langle T_{kk}\rangle \geq -\frac{B}{G_N \tau^2},$$

where $$\tau^{-2}\equiv \int d\lambda\,\frac{(f'(\lambda))^2}{f(\lambda)}$$, $G_N$ is Newton's constant, and $B$ is an order-one numerical constant. SNEC prohibits strongly gravitating, isolated negative-energy regions and interpolates between quantum energy inequalities for finite smearing (allowing negative averages) and the global ANEC, recovered in the infinite smearing limit (Freivogel et al., 2018). Applications to cosmological Genesis scenarios further show that SNEC imposes semi-local, quantitative restrictions on the magnitude and duration of negative null energy in early universe models (Yu et al., 4 Dec 2025).

• Double-Smeared Null Energy Condition (DSNEC): DSNEC further addresses the problem that even smeared null energy in free theories is unbounded from below when averaged over a single direction. By smearing in two null directions with smooth profiles $f(u)$ and $g(v)$, DSNEC establishes a state-independent, finite lower bound

$$I_{f,g} = \int du\,dv\,f(u)g(v)\langle T_{--}(u,v)\rangle_\psi \ge -B[f,g],$$

where $B[f,g]$ is a calculable functional of the sampling functions and theory-dependent data. DSNEC forms a robust foundation for singularity theorems in the semi-classical regime (Fliss et al., 2021).

3. Holographic and Interacting Field Theory Realizations

In the context of strongly coupled conformal field theories (CFTs) and AdS/CFT correspondence, holographic principles can enforce ANEC-type constraints:

• Holographic Proofs: In theories with an Einstein gravity dual, the ANEC along any null geodesic in the boundary CFT is a consequence of bulk causality: acausal propagation in the bulk from negative average null energy on the boundary is forbidden, leading to a holographic proof of

$$\int_{-\infty}^{+\infty}\langle T_{uu} \rangle \, du \ge 0$$

for all complete boundary null geodesics (Kelly et al., 2014). Analogous techniques apply in curved boundary spacetimes, where the bound involves weighted averages and additional curvature terms, potentially allowing for negative averages when curvature anomalies are present (Iizuka et al., 2020).

• Scaling in Large $N$ CFTs: In large $N$ CFTs, negative smeared null energy in states built from superpositions or products of primary operators remains bounded from below. The lower bound on negative null energy scales at worst linearly in the central charge, $C_T$, of the theory: $$\int g^2(x)\,\langle T_{--}(x) \rangle_\psi \ge -k[g]\,C_T,$$ where $k[g]$ is a positive, profile-dependent constant. This represents a significant improvement over free fields (unbounded negativity) and reflects the role of interaction as an effective regulator. Product CFTs and holographic (AdS gravity) arguments reinforce the $-O(C_T)$ scaling (Fliss et al., 2024).

4. Mechanisms and Physical Contexts for Negative Average Null Energy

Negative averaged null energy arises through several mechanisms:

• Quantum Coherence and Casimir Configurations: Quantum superpositions and boundary conditions (e.g., plates, compactifications) create local or even segment-averaged negative $T_{kk}$.
• Gravitational Anomalies and Curved Spacetimes: In even-dimensional QFTs with gravitational conformal anomaly, geometric terms can generate negative weighted averages for the null-projected stress tensor, determined explicitly by curvature invariants and expansion parameters (Iizuka et al., 2020).
• Non-minimal Couplings: Classically, non-minimally coupled scalars allow local violation of NEC but (under effective field theory constraints on field values) cannot support net negative average null energy. In these models, ANEC and smeared bounds such as SNEC or DSNEC are restored if the field amplitude stays within the EFT cutoff (Fliss et al., 2023).

5. Constraints and Theoretical Implications

A full understanding of negative averaged null energy centers on the distinction between local, segment, and global averages, and on the necessity of state-dependent versus universal bounds. Key implications include:

• Prohibition of Exotic Spacetimes: Self-consistent achronal ANEC is sufficient to rule out traversable wormholes, time machines, and other non-causal geometries in semiclassical gravity (0705.3193, Kontou et al., 2012, Kontou et al., 2015).
• Model-Building in Cosmology and Gravity: Bounds such as SNEC and DSNEC place explicit constraints on the magnitude and duration of negative energy in NEC-violating cosmologies, affecting Genesis and bounce models (Yu et al., 4 Dec 2025).
• Role of Quantum Inequalities: Quantum energy inequalities generically force any local negative energy pulse to be temporary or bounded in amplitude and duration, and these bounds weaken in the classical or large-$N$ limit (Kontou, 2015, Lee et al., 2016).
• Limits of Averaging Procedures: No procedure for averaging over submanifolds or the entire spacetime rescues universal positivity; construction of counterexamples shows that purely quantum anomalies can lead to negative spacetime-averaged null energy in prescribed backgrounds unless self-consistency is enforced (Urban et al., 2010).

6. Open Directions and Generalizations

Fundamental questions remain as to the optimal generalization of ANEC for quantum gravity. Existing work suggests:

• State-Independent Bounds Depend on Interactions and Central Charge: Non-interacting theories permit unbounded negativity via particle-number scaling, while genuinely interacting CFTs and holographic theories enforce a scaling with $C_T$, possibly subject to further universal constraints (Fliss et al., 2024).
• Anomalous and Curved Geometries: Even-dimensional and curved spacetimes require care due to anomalies, and only weighted ANECs (with geometrical weight functions) may hold in these settings (Iizuka et al., 2020).
• Quantum Gravity and Planckian Regimes: At Planck-scale curvatures, violations or modifications of smeared/averaged null energy conditions could arise, potentially linked to a finite quantum gravity cutoff. Ultimately, the semiclassical regime (where SNEC/DSNEC and achronal ANEC hold) may emerge as the low-energy remnant of a more fundamental principle in quantum gravity (0705.3193).

Negative average null energy, while prohibited classically, appears as a subtle, regulated feature in quantum and semiclassical regimes, subject to robust inequalities and deeply connected to the consistency of the causal and geometric structure of both quantum field theories and gravity.