Energy Conditions, Causality, and Regge Consistency

Microcausality, ANEC, collider positivity, detector commutativity, and Regge boundedness are related, but they are not equivalent formulations of one condition. Each implication uses different states, smearings, Lorentzian sheets, or growth estimates. The safest comparison follows a single stress-tensor matrix element through those changes and stops when a new hypothesis is required.

Required background. Conformal collider bounds supplies the normalized energy matrices and their dimension-specific tensor sectors. The CFT Regge limit and boundedness fixes the Lorentzian continuation, boost variables, intercept, and arc criterion.

Helpful background. CFT dispersion relations and subtractions explains why growth control determines subtraction data rather than merely improving convergence.

Evidence cutoff. Direction-of-implication claims and current applications here were checked against primary sources available through 2026-08-09. Later gap bounds, dispersive improvements, or proposed converse theorems are outside this account.

Three Lorentzian questions, not one limit

The following regimes must be distinguished.

ANEC from causality. Two probe operators approach a lightcone in a Rindler-reflected four-point function. A complete null integral of $T_{uu}$ is isolated, and half-disk analyticity plus Rindler positivity fixes its sign. The proof uses a normalizable or regulated state, a specific Wightman ordering, and a Euclidean quarter-rotation Hartman, Kundu, and Tajdini 2017, §§2–4.

CFT Regge limit. Start from the Euclidean correlator, continue $\bar z$ clockwise around $1$ so $(1-\bar z)\mapsto e^{-2\pi i}(1-\bar z)$ while $z$ stays on its first sheet, and then take

$$z=\sigma e^\rho, \qquad \bar z=\sigma e^{-\rho}, \qquad \sigma\to0^+, \qquad \rho\ \text{fixed}.$$

A normalized smeared correlator may obey

$$\lvert\mathcal G^{\circlearrowleft}(\sigma,\rho)\rvert \le C(\rho)\sigma^{1-j_0}.$$

This statement concerns that sheet, ordering, and smearing. It is not the Euclidean OPE limit and not the fixed-$\bar z$ lightcone limit.

The same sheet discipline is essential in shockwave four-point-function causality arguments: analyticity and crossing constrain the continued correlator, not an arbitrarily chosen branch Hartman, Jain, and Kundu 2016, §§3–6.

Semiclassical shockwave limit. In a large-$N$ CFT with a weakly coupled gravitational description and a declared high-spin single-trace gap, the Regge correlator can admit an eikonal phase-shift interpretation. Time-delay constraints then restrict higher-derivative graviton couplings. Without large-$N$ factorization, the gap, and a controlled semiclassical regime, the same words do not define a theorem about a general CFT.

The ANEC causality proof uses analytic techniques related to Regge or chaos arguments, but its lightcone and large-boost limits need not commute. Hartman, Kundu, and Tajdini explicitly separate the small-$N$ lightcone argument from the large-$N$ Regge/chaos regime Hartman, Kundu, and Tajdini 2017, discussion following eq. (2.9).

A hypothesis-labeled implication table

From	To	Additional hypotheses used by the arrow	What the arrow does not prove
Relative-entropy monotonicity	ANEC	Unitary relativistic QFT, nested half-spaces, common state domain, first-order null deformation, boundary control	A Regge intercept or dispersion relation
Microcausality	ANEC	$d>2$ interacting CFT in the original route, Rindler positivity, normalizable smeared insertions, lightcone OPE projection, half-disk analyticity, fixed ordering	Pointwise $T_{uu}\ge0$
ANEC	Collider inequalities	Conformal transformation to null infinity, state and tensor Jacobians, $\int\mathcal E=P^0$, declared dimension/parity/polarization sector	Existence of a full CFT with those three-point coefficients
Regge bound $j_0<3$	Separated energy-detector commutativity and a spin-three light-ray OPE	Existence of the detector product, Sommerfeld–Watson/inversion continuation, no relevant spin-three obstruction, contacts excluded	Coincident-angle commutativity or absence of contact terms
Regge growth estimate	Inversion/dispersion arc control	Uniform control in $\rho$ and other endpoints, declared sheet and kernel	Vanishing of every arc or an unsubtracted formula
Large-$N$, large gap, semiclassical causality	Suppression of non-Einstein stress-tensor structures	Weakly coupled gravity, eikonal regime, higher-spin gap, appropriate polarizations and impact parameter	A statement for arbitrary finite-$N$ CFTs

Every arrow is one-way as written. The target may be established by a different proof, but that does not remove the listed assumptions from this route.

Detector commutativity and the intercept

Each energy detector is the light transform of a spin-two stress tensor. Their product selects parent spin

$$J=2+2-1=3.$$

The light-ray argument shows that a sufficient condition for distinct-angle commutativity is

$$j_0<3.$$

Under that condition, a possible odd-signature spin-three contribution is related to a local $T\times T$ structure; conservation and Ward identities remove the prohibited obstruction. The argument excludes coincident directions, where contact terms can remain Koloğlu et al. 2021, §§4.1–4.2.

This threshold also diagnoses perturbation theory. A fixed order in a large-$N$ expansion can grow more rapidly than the resummed correlator and violate $j_0<3$ term by term. Performing the null transforms at that order can therefore produce an ill-defined event shape even when the nonperturbative observable exists Koloğlu et al. 2021, §7.9 and §8.2. Resummation and the detector limit must be ordered according to the proven bound.

Collider positivity through a causal correlator

Consider the four-dimensional parity-even stress-tensor state and choose $\mathbf n=\hat z$. In the transverse helicity-two sector, collider positivity is

$$\lambda_{(2)} \equiv1-\frac{t_2}{3}-\frac{2t_4}{15}\ge0.$$

To compare this with causality, keep the following data fixed:

1. prepare the same Ward-normalized stress-tensor polarization with a smooth, normalizable wave packet;
2. insert two probe operators in the Rindler-reflected configuration and choose their $i\epsilon$ ordering;
3. isolate the complete-null-line stress-tensor contribution in the lightcone limit;
4. use half-disk analyticity and Rindler positivity to obtain a positive contour coefficient;
5. perform the Euclidean quarter-rotation and conformal transformation that turn the null line into the physical detector;
6. divide by the same state norm and impose $\int d\Omega_2\,\mathcal E=P^0$.

With this smearing, the positive causal sum-rule coefficient is a positive normalization factor times $\lambda_{(2)}$. The same construction in the helicity-one and helicity-zero sectors yields the other collider eigenvalues. The explicit matching of causal smearing to conformal-collider states is given in Hartman, Kundu, and Tajdini 2017, §5; the target energy matrix is Hofman and Maldacena 2008, eqs. (2.37)–(2.38).

This equivalence is between two computations of a smeared matrix element. It does not identify an unsmeared local commutator with $\mathcal E(\mathbf n)$, nor does it assert that every Regge coefficient equals a collider parameter.

Growth, arcs, and subtractions

Suppose a complex-spin or dispersive kernel contributes a factor $\sigma^{J-1}$ on the Regge arc. Combining it with $\mathcal G^{\circlearrowleft}=O(\sigma^{1-j_0})$ gives an arc integrand with schematic power $\sigma^{J-j_0}$. The simple closure criterion is then

$$\operatorname{Re}J>j_0,$$

plus uniform control of the $\rho$ endpoints and all other contour pieces. For the energy-detector OPE, the relevant parent spin is $J=3$, hence the appearance of $j_0<3$.

A dispersion relation asks a different question: does the large contour of the correlator itself vanish? If not, one performs enough subtractions and retains the corresponding constants or contact structures. Schematically,

$$\mathcal G =\mathcal D[\operatorname{Disc}\mathcal G] +\sum_{a=1}^{N_{\rm sub}}c_a\,\mathcal C_a.$$

The growth bound determines $N_{\rm sub}$ together with the kernel and endpoint powers. ANEC positivity may constrain the discontinuity in a suitable channel, but it neither fixes the subtraction count nor determines the $c_a$. An “unsubtracted ANEC dispersion relation” is therefore incomplete unless its arc estimate is stated.

What a large higher-spin gap adds

Camanho, Edelstein, Maldacena, and Zhiboedov analyze a weakly coupled gravitational theory in a high-energy shockwave regime. Non-Einstein graviton three-point structures can produce polarization-dependent time advances; repairing them requires new higher-spin states. In the AdS/CFT setting and under the weak-gravity, large-$N$, and high-spin-gap assumptions, their four-dimensional anomaly estimate is parametrically

$$\left|\frac{a-c}{c}\right| \lesssim\frac{1}{\Delta_{\rm gap}^2}.$$

This is a semiclassical large-gap conclusion, not a consequence of the conformal-collider triangle alone Camanho et al. 2016, §§2–5. The collider inequalities allow a finite region of $(t_2,t_4)$; the large-gap argument can force a much smaller neighborhood of the Einstein-like structure because it imports an entire dynamical regime.

Conversely, observing small non-Einstein collider parameters does not prove a large gap or a gravity dual. Many microscopic mechanisms can make a coefficient small.

Converse checks

Tempting converse	Why it fails
Collider positivity $\Rightarrow$ microcausality	Collider tests only selected smeared three-point matrices; microcausality constrains all spacelike commutators.
ANEC $\Rightarrow$ a Regge bound	ANEC is one positive null integral; a Regge bound controls a full second-sheet four-point function uniformly in a boost limit.
Regge boundedness $\Rightarrow$ ANEC	The sign also requires unitarity/reflection or Rindler positivity and the correct lightcone projection.
A vanishing collider eigenvalue $\Rightarrow$ a free theory	Saturation is one polarization selection rule; identification needs spectrum and higher-point data.
A large gap $\Rightarrow$ Einstein-like $TTT$ in any CFT	The suppression theorem also assumes large $N$, weak gravitational coupling, and a controlled eikonal regime.
A dispersion reconstruction $\Rightarrow$ positivity	Analytic reconstruction fixes cut-sensitive data up to subtractions; positivity is a separate Hilbert-space input.

Common pitfalls

Calling the ANEC lightcone limit “the Regge limit.” The analytic tools overlap, but the scaling regimes and order of limits differ. Name the variables and sheet.

Quoting $j_0<3$ at coincident angles. It controls the separated detector product. Contact distributions require a separate analysis.

Dropping the large contour after proving a sign. Positivity of a discontinuity does not make the arc vanish. Estimate growth and record subtractions.

Using a gap without a regime. A spectral gap alone does not supply factorization, an eikonal phase, or a semiclassical time-delay interpretation.

Exercises

1. Spin threshold. A detector product selects parent spin $J=3$, while a correlator obeys $\mathcal G^{\circlearrowleft}=O(\!\left(\sigma^{-3/2}\right))$. Does the simple inversion arc close at $J=3$?

Solution

Matching $\sigma^{-3/2}=\sigma^{1-j_0}$ gives $j_0=5/2$. Since $3>5/2$, the simple spin criterion is satisfied. One must still check uniformity in $\rho$, other endpoints, the selected sheet, and contacts.

2. Identify the missing hypothesis. A positive collider matrix is used to claim an unsubtracted dispersion relation. What additional check is indispensable?

Solution

One must bound the correlator on the full large contour in the declared Lorentzian analytic domain, including endpoint behavior, strongly enough that the arc vanishes. If it does not, the required subtraction structures and constants must be retained.

References

• Camanho, Xian O., José D. Edelstein, Juan Maldacena, and Alexander Zhiboedov. “Causality Constraints on Corrections to the Graviton Three-Point Coupling.” Journal of High Energy Physics 2016, no. 02 (2016): 020. doi:10.1007/JHEP02(2016)020.
• Hartman, Thomas, Sachin Jain, and Sandipan Kundu. “Causality Constraints in Conformal Field Theory.” Journal of High Energy Physics 2016, no. 05 (2016): 099. doi:10.1007/JHEP05(2016)099.
• Hartman, Thomas, Sandipan Kundu, and Amirhossein Tajdini. “Averaged Null Energy Condition from Causality.” Journal of High Energy Physics 2017, no. 07 (2017): 066. doi:10.1007/JHEP07(2017)066.
• Hofman, Diego M., and Juan Maldacena. “Conformal Collider Physics: Energy and Charge Correlations.” Journal of High Energy Physics 2008, no. 05 (2008): 012. doi:10.1088/1126-6708/2008/05/012.
• Koloğlu, Murat, Petr Kravchuk, David Simmons-Duffin, and Alexander Zhiboedov. “The Light-Ray OPE and Conformal Colliders.” Journal of High Energy Physics 2021, no. 01 (2021): 128. doi:10.1007/JHEP01(2021)128.