Interactions, Gravitational Dressing, and Microcausality

Interactions correct a free HKLL field by boundary multi-trace operators; gravitational gauge invariance then requires a dressing that reaches an asymptotic reference structure. These are separate effects. Scalar interactions can restore perturbative bulk microcausality order by order, whereas a gravitationally dressed observable generally retains dressing-dependent long-range commutators required by the constraints. Throughout this page the bulk is Lorentzian asymptotically AdS, the scalar uses standard quantization, boundary Wightman functions carry the vacuum $i0$, and statements are made inside a low-energy large-$N$ code sector through the displayed order.

Required background. Free HKLL reconstruction supplies the leading operator, and exchange Witten diagrams supply the perturbative relation between bulk exchange and CFT operator data.

Helpful background. Gauge constraints and centers explain why gauge-invariant algebras need boundary data; bulk field redefinitions distinguish convention-dependent contact terms; and causal composition supplies the operational signaling test.

Interacting bulk equations generate multi-trace corrections

Consider a scalar in fixed $\mathrm{AdS}_{d+1}$ of radius $L$. In the inherited $(+,-,\ldots,-)$ convention, take

$$S=\int d^{d+1}X\sqrt{\lvert g\rvert} \left[ \frac12(\nabla\phi)^2-\frac12m^2\phi^2 -\frac{\lambda}{3!}\phi^3 \right], \qquad m^2L^2=\Delta(\Delta-d).$$

The Lorentzian equation is

$$(\Box+m^2)\phi=-\frac{\lambda}{2}\phi^2.$$

The signs follow directly from the site’s mostly-minus scalar convention with $\mathcal V_{\mathrm{int}}=\lambda\phi^3/3!$. Expand $\phi=\phi^{(0)}+\lambda\phi^{(1)}+O(\lambda^2)$. With normalizable boundary conditions and a chosen Green function for $\Box+m^2$,

$$\phi^{(1)}(X)=-\frac12\int d^{d+1}Y\sqrt{\lvert g(Y)\rvert}\, G(X,Y)\,{:}\phi^{(0)}(Y)^2{:}.$$

Substituting the HKLL formula for each free field produces a bilocal boundary operator,

$$\phi(X)=\int d^dx\,K_\Delta(X|x)\mathcal O(x) -\frac{\lambda}{2}\int d^dx_1d^dx_2\, K^{(2)}(X|x_1,x_2){:}\mathcal O(x_1)\mathcal O(x_2){:} +O(\lambda^2).$$

In a large-$N$ normalization where connected three-point functions are $O(1/N)$, the second term is of that order. Expressing it in CFT conformal families gives a tower of double-trace primaries $[\mathcal O\mathcal O]_{n\ell}$. Their coefficients are not optional decorations: their anomalous dimensions and OPE data encode the exchange and contact terms needed for the interacting equation. Kabat, Lifschytz, and Lowe showed explicitly how such higher-dimension operators remove unwanted nonlocal singularities order by order (Kabat, Lifschytz, and Lowe 2011, §§2–4; Kabat, Lifschytz, and Lowe 2014, §§2–3).

Cancelling a spacelike commutator at controlled order

Let $X$ be spacelike to a boundary point $y$. The free field obeys

$$[\phi^{(0)}(X),\mathcal O(y)]=0$$

as a distribution away from coincident or null support. At order $\lambda$, the naive free smearing evaluated in an interacting CFT develops an unwanted contribution determined by the connected three-point function. The corrected commutator is schematically

$$\begin{aligned} \langle[\phi(X),\mathcal O(y)]\mathcal O(z)\rangle ={}&\langle[\phi^{(0)}(X),\mathcal O(y)]\mathcal O(z)\rangle_{1/N}\\ &+\sum_{n,\ell}a_{n\ell} \langle[\Phi_{n\ell}^{(0)}(X),\mathcal O(y)]\mathcal O(z)\rangle_0 +O(N^{-2}), \end{aligned}$$

where $\Phi_{n\ell}^{(0)}$ is the free smearing of $[\mathcal O\mathcal O]_{n\ell}$. Matching the discontinuity across the spacelike branch cut fixes the coefficients $a_{n\ell}$ up to terms corresponding to local bulk field redefinitions. With the complete double-trace tower, the noncausal discontinuity cancels and the result agrees with the bulk retarded solution through $O(1/N)$.

This is the page’s first application: a cubic interaction changes a single-trace HKLL operator into a single-plus-multi-trace expansion, and bulk scalar microcausality is recovered only to the perturbative order and in the correlators used to match it. Truncating the tower at dimension $\Delta_{\max}$ leaves a resolution-dependent remainder; it does not define an exactly local operator.

Gravity changes what locality can mean

A coordinate scalar $\phi(x)$ is not invariant under an infinitesimal diffeomorphism $x^\mu\mapsto x^\mu-\kappa\xi^\mu$:

$$\delta_\xi\phi(x)=\kappa\xi^\mu(x)\partial_\mu\phi(x).$$

Introduce a metric-dependent displacement $V^\mu[h](x)$ with $\delta_\xi V^\mu=-\kappa\xi^\mu$, and define

$$\Phi_V(x)=\phi\!\left(x+V[h](x)\right) =\phi(x)+V^\mu[h](x)\partial_\mu\phi(x)+O(\kappa^2).$$

The invariant operator is labeled not just by $x$, but by the dressing $V$ and its asymptotic anchor. A line dressing, a Coulomb-like dressing, and a geodesic dressing can create the same local matter excitation while carrying different gravitational fields. Their commutators can therefore differ at spacelike separation. This does not automatically signal acausal communication: changing the dressing changes the physical long-range field and often the boundary charges being manipulated. Donnelly and Giddings derive such nonlocal commutators for perturbatively dressed observables (Donnelly and Giddings 2016, §§II–IV).

Dressing and field-redefinition adversarial check

Suppose one changes the scalar variable by

$$\phi\longrightarrow\phi'=\phi+c\lambda\phi^2$$

and independently changes $V\to V'=V+\Delta V$, with $\Delta V$ gauge invariant. Three effects must not be conflated.

• The scalar redefinition shifts contact interactions and the coefficients of boundary double-trace terms. Separated on-shell amplitudes and properly transformed correlators are unchanged to the same order.
• The dressing change adds a genuine homogeneous gravitational field. If it changes the asymptotic tail or soft data, the two operators are physically distinct even though their undressed matter cores agree.
• Terms beyond the retained order are unresolved. A commutator of size $O(\lambda^2,\kappa^2,\lambda\kappa)$ cannot be interpreted from a first-order construction.

An operational test couples localized probes to two dressed observables while holding their asymptotic preparation fixed. A difference removable by the simultaneous field redefinition of observables and couplings is conventional. A difference in boundary charge, radiative data, or a gauge-invariant response is physical. A bare coordinate-field commutator is neither test.

Controlled limits and handoff

The scalar cancellation is controlled for fixed low-point correlators, low bulk energy, a complete perturbative multi-trace basis, and a specified Green-function contour. The large-$N$ and weak-coupling expansions must be truncated with an error estimate. Once gravity is dynamical, exact compact localization is incompatible with Gauss constraints; the appropriate target is a relational, boundary-anchored observable whose commutators are small in a declared state set and norm. Gravitational Gauss laws derive that obstruction, and relational observables compare useful dressings. Finite-$N$ and horizon claims require the later precision analysis.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Donnelly, W., and Giddings, S. B. (2016). “Observables, gravitational dressing, and obstructions to locality and subsystems.” Physical Review D 93, 024030. DOI.
• Kabat, D., Lifschytz, G., and Lowe, D. A. (2011). “Constructing local bulk observables in interacting AdS/CFT.” Physical Review D 83, 106009. DOI.
• Kabat, D., Lifschytz, G., and Lowe, D. A. (2014). “Multi-trace operators and bulk locality in AdS/CFT.” Fortschritte der Physik 62, 391–399. DOI.