[Paper Review] S-Matrices from AdS Spacetime

Joseph Polchinski|ArXiv.org|Jan 18, 1999

Black Holes and Theoretical PhysicsPhysics and Astronomy5 references111 citations

TL;DR

This paper proposes a method to extract the flat-space string S-matrix from the large-N limit of N=4 super Yang-Mills theory on S³×R, using wavepacket states in AdS₅×S⁵ that localize scattering at the origin. By holding external momenta fixed while taking R→∞ (via N→∞), the S-matrix in the dual gauge theory reproduces the flat spacetime scattering amplitude, establishing a concrete holographic realization of the S-matrix in AdS/CFT.

ABSTRACT

In the large-N limit of d=4, N=4 gauge theory, the dual AdS spacetime becomes flat. We identify a gauge theory correlator whose large-N limit is the flat spacetime S-matrix.

Motivation & Objective

• To identify a gauge theory correlator in the large-N limit of N=4 SYM that corresponds to the flat space string S-matrix.
• To address the non-universal and non-covariant nature of the flat spacetime limit in AdS/CFT by constructing a specific, explicit realization.
• To demonstrate that wavepackets in AdS₅×S⁵, with proper momentum uncertainty scaling as R⁻¹ω⁻¹/², yield a well-localized scattering process approaching flat space kinematics.
• To show that the large-N limit with fixed gₛ and s, and R∼N¹/⁴, reproduces the correct flat spacetime S-matrix via boundary-to-boundary scattering.

Proposed method

• Use of wavepacket solutions φ_ωe to the free scalar wave equation in AdS₅, localized along classical geodesics with width ω⁻¹/² in coordinate space.
• Employment of a WKB approximation for ω≫1, with phase f=ρ−t and envelope functions g₁, g₂, h derived from solving the d’Alembertian equation order-by-order in ω.
• Matching of the WKB wavefunction to the large-r asymptotic form involving Hankel functions H²,¹₂(ω/r), enabling extraction of boundary wavepackets G±(τ,θ).
• Derivation of boundary wavepacket profiles G±(τ,θ)∝exp(−ω/2[τ²+θ²]) for massless scalars, and generalized to massive modes with Bessel function order ν.
• Use of LSZ-like prescription with sources and detectors on the AdS boundary at t=±π/2, integrating over angles and times to extract the S-matrix.
• Scaling limit: N→∞, gₛ fixed, R=(4πα′²gₛN)¹/⁴, ω∝Rs¹/², ensuring fixed s and vanishing proper momentum uncertainty.

Experimental results

Research questions

• RQ1 What gauge theory correlator in the large-N limit of N=4 SYM corresponds to the flat space string S-matrix?
• RQ2 How can wavepackets in AdS₅×S⁵ be used to define a scattering process with well-localized kinematics in the large-N limit?
• RQ3 What scaling of parameters (N, gₛ, α′, s) is required to recover the flat spacetime S-matrix from AdS/CFT?
• RQ4 How does the proper momentum uncertainty scale in the large-N limit, and does it vanish as required for a well-defined S-matrix?
• RQ5 Can the S-matrix be extracted from AdS spacetime despite periodicity and position-dependent metric effects?

Key findings

• The boundary wavepacket profile for massless scalars is G±(τ,θ) = −ie^{±iπω/2}(πω/2)^{1/2} exp(−ω/2[θ² + τ²]), matching the expected form for flat space scattering.
• For massive scalars with Kaluza–Klein mass m∼R⁻¹, the wavepacket becomes G±(τ,θ) = e^{±iπ(ω+ν)/2}(2/ω)^{ν−1/2}Γ(ν)π^{-1/2} exp(−ω/2[τ² + θ²]), with ν=√(m²R²+1/4).
• The proper momentum uncertainty scales as R⁻¹ω⁻¹/², which vanishes in the large-N limit, ensuring sharp kinematics.
• The S-matrix is recovered in the limit N→∞ with fixed gₛ, s, and R∼N¹/⁴, while ω∝Rs¹/².
• The construction shows that the S-matrix can be extracted from AdS spacetime, countering claims of its inaccessibility in the large-N limit.
• The wavepacket approach resolves the issue of position dependence in AdS by localizing scattering at the origin, with uncertainty shrinking as ω⁻¹/².