Shows Undecidability in AdS/CFT Correspondence Limits Computability of Spacetime Geometry

Researchers have discovered a surprising link between mathematical undecidability and the very fabric of spacetime.

Sameer Ahmad Mir (Jamia Millia Islamia and Asian School of Business), Francesco Marino (CNR-Istituto Nazionale di Ottica and INFN), and Arshid Shabir (Canadian Quantum Research Center) alongside Lawrence M. Krauss and Mir Faizal (University of British Columbia Okanagan, Durham University, and Hasselt University) demonstrate how Gödel’s incompleteness theorems, typically associated with formal logic, can manifest in gravitational theories through the holographic principle known as the AdS/CFT correspondence. Their work reveals that, for certain scenarios, selecting the correct spacetime geometry, whether it resembles standard Poincaré AdS space or an AdS soliton, becomes fundamentally impossible, even with standard computational methods. This finding suggests a profound limitation on our ability to fully understand and predict the behaviour of gravity, potentially reshaping our understanding of spacetime itself. Gödel’s Incompleteness and Undecidability in Holographic Spacetime Selection presents significant theoretical challenges Scientists have demonstrated a surprising link between the limits of computation and the fundamental nature of spacetime geometry, revealing that determining the shape of spacetime itself can be, in principle, undecidable. Consequently, under standard holographic assumptions, even identifying which smooth spacetime emerges from quantum gravity may lie beyond the reach of computation. This breakthrough reveals that the problem of selecting the dominant bulk geometry is fundamentally connected to the spectral gap problem in quantum many-body systems. The study unveils that determining whether a system is gapped or gapless is, in general, computationally impossible, mirroring the unsolvability of the halting problem. Researchers mapped a translationally invariant spin Hamiltonian with an undecidable gap status into a Euclidean path integral, decoupling interactions with a Hubbard-Stratonovich transformation and introducing an auxiliary scalar field. This scalar field encodes the halting bit of a Turing machine, persisting even in the continuum limit, and is then recast as an adjoint matrix model with a non-Abelian gauge structure. The key observation is that the sign of the scalar effective mass term, dictated by the undecidable halting predicate, determines the dominant smooth geometry: Poincaré AdS representing a gapless phase, or the AdS soliton indicating a gapped phase. Preliminary work established the undecidability of the spectral gap problem by mapping a universal Turing machine to a frustration-free Hamiltonian on a 2D lattice. If the computation halts, the Hamiltonian carries a weight of 1/(T+1) on the halting configuration, shifting the ground state energy, while a non-halting computation results in zero shift. This defines a computable map assigning a Hamiltonian to each input, with the spectral gap determining whether the system is gapped or gapless, a distinction that cannot be algorithmically decided.

This research establishes a profound connection between the foundations of mathematics, quantum physics, and gravity, opening avenues for exploring the limits of knowledge in our universe. Holographic transmission of undecidability via spectral gap competition and vacuum stress tensor analysis reveals emergent spacetime properties Scientists demonstrated how undecidability, originating from Gödel’s incompleteness theorems, can be transmitted holographically to gravitational theories using the AdS/CFT correspondence. Consequently, determining the emergent spacetime geometry becomes computationally intractable under standard semiclassical holographic assumptions. The study employed compactification on a Euclidean time circle of circumference β to investigate the vacuum stress tensor. If the theory is gapless, the vacuum exhibits scale-and translation-invariance with a vanishing stress tensor, corresponding to a Poincaré AdS4 saddle. Conversely, a gapped spectrum compactified on S1 β yields a negative Casimir energy, calculated using zeta-function methods, specifically εCas(β, m∗) = −m2 ∗ 2π2β ∞ X n=1 K1(m∗nβ) n, where K1 represents the modified Bessel function. The corresponding bulk saddle is the AdS4 soliton, with a Brown-York tensor matching the Casimir energy at leading order in G4, given by ⟨Tττ⟩sol = − r3 0 16πG4L4, and ⟨Txx⟩= ⟨Tyy⟩= −1 2⟨Tττ⟩, where r0 = 4πL2 3Lθ. The work constructed a computable family of large N theories where an undecidable input selects a relevant deformation, analysing the resulting competition of smooth Euclidean bulk fillings under the GKPW hypothesis. This approach provides a prototype for analogous computations in holographic settings, including planar N = 4 super-Yang-Mills theory and ABJM theory. Undecidability in quantum spin systems maps to intractable geometry selection in holography, suggesting a deep connection Scientists have demonstrated a holographic transmission of undecidability, a concept originating from Gödel’s incompleteness theorems, into gravitational theories. The research team embedded a translationally invariant spin Hamiltonian, known to possess an undecidable spectral gap status, into a large-N gauge theory. This generated an AdS dual where selecting the dominant bulk saddle point, either Poincaré AdS or AdS soliton, is itself undecidable. Consequently, determining the emerging smooth spacetime geometry becomes computationally intractable under standard semiclassical holographic assumptions. Experiments revealed that the selection of the dominant bulk geometry is dual to the spectral-gap problem of the boundary quantum many-body system. Researchers mapped a local, translationally invariant Hamiltonian into a Euclidean path-integral formulation using a Trotter decomposition, decoupling interactions with a Hubbard-Stratonovich transformation. This transformation encodes the halting bit of a Turing machine via a coupling that persists even in the continuum limit. Measurements confirm the resulting scalar field theory can be recast as an adjoint matrix model with a non-Abelian gauge structure, allowing definition of a large-N limit. The key observation is that the sign of the scalar effective mass term, determined by the undecidable halting predicate, dictates the dominant geometry. Poincaré AdS emerges in the gapless phase, while the AdS soliton dominates in the gapped phase. Scientists established that the spectral gap, defined as γL = E1(L) − E0(L) for a 2D lattice of size L × L, is crucial for determining the system’s stability. A uniform spectral gap implies exponential decay of correlations, while a vanishing limit indicates a gapless system with low-energy excitations. Using a Feynman-Kitaev history state, the team mapped a universal Turing machine onto a frustration-free k-local Hamiltonian. If the computation halts after runtime T, the history state carries a weight of 1/(T + 1) on the halting configuration, introducing a ground state energy shift of δ(T)/(T + 1). This construction embeds the computational layer into the lattice, coupling it to a spectral scaling mechanism. The work demonstrates that for certain Hamiltonians, determining whether they are gapped or gapless is formally independent of any consistent, computably axiomatizable theory. Researchers utilized a Suzuki-Trotter decomposition to map the system to a (2+1)D classical statistical model, accurate to O(a2 τ) with aτ = β/Nτ. The Lieb-Robinson bound ensures a finite signal velocity, and the team restricted analysis to microscopic models flowing to a Lorentz-invariant fixed point. The Hubbard-Stratonovich decoupling yielded an O(Nv) scalar theory, further solidifying the connection between the boundary quantum system and the emergent gravitational geometry. Undecidability in Quantum Spin Systems Constrains Emergent Spacetime Geometry and its potential observation Scientists have demonstrated a holographic transmission of undecidability, a concept originating in Gödel’s incompleteness theorems, into gravitational theories using the AdS/CFT correspondence. The findings establish a link between logical undecidability and the emergence of spacetime in a holographic context, highlighting a potential limitation of semiclassical holography. Authors acknowledge that their conclusions are conditional, relying on the assumptions of large N and strong coupling, alongside additional CFT structure necessary for a reliable Einstein gravity regime. The model presented serves as a prototype applicable to other holographic settings, such as planar N=4 super-Yang-Mills and ABJM theory, provided a relevant boundary coupling is appropriately chosen. Future research could explore exporting this mechanism to top-down string theory settings by carefully selecting boundary couplings and examining scenarios with multiple admissible Euclidean bulk fillings, like those found in mass deformations of N=4 SYM or ABJM theories. 👉 More information 🗞 Undecidability in Spacetime Geometry via the AdS/CFT Correspondence 🧠 ArXiv: https://arxiv.org/abs/2601.22761 Tags: