In the de Sitter/conformal field theory (dS/CFT) correspondence, the wave function of an asymptotically de Sitter universe is related to the partition function of a Euclidean boundary quantum field theory deformed by external sources. Hertog, Monten and Vreys (JHEP 01 (2017) 060) used a canonical transformation to the source variables to derive a sufficient set of 'asymptotic' classicality conditions — approximate reality of the operator expectation values (vevs) dual to the sources — under which the boundary theory predicts classical Lorentzian bulk evolution at large spatial volume. In a four-dimensional, single-scalar, no-boundary minisuperspace model (a finite-dimensional truncation of the configuration space of gravity and matter) they showed these conditions are strictly stronger than the usual semiclassical (WKB) classicality criteria, excluding branches that the latter accepts. Whether the vev-reality conditions are necessary, and how the implied restriction on the allowed sources and on the boundary path-integral contour can be derived from first principles in the dual theory, has remained open. Later work supplies model-specific evidence — in a top-down dual built on complex deformations of ABJM theory, a three-dimensional supersymmetric conformal field theory with a known string-theory gravity dual, the regime where the wave function fails to behave classically is excluded from the configuration space by the quantum field theory's own consistency — but no general criterion, no necessity result, and no settled interpretation of the non-classical branches. The problem is to make the 'Lorentzian condition' of dS/CFT precise.

Background

The de Sitter/conformal field theory correspondence (dS/CFT) conjectures that the wave function of the universe with asymptotically de Sitter boundary conditions — de Sitter space being the maximally symmetric solution of general relativity with positive cosmological constant, relevant both to inflation and to the observed accelerated expansion — is computed by the partition function of a Euclidean quantum field theory on the future boundary of de Sitter space, deformed by external sources (background values coupled to its operators). A concrete semiclassical realization is the holographic form of the Hartle–Hawking or no-boundary wave function (NBWF) of Hertog and Hartle: the wave function Ψ\Psi on superspace (the configuration space of three-geometries and matter fields on a spatial slice) is a sum over regular, in general complex, Euclidean geometries with a single boundary; at large spatial volume this sum is dominated by saddle points whose Euclidean anti-de Sitter (AdS) regions are related, via Euclidean AdS/CFT (the anti-de Sitter/conformal field theory correspondence), to partition functions of dual field theories. In this dictionary the leading coefficients of the asymptotic radial expansion of the bulk fields near the boundary (the Fefferman–Graham expansion, the standard near-boundary expansion of anti-de Sitter holography) are the sources of the dual, and the subleading coefficients are the expectation values (vevs) of the dual operators.

In canonical quantum cosmology the wave function obeys the Wheeler–DeWitt (WDW) equation, the Hamiltonian constraint of general relativity promoted to an operator. At large volume, oscillatory WKB-type solutions predict classical Lorentzian evolution: the standard classicality conditions require the phase of the wave function to vary much more rapidly than the real part of the Euclidean action, so that histories — the integral curves of the phase in superspace — follow the classical Einstein equations. The requirement that a quantum state describe a Lorentzian universe in this sense goes back to Bousso and Hawking, who called the corresponding condition on the wave function — in their conjugate representation, where the wave function depends on the momentum variables rather than the boundary metric, the requirement that its argument be purely imaginary — the 'Lorentzian condition'.

Hertog, Monten and Vreys asked how classicality is seen from the dual boundary theory. The large phase of the bulk wave function is a local surface term that is canceled by the holographic counterterms, so classicality cannot be read off directly from the partition function. Performing a canonical transformation of the wave function to asymptotically well-defined variables — the sources of the dual — they constructed a new wave function Ψ~\tilde\Psi and derived a sharper set of asymptotic classicality conditions: the vevs corresponding to the sources must be approximately real. In a homogeneous isotropic minisuperspace model (a finite-dimensional truncation of superspace) of four-dimensional gravity coupled to a single scalar field in the no-boundary state, they showed these conditions are strictly stronger than the usual ones: several branches of saddle-point histories, including small perturbations of empty de Sitter, pass the usual classicality conditions but fail the vev-reality test, and the authors argue the stronger conditions are the physically appropriate ones in quantum gravity. The conditions were established only as sufficient, WKB-level criteria, for four dimensions, a single scalar, and the Hartle–Hawking state, and the analysis implies — but does not derive — a restriction on the allowed sources and on the boundary path-integral contour if the dual is to predict classical bulk evolution. The paper also notes (footnote 13) that one numerically obtained branch (the 'green' branch for scalar mass m=1.4m=1.4) suffers numerical instability whose physical significance is unclear.

Problem Statement

Within dS/CFT, determine precisely when a Euclidean boundary quantum field theory predicts classical Lorentzian bulk evolution at large spatial volume, upgrading the sufficient vev-reality classicality conditions of Hertog, Monten and Vreys (2017) to a sharp structural statement about the dual theory. Concretely:

  1. Necessity. Establish whether the asymptotic (vev-reality) classicality conditions are necessary as well as sufficient: either prove that whenever the vevs dual to the sources fail to be approximately real, the wave function fails to predict classical bulk evolution (with 'predicts classical evolution' made precise), or exhibit a controlled regime in which the conditions fail yet the bulk still evolves classically.

  2. First-principles boundary formulation. Derive, for at least one explicit dS/CFT dual, the exact configuration space of deformations: the restriction on the allowed values of the sources and on the path-integral measure or integration contour of the boundary Euclidean QFT that is required for compatibility with the asymptotic semiclassical structure implied by the WDW equation — rather than inferring this restriction bottom-up from the bulk theory.

  3. Physical interpretation of non-classical branches. Decide whether histories that satisfy the usual bulk classicality conditions but fail the vev-reality conditions represent genuinely quantum-gravitational regimes of the bulk, with no classical geometric description, or an artifact of the criterion — i.e. answer the paper's own question of which of the two notions of classicality is the physical one in quantum gravity.

These three components jointly serve a single overarching objective — making the 'Lorentzian condition' of dS/CFT precise. Component 3 is the interpretive reading of component 1's dichotomy; component 2 is independent of the other two.

References

  1. Thomas Hertog, Ruben Monten, Yannick Vreys, Lorentzian Condition in Holographic Cosmology, JHEP 01 (2017) 060, doi:10.1007/JHEP01(2017)060, arXiv:1607.07471, https://arxiv.org/abs/1607.07471
  2. James B. Hartle, S. W. Hawking, Thomas Hertog, The Classical Universes of the No-Boundary Quantum State, Phys. Rev. D 77 (2008) 123537, doi:10.1103/PhysRevD.77.123537, arXiv:0803.1663, https://arxiv.org/abs/0803.1663
  3. Thomas Hertog, James Hartle, Holographic No-Boundary Measure, JHEP 05 (2012) 095, doi:10.1007/JHEP05(2012)095, arXiv:1111.6090, https://arxiv.org/abs/1111.6090
  4. Raphael Bousso, Stephen W. Hawking, Lorentzian condition in quantum gravity, Phys. Rev. D 59 (1999) 103501, doi:10.1103/PhysRevD.59.103501, arXiv:hep-th/9807148, https://arxiv.org/abs/hep-th/9807148
  5. Gabriele Conti, Thomas Hertog, Ellen van der Woerd, Holographic Tunneling Wave Function, JHEP 12 (2015) 025, doi:10.1007/JHEP12(2015)025, arXiv:1506.07374, https://arxiv.org/abs/1506.07374
  6. Gabriele Conti, Thomas Hertog, Yannick Vreys, Squashed Holography with Scalar Condensates, JHEP 09 (2018) 068, doi:10.1007/JHEP09(2018)068, arXiv:1707.09663, https://arxiv.org/abs/1707.09663
  7. Thomas Hertog, Gabriele Tartaglino-Mazzucchelli, Thomas Van Riet, Victoria Venken, Supersymmetric dS/CFT, JHEP 02 (2018) 024, doi:10.1007/JHEP02(2018)024, arXiv:1709.06024, https://arxiv.org/abs/1709.06024
  8. Ram Brustein, A. J. M. Medved, A correspondence between strings in the Hagedorn phase and asymptotically de Sitter space, Phys. Rev. D 102 (2020) 086002, doi:10.1103/PhysRevD.102.086002, arXiv:2005.09321, https://arxiv.org/abs/2005.09321
  9. Jean-Luc Lehners, Review of the no-boundary wave function, Physics Reports 1022 (2023) 1-82, doi:10.1016/j.physrep.2023.06.002, arXiv:2303.08802, https://arxiv.org/abs/2303.08802
  10. Thomas Hertog, Jef Pauwels, Victoria Venken, Holographic (eternal) inflation, JHEP 07 (2025) 061, doi:10.1007/JHEP07(2025)061, arXiv:2411.18396, https://arxiv.org/abs/2411.18396
  11. Jonathan J. Halliwell, James B. Hartle, Thomas Hertog, What is the No-Boundary Wave Function of the Universe?, Phys. Rev. D 99 (2019) 043526, doi:10.1103/PhysRevD.99.043526, arXiv:1812.01760, https://arxiv.org/abs/1812.01760
  12. Thomas Hertog, Oliver Janssen, Joel Karlsson, Kontsevich-Segal Criterion in the No-Boundary State Constrains Inflation, Phys. Rev. Lett. 131 (2023) 191501, doi:10.1103/PhysRevLett.131.191501, arXiv:2305.15440, https://arxiv.org/abs/2305.15440
  13. Thomas Hertog, Oliver Janssen, Joel Karlsson, Kontsevich-Segal criterion in the no-boundary state constrains anisotropy, Phys. Rev. D 111 (2025) 046008, doi:10.1103/PhysRevD.111.046008, arXiv:2408.02652, https://arxiv.org/abs/2408.02652
  14. Juan Diaz Dorronsoro, Jonathan J. Halliwell, James B. Hartle, Thomas Hertog, Oliver Janssen, Real no-boundary wave function in Lorentzian quantum cosmology, Phys. Rev. D 96 (2017) 043505, doi:10.1103/PhysRevD.96.043505, arXiv:1705.05340, https://arxiv.org/abs/1705.05340
  15. Yannick Vreys, Beyond Classical Cosmology with the No-Boundary Wave Function, Ph.D. thesis, KU Leuven, 2017, https://inspirehep.net/literature/1638840
  16. Ruben Monten, De Sitter Space and Holography, Ph.D. thesis, KU Leuven, 2017, https://inspirehep.net/literature/1646171
  17. Victoria Venken, Quantum gravity aspects of de Sitter space, Ph.D. thesis, KU Leuven, 2020, https://inspirehep.net/literature/1817615