Afshordi and Stojkovic (Phys. Lett. B 739, 2014) proposed that spacetime dimensionality is energy-dependent: target-space coordinates are treated as fields on the string worldsheet with temperature-dependent mass terms, so that dimensions open up one by one as the universe cools, and a self-intersecting string network kinematically weaves an effective (3+1)-dimensional spacetime with an induced Einstein–Hilbert action and a Planck mass tied directly to the string scale. The construction, however, rests on phenomenological inputs — a conformal-symmetry-breaking mass hierarchy added by hand, a non-locality kernel chosen on physical grounds, and an incalculable regulator for a logarithmic divergence — and its central dynamical question, whether the string network forms a stable three-dimensional condensate that survives the opening of further dimensions, was left unresolved. The problem is to replace these inputs with a microscopic derivation: embed the thermally induced masses in a UV-complete, anomaly-consistent string framework, derive the non-locality kernel and string-intersection smearing from first principles, compute the regulator (hence the induced Planck mass) from worldsheet quantum statistics, determine the condensate-formation dynamics, establish quantitatively whether stochastic averaging protects Lorentz invariance, complete the quantum-field-theoretic map from worldsheet to spacetime fields, and extract the model's parameter-specified phenomenological forecasts (gravitational-wave cutoff, planar cosmic-ray events, collider cross-over signatures).

Background

A recurring hint across approaches to quantum gravity is that spacetime becomes effectively lower-dimensional at very short distances, which would tame ultraviolet divergences and help quantize gravity; complementary proposals let dimensionality grow at cosmological scales to attack problems such as the cosmological constant. The "vanishing" or "evolving" dimensions program of Stojkovic and collaborators made this concrete phenomenologically: dimensions are degrees of freedom that open up sequentially as the temperature of the universe drops, so that the dimensionality probed depends on energy or temperature.

Afshordi and Stojkovic (2014) gave this program a worldsheet realization. In string theory, the target-space coordinates Xμ(ξa)X^{\mu}(\xi^{a}) (with worldsheet coordinates ξa=(τ,σ)\xi^{a}=(\tau,\sigma)) can themselves be regarded as fields living on the two-dimensional string worldsheet; α′\alpha' denotes the inverse string tension, the fundamental length scale of the string. They added temperature-dependent mass terms Lmass=m02e−mi/TXiXi\mathcal{L}_{\rm mass}=m_0^{2} e^{-m_i/T} X^{i}X_{i} for these worldsheet fields, i.e. a squared mass m02e−mi/Tm_0^{2} e^{-m_i/T} (effective mass m0e−mi/(2T)m_0 e^{-m_i/(2T)}) for each field XiX^{i}, where the mim_i are fixed constant mass scales with the hierarchy m1≫m2≫⋯≫mnm_1\gg m_2\gg\cdots\gg m_n and m0m_0 is an overall mass scale of order the fundamental energy scale, so at the high temperatures of the Big Bang all "dimensions" are massive and unexcited, and as the universe cools the fields XμX^{\mu} become light one by one — dimensions open. A symmetron (chameleon-type) variant sketched in the same paper instead couples these masses to the environmental energy density, giving a dimensional cross-over that depends polynomially rather than exponentially on temperature. Adding masses breaks worldsheet conformal invariance (the local symmetry underlying standard string quantization), which the authors justify only at the effective, finite-temperature level: at zero temperature all masses vanish and the theory is conformal.

The heart of the construction is kinematic. A long string that intersects itself many times is fundamentally a (1+1)-dimensional object but effectively fills a higher-dimensional region; string intersections play the role of sites of a random lattice or causal set (a discrete, locally finite set of events ordered by causality). This realizes geometrically the Brandenberger–Vafa mechanism, the argument that strings in more than three spatial dimensions almost never intersect and so cannot unwind, which singles out three large spatial dimensions. To recover locality at large distances, the authors coupled worldsheet matter fields ϕ(ξa)\phi(\xi^{a}) through a non-local kernel K(ΔX2)K(\Delta X^{2}) that suppresses interactions between far-separated points in target space; expanding for infrared fluctuations maps the worldsheet theory onto a continuum (3+1)-dimensional Lorentz-invariant scalar action, with an overall coefficient NcZN_c Z, where NcN_c is the spacetime density of string crossings and ZZ is a geometrical factor averaged over string intersections. Integrating out ϕ\phi then yields a geometric term proportional to the (3+1)-dimensional Ricci scalar RR (the standard curvature invariant of general relativity), i.e. an Einstein–Hilbert action, the usual action of general relativity, with Planck mass Mp,4d2∼(ln⁡Λ)/α′M^{2}_{p,4d}\sim(\ln\Lambda)/\alpha', relating the Planck and string scales without compactification; here Λ\Lambda is proportional to the ratio of the ultraviolet cutoff (Nc1/4N_c^{1/4}) to the infrared cutoff (or mass), and the ultraviolet worldsheet expectation value ⟨ϕ2⟩UV\langle\phi^{2}\rangle_{\rm UV} combines with NcZN_c Z into the single logarithmic factor, NcZ⟨ϕ2⟩UV∼(ln⁡Λ)RN_c Z\langle\phi^{2}\rangle_{\rm UV}\sim(\ln\Lambda) R. The geometrical factor Z≃4πln⁡(2/ϵ)Z\simeq 4\pi\ln(2/\epsilon) is itself logarithmically divergent and hides a regulator ϵ∼Kˉα′\epsilon\sim\bar{\mathcal{K}}\sqrt{\alpha'} set by the typical extrinsic curvature of strings over the string length, so the numerical prediction for the Planck mass is incalculable without a microscopic theory of the worldsheet geometry. When a fourth spatial dimension eventually becomes light, the network either disintegrates (intersections become rare and non-persistent in (4+1) dimensions) or survives as a three-dimensional bound state — a condensate — and the paper does not determine which occurs.

Phenomenologically, the model predicts a dimensional cross-over: thermal (temperature-driven, relevant for heavy-ion and cosmic-ray collisions of high multiplicity) or kinematic (energy-driven, relevant for colliders). In the thermal limit the LHC would be essentially blind to new physics even at the TeV scale; in the rigid-lattice limit high-energy events become effectively (2+1)-dimensional, producing planar events and elliptic jets (collimated particle sprays flattened into a plane) of the kind possibly seen in old high-altitude cosmic-ray super-families (unusually dense multiparticle events recorded in emulsion chambers), a signature worked out for the lattice version by Anchordoqui et al. (2010, published 2012). Lower-dimensional early epochs cannot support propagating gravitational waves, so primordial gravitational-wave spectra could show a cutoff frequency, an idea proposed for the lattice model by Mureika and Stojkovic (2011) and criticized as misleading in that setting by Sotiriou, Visser and Weinfurtner (2011). The stochastic (random-lattice/causal-set) version proposed in 2014 could also conflict with Lorentz invariance: Hossenfelder (2015) proved that no network with a locally finite distribution of nodes and links can be exactly Poincaré-invariant, so the model must rely on stochastic averaging of Lorentz-violating fluctuations — asserted but never computed in the source. Reviews by Stojkovic (2013) and Carlip (2017) situate the model within the broader landscape of dimensional reduction in quantum gravity, where the origin of the dimensional reduction remains unexplained.

Problem Statement

Give the evolving-dimensions worldsheet model of Afshordi and Stojkovic (2014) a microscopic foundation and extract its quantitative content, or demonstrate a definitive obstruction. All seven requested results serve one objective — replacing the model's phenomenological inputs with derived physics — and are to be understood in the framework of that paper: (1) a UV-complete, anomaly-consistent (free of quantum obstructions to its gauge and conformal symmetries) string (or equivalently conformal worldsheet) framework whose finite-temperature state reproduces the thermally induced mass hierarchy m02e−mi/Tm_0^{2} e^{-m_i/T} for the target-space coordinate fields Xμ(ξa)X^{\mu}(\xi^{a}), so that the breaking of worldsheet conformal invariance is derived rather than imposed; (2) a dynamical, first-principles derivation of the non-locality kernel K(ΔX2)K(\Delta X^{2}) and of the string-intersection smearing (the α′\alpha' corrections), where the source only argued on power-counting grounds which kernels yield renormalizable infrared behavior; (3) a microscopic quantum-statistical calculation of worldsheet geometry fixing the regulator ϵ∼Kˉα′\epsilon\sim\bar{\mathcal{K}}\sqrt{\alpha'}, and thereby of the factor Z≃4πln⁡(2/ϵ)Z\simeq 4\pi\ln(2/\epsilon) and the induced Planck mass Mp,4d2∼(ln⁡Λ)/α′M^{2}_{p,4d}\sim(\ln\Lambda)/\alpha', as a number rather than a parametric relation; (4) a dynamical determination of whether the interacting string network forms a stable three-dimensional condensate that preserves the emergent (3+1)-dimensional geometry when additional spatial coordinates become light, or whether the causal-set structure disintegrates, including the conditions governing the outcome; (5) a quantum-statistical analysis of whether stochastic averaging over the random intersection structure suffices to keep Lorentz-invariance violations within current observational constraints (photon time-of-flight and dispersion bounds); (6) completion of the map from worldsheet matter fields ϕ(ξa)\phi(\xi^{a}) to spacetime fields ϕ(Xμ)\phi(X^{\mu}) as a full quantum field theory, including interactions, renormalization, and vacuum-state dependence; and (7) the resulting quantitative phenomenological forecasts — the primordial gravitational-wave cutoff frequency, expected rates and kinematic signatures of planar cosmic-ray events, and collider predictions interpolating between the thermal and kinematic cross-over limits — from a specified mass hierarchy mim_i, intersection density NcN_c, and numerically evaluated logarithmic factors. An answer of obstruction type (for example, a proof that no anomaly-free conformal completion can generate the required thermal masses, or that stochastic Lorentz violation necessarily violates observational bounds) settles the central objective negatively and is equally admissible. All seven results are jointly required on the affirmative route: a rigorous result confined to a subset of (1)-(7), however strong, advances the objective without resolving it. An obstruction resolves the objective only when it is decisive for the construction as a whole, i.e. when it shows that the model cannot yield the emergent (3+1)-dimensional spacetime or is observationally excluded. The negative branches of (1), (4) and (5) are decisive in this sense — the required thermal masses cannot exist, the three-dimensional condensate cannot survive the opening of further dimensions, or the network cannot be Lorentz-safe — whereas a negative or partial outcome confined to one of the remaining items resolves only that item and leaves the objective open.

References

  1. Niayesh Afshordi and Dejan Stojkovic, Emergent spacetime in stochastically evolving dimensions, Physics Letters B 739 (2014) 117–124, DOI 10.1016/j.physletb.2014.10.048, arXiv:1405.3297, https://arxiv.org/abs/1405.3297
  2. Dejan Stojkovic, Vanishing Dimensions: A Review, Modern Physics Letters A 28 (2013) 1330034, DOI 10.1142/S0217732313300346, arXiv:1406.2696, https://arxiv.org/abs/1406.2696
  3. Sabine Hossenfelder, A no-go theorem for Poincaré-invariant networks, Classical and Quantum Gravity 32 (2015) 207001, DOI 10.1088/0264-9381/32/20/207001, arXiv:1504.06070, https://arxiv.org/abs/1504.06070
  4. S. Carlip, Dimension and dimensional reduction in quantum gravity, Classical and Quantum Gravity 34 (2017) 193001, DOI 10.1088/1361-6382/aa8535, arXiv:1705.05417, https://arxiv.org/abs/1705.05417
  5. Deepak Vaid and Lin Teixeira de Sousa, A loop quantum gravity inspired action for the bosonic string and emergent dimensions at large scales, Annals of Physics 483 (2025) 170240, DOI 10.1016/j.aop.2025.170240, arXiv:2407.13807, https://arxiv.org/abs/2407.13807
  6. Joshua Erlich, The Minkowski vacuum of stochastic composite gravity, Modern Physics Letters A 39 (2024) 2450179, DOI 10.1142/S0217732324501797, https://doi.org/10.1142/S0217732324501797
  7. Sam Powers and Dejan Stojkovic, Emergent Space, Time, and the Speed of Light from Binary Sequences, arXiv:2608.23586 (2026), https://arxiv.org/abs/2608.23586
  8. Jonas R. Mureika and Dejan Stojkovic, Detecting Vanishing Dimensions Via Primordial Gravitational Wave Astronomy, Physical Review Letters 106 (2011) 101101, DOI 10.1103/PhysRevLett.106.101101, arXiv:1102.3434, https://arxiv.org/abs/1102.3434
  9. Thomas P. Sotiriou, Matt Visser and Silke Weinfurtner, Comment on: Detecting Vanishing Dimensions Via Primordial Gravitational Wave Astronomy, Physical Review Letters 107 (2011) 169001, DOI 10.1103/PhysRevLett.107.169001, arXiv:1104.1223, https://arxiv.org/abs/1104.1223
  10. Luis Anchordoqui, De-Chang Dai, Malcolm Fairbairn, Greg Landsberg and Dejan Stojkovic, Vanishing Dimensions and Planar Events at the LHC, Modern Physics Letters A 27 (2012) 1250021, DOI 10.1142/S0217732312500216, arXiv:1003.5914, https://arxiv.org/abs/1003.5914