Spin Foams and Group Field Theory Reveal Dynamic Spacetime Dimensions.

The fundamental structure of spacetime remains one of the most challenging questions in theoretical physics, with current models struggling to reconcile general relativity with quantum mechanics. Recent research explores the possibility that spacetime itself emerges from more fundamental, discrete building blocks, potentially resolving inconsistencies at extremely small scales. This approach utilises techniques from spin-foam models and group field theories, mathematical frameworks that attempt to define quantum gravity by summing over possible spacetime geometries. A study published recently by Alexander Florian Jercher, from Friedrich-Schiller-Universität Jena, alongside Sebastian Steinhaus, also of Jena, and Edward Wilson-Ewing from the University of New Brunswick, with Steffen Gielen from the University of Sheffield, details investigations into the emergent properties of Lorentzian geometries within these frameworks, as presented in their article, “Emergent Lorentzian Geometries from Spin-Foams and Group Field Theories”. Their work examines spectral dimensions, causal regularity, and cosmological perturbations, offering insights into the potential for a consistent quantum description of gravity and spacetime.

The emergence of Lorentzian geometries, those conforming to the principles of special relativity where spacetime intervals can be positive, negative, or zero, constitutes a central challenge in quantum gravity. Current research investigates this phenomenon within the frameworks of spin foam and group field theory, both approaches attempting to define a quantum theory of spacetime. Analyses reveal that the spectral dimension, a measure of how volume scales with distance at different scales, of periodic Euclidean spin-foam frusta, geometric building blocks resembling polyhedra, generically approaches four at large scales, suggesting a recovery of the classical four-dimensional spacetime we observe. However, at smaller scales, a more complex behaviour emerges, with the spectral dimension exhibiting a flow sensitive to curvature and the underlying parameters of the theory.

Lorentzian Regge calculus, a method approximating spacetime with discrete geometric pieces, is applied to (3+1)-dimensional cosmology, modelling the universe with Lorentzian 4-frusta – four-dimensional analogues of the aforementioned polyhedra – coupled to a massless free scalar field, a fundamental field describing particles with no mass. Crucially, this research establishes that maintaining causal regularity, ensuring that cause precedes effect, obtaining solutions to the Regge equations governing the geometry, and achieving a well-defined continuum limit, essential for a consistent physical theory, requires that the connections between neighbouring spacetime slices are timelike, meaning they allow for signals to travel between them.

Further investigation focuses on effective (2+1)-dimensional spin-foam cosmology, a simplified model reducing the complexity of calculations, incorporating a minimally coupled massive scalar field. This reveals that the mass of the scalar field plays a critical role in ensuring the convergence of the path integral, a mathematical tool summing over all possible spacetime geometries to calculate probabilities. Convergence is vital for obtaining finite and meaningful physical predictions.

The research demonstrates a strong connection between the classicality of expectation values, causal regularity, and the path integral measure, highlighting the interplay between quantum and classical descriptions of the universe. A significant contribution lies in the development of a causal completion of the Barrett-Crane group field theory model, a specific formulation of the theory. Amplitudes, representing the probability of a particular spacetime configuration, are explicitly calculated using techniques from integral geometry, a branch of mathematics dealing with geometric objects defined by integrals. A Landau-Ginzburg analysis, a method used to study phase transitions and critical phenomena, assesses the model’s stability. The analysis reveals that mean-field theory, an approximation simplifying complex interactions, remains self-consistent, with timelike faces not significantly influencing the critical behaviour, providing a rigorous test of the model’s internal consistency.

The research successfully couples a physical Lorentzian reference frame to the complete Barrett-Crane model, extracting scalar cosmological perturbations, small deviations from uniformity in the early universe, from group field theory coherent states, quantum states representing a definite geometry. These perturbations demonstrate agreement with classical results, albeit with minor corrections, demonstrating the potential for connecting the theoretical framework to observable cosmological phenomena. This connection represents a crucial step towards testing quantum gravity predictions against cosmological observations.