Quantum spacetime and gravity from the IKKT matrix model: an invitation

For most of the twentieth century, physicists operated under two conflicting rules. One rule, quantum mechanics, governs the very small, describing how particles behave like waves and exist in multiple states at once. The other rule, general relativity, governs the very large, describing gravity as the smooth bending of space and time by massive objects. For decades, these two rules worked perfectly in their own domains, but they refused to work together. When scientists tried to combine them to describe the center of a black hole or the very first moment of the universe, the math broke down, producing infinite values that made no sense. This failure suggests that our current understanding of space and time is incomplete, likely because we are treating them as a smooth, continuous fabric when they might actually be made of something more fundamental and discrete.

Harold C. Steinacker, a physicist at the University of Vienna, offers a new way to think about this problem. He explores a framework based on a mathematical structure known as the IKKT matrix model. Instead of starting with a pre-existing stage of space and time where particles perform, this approach suggests that space and time themselves emerge from the interactions of giant, abstract grids of numbers called matrices. In this view, the universe is not a container but a result. The paper serves as a guide to how this model can generate a universe that looks like our own, complete with expanding space, gravity, and the specific laws of physics we observe, without needing to force the theory to fit into a pre-defined shape.

The journey begins with a simple idea: imagine a collection of matrices, which are essentially grids of numbers, acting as the fundamental building blocks of reality. In the standard view of physics, we assume space exists first, and then we place fields and particles into it. Steinacker's work flips this script. He proposes that if you arrange these matrices in a specific, non-trivial way, the patterns they form naturally create the geometry of space and time. These matrices do not sit in a pre-existing void; their relationships define what "here" and "there" mean. When the matrices interact, they generate a structure that behaves like a four-dimensional universe with three dimensions of space and one of time.

A major hurdle in previous attempts to unify physics has been the "landscape" problem, particularly in string theory. This issue arises when a theory allows for a vast, almost infinite number of possible universes, each with different laws of physics, making it impossible to predict which one we live in. Steinacker's approach avoids this trap, but not by proving that a single vacuum is inevitable. Instead, the landscape problem is avoided because this specific approach does not require the compactification of extra dimensions, a process that in string theory introduces vast ambiguities. While it remains an open question whether one specific matrix configuration will dominate the physics or if many contribute significantly, the model focuses on a class of candidate vacua that describe an expanding universe without needing to navigate the complex landscape of string compactification.

The paper details how this matrix universe evolves. It describes a background that expands over time, similar to the Big Bang theory, but with a crucial difference: it includes a "Big Bounce." Instead of starting from a singular point of infinite density where the laws of physics break down, the universe in this model contracts to a minimum size and then expands again. This avoids the problematic singularity that plagues classical theories. The mathematics shows that this expanding space is stable and contains a finite number of degrees of freedom per volume, meaning the universe is not infinitely divisible but has a fundamental graininess. This graininess is key to resolving the inconsistencies that arise when trying to merge gravity with quantum mechanics.

Gravity, in this framework, is not a fundamental force but a quantum effect that arises from the behavior of the matrices. When the matrices fluctuate or wiggle around their stable arrangement, these fluctuations generate an action that leads to the Einstein-Hilbert action, the mathematical formula that describes gravity in general relativity. However, this gravity is not identical to Einstein's theory; the resulting gravitational sector is considerably richer and remains to be understood in detail. The model predicts that at very large scales, gravity behaves differently than Einstein predicted. It suggests the existence of "mirage matter," a phenomenon where the effects of ordinary matter are reflected in a way that mimics the presence of invisible mass or dark energy. This could explain why galaxies rotate faster than expected and why the universe's expansion is accelerating, without needing to invent new, unseen particles.

The model also addresses the issue of extra dimensions. String theory often requires ten or eleven dimensions, with the extra ones curled up so tightly they are invisible. Steinacker's work shows that the extra dimensions in the matrix model can be stabilized in a way that keeps them compact but does not destroy the non-perturbative power of the theory. This means the extra dimensions are not just mathematical artifacts but play a real role in stabilizing the universe and giving mass to particles. The theory suggests that the particles we see, including those that make up matter and forces, are actually vibrations of the matrices in these extra dimensions.

One of the most significant findings is how the model handles the passage of time. In many quantum gravity theories, time is a difficult concept to define because the equations are static. Here, time emerges naturally from the evolution of the matrix configurations. The model defines a physical Hilbert space, which is a mathematical way of describing all possible states of the system, and shows that this space evolves in a way that preserves the probability of events, a property known as unitarity. This ensures that the theory is consistent and that information is not lost, a problem that has long troubled physicists studying black holes.

The paper acknowledges that while the framework is promising, it is still a work in progress. The author has shown that the basic ingredients for a realistic universe can emerge from the matrix model, including the correct number of dimensions, the expansion of space, and the emergence of gravity. However, the detailed dynamics of how gravity behaves at all scales, especially in the non-linear regime where the math becomes extremely complex, still requires further investigation. The concept of "mirage matter" and its role in the universe's evolution is a suggestion that needs more rigorous testing. The stability of the cosmic background also depends on specific conditions that have been identified but not yet fully explored in all their complexity.

Ultimately, this work offers a fresh perspective on the nature of reality. It suggests that the smooth, continuous space and time we experience are not the fundamental layer of the universe. Instead, they are a large-scale illusion created by the intricate dance of numbers. By starting with these numbers and letting the mathematics dictate the outcome, the model provides a path toward a theory of everything that is free from the infinities and singularities that have plagued physics for a century. It does not claim to have solved every mystery, but it provides a coherent and mathematically sound foundation for understanding how a universe like ours could arise from the simplest possible ingredients. The path forward involves refining these calculations and testing whether the predicted modifications to gravity can be observed, potentially offering a new way to understand the dark energy and dark matter that dominate our cosmos.