Matrix Models in Quantum Field Theory

Matrix models constitute a powerful nonperturbative framework in which quantum fields and the geometry of space–time are encoded in large matrices. Originally motivated by attempts to formulate string theory and quantum gravity without reliance on perturbative expansions, these models treat space–time itself as an emergent phenomenon arising from the dynamics of matrix degrees of freedom. Key developments have demonstrated that classical space–time manifolds can arise as stable solutions—often termed “fuzzy” geometries—within the matrix action, with fluctuations around these backgrounds yielding effective gauge fields, gravity-like interactions and a tower of higher-spin modes. Matrix models thus provide a unifying language for exploring the quantum structure of space–time, the interplay of gauge symmetries and the emergence of gravitational dynamics. Recent advances have refined our understanding of how Lorentz invariance can be preserved or broken in noncommutative settings, clarified the coupling of matter fields—such as fermions—to emergent geometries, and highlighted novel cosmological and black-hole–like solutions. By bridging concepts from noncommutative geometry, higher-spin theory and numerical simulation, matrix models continue to shed light on longstanding challenges in quantum field theory and quantum gravity, offering concrete mechanisms for space–time emergence, novel ultraviolet completions and testable phenomenological signatures.

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Matrix Models in Quantum Field Theory publication trend

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Technical terms

Matrix model: A theoretical framework in which fields and space–time coordinates are represented by large Hermitian matrices.

Nonperturbative formulation: An approach that captures full quantum dynamics without relying on small-coupling expansions.

Fuzzy geometry: A quantised version of a classical manifold approximated by finite-dimensional matrices, smoothing out singularities.

Higher-spin gauge theory: An extension of gauge theory including massless fields of spin greater than two, interacting consistently.

Torsion: The antisymmetric part of a connection on a manifold, which can couple to spin density of matter fields.

Lorentz symmetry: Invariance of physical laws under rotations and boosts in space–time, a cornerstone of relativity.

References

1. Fermions on curved backgrounds of matrix models. Physical Review D (2023).
2. Unification of conformal gravity and internal interactions. European Physical Journal C (2024).
3. On the quantum structure of space-time, gravity, and higher spin in matrix models. Classical and Quantum Gravity (2020).
4. Covariant 4-dimensional fuzzy spheres, matrix models and higher spin. Journal of Physics A: Mathematical and Theoretical (2017).