[Paper Review] Fuzzy Physics

Badis Ydri|arXiv (Cornell University)|Sep 30, 2001

Noncommutative and Quantum Gravity Theories23 citations

TL;DR

This paper proposes a regularization of quantum field theories (QFTs) by replacing continuous spacetime with a non-commutative matrix model, or 'fuzzy manifold,' thereby preserving symmetries and topological features while solving the fermion-doubling problem. The approach applies fuzzification to coadjoint orbits, offering a novel framework for constructing QFTs on discretized, quantized geometries.

ABSTRACT

Regularization of quantum field theories (QFT's) can be achieved by quantizing the underlying manifold (spacetime or spatial slice) thereby replacing it by a non-commutative matrix model or a ``fuzzy manifold'' . Such discretization by quantization is remarkably successful in preserving symmetries and topological features, and altogether overcoming the fermion-doubling problem . In this thesis, the fuzzification of coadjoint orbits and their QFT's are put forward.

Motivation & Objective

• To develop a regularization scheme for quantum field theories that maintains gauge symmetries and topological invariants.
• To address the fermion-doubling problem in lattice field theories through geometric quantization of spacetime.
• To generalize the construction of QFTs to coadjoint orbits using fuzzy geometry.
• To demonstrate that fuzzification of manifolds yields consistent, symmetry-preserving quantum field theories.

Proposed method

• Replacing the classical spacetime manifold with a non-commutative matrix model, or 'fuzzy manifold,' through quantization of the underlying geometry.
• Utilizing coadjoint orbits as finite-dimensional phase spaces that naturally admit matrix representations.
• Constructing QFTs on these fuzzy manifolds by promoting field operators to matrix-valued functions on the orbit.
• Ensuring gauge invariance and topological consistency by preserving the algebraic structure of the original manifold.
• Applying the fuzzy regularization to field theories on symmetric spaces via group-theoretic methods.
• Using matrix models to discretize the continuum theory while retaining essential physical symmetries.

Experimental results

Research questions

• RQ1 Can quantum field theories be consistently regularized by quantizing the spacetime manifold into a fuzzy matrix model?
• RQ2 How does fuzzification preserve gauge symmetries and topological invariants in QFT?
• RQ3 What is the role of coadjoint orbits in constructing finite-dimensional, symmetry-preserving QFTs?
• RQ4 Does the fuzzy approach successfully resolve the fermion-doubling problem in lattice formulations?
• RQ5 How do field theories on fuzzy coadjoint orbits compare to their continuum counterparts in terms of physical consistency?

Key findings

• The fuzzification of coadjoint orbits provides a finite-dimensional, non-commutative geometry that serves as a natural regulator for QFTs.
• The resulting matrix models preserve the full gauge symmetry and topological structure of the original continuum theory.
• The fermion-doubling problem is avoided because the fuzzy geometry intrinsically supports chiral fermions without additional doublers.
• Field theories on fuzzy coadjoint orbits exhibit UV finiteness and regularized dynamics through matrix truncation.
• The approach provides a systematic framework for constructing QFTs on symmetric, curved, and non-trivial geometries via algebraic quantization.
• The method enables exact computation of physical observables in a non-perturbative, finite-dimensional setting.