[Paper Review] Simulation Results for U(1) Gauge Theory on Non-Commutative Spaces

Wolfgang Bietenholz, Antonio Bigarini+4 more|ArXiv.org|Aug 14, 2007

Noncommutative and Quantum Gravity Theories2 references3 citations

TL;DR

This paper presents non-perturbative lattice simulations of U(1) gauge theory on non-commutative (NC) spaces using a twisted matrix model formulation, enabling numerical study of NC QED in 2D and 4D. The key result is that in 4D, the Double Scaling Limit leads to a phase with broken translation symmetry that is IR-stable, suggesting photons can survive in a non-commutative world, while in 2D, Wilson loop invariance under area-preserving diffeomorphisms is broken non-perturbatively.

ABSTRACT

We present numerical results for U(1) gauge theory in 2d and 4d spaces involving a non-commutative plane. Simulations are feasible thanks to a mapping of the non-commutative plane onto a twisted matrix model. In d=2 it was a long-standing issue if Wilson loops are (partially) invariant under area-preserving diffeomorphisms. We show that non-perturbatively this invariance breaks, including the subgroup SL(2,R). In both cases, d=2 and d=4, we extrapolate our results to the continuum and infinite volume by means of a Double Scaling Limit. In d=4 this limit leads to a phase with broken translation symmetry, which is not affected by the perturbatively known IR instability. Therefore the photon may survive in a non-commutative world.

Motivation & Objective

• To investigate the non-perturbative behavior of U(1) gauge theory on non-commutative spaces, where perturbative methods fail due to UV/IR mixing.
• To resolve the longstanding question of whether Wilson loops in 2D NC U(1) gauge theory are invariant under area-preserving diffeomorphisms (APD).
• To determine whether a stable, physically viable phase exists in 4D NC QED, particularly whether the photon survives despite perturbative IR instabilities.
• To establish a non-perturbative framework for studying NC gauge theories using a mapping to a twisted Eguchi-Kawai (TEK) matrix model.
• To extrapolate results to the continuum and infinite volume via the Double Scaling Limit (DSL), ensuring a well-defined NC geometry.

Proposed method

• Map the non-commutative plane to a twisted Eguchi-Kawai (TEK) matrix model, enabling numerical simulations via unitary matrices $ U_{ u} $ of size $ N \times N $.
• Use the star product formalism to define the NC gauge action: $ S[A] = \frac{1}{4} \int d^2x\, F_{\mu\nu} \star F_{\mu\nu} $, with $ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + ig[A_\mu, A_\nu]_\star $.
• Define the NC Wilson loop in the matrix model as $ W_{\mu\nu}(I\times J) = \frac{1}{N} Z_{\mu\nu}^{IJ} \mathrm{Tr}[U_\mu^I U_\nu^J U_\mu^{\dagger I} U_\nu^{\dagger J}] $, which is star-gauge invariant and numerically tractable.
• Implement the Double Scaling Limit (DSL) by taking $ a \to 0 $, $ N \to \infty $, with $ Na^2 = \text{const.} $, to reach a continuous, infinite NC plane with fixed non-commutativity parameter $ \theta \propto N / \beta^2 $.
• Measure the open Polyakov line as an order parameter for translation symmetry breaking: $ P_\mu(n) = \mathcal{P} \exp_\star\left(ig \int_x^{x+\tilde{p}_\mu} A_\mu(\xi) d\xi_\mu \right) $.
• Perform simulations across varying $ \beta $, $ N $, and $ a $, and extrapolate to the DSL using controlled scaling, ensuring consistency with the continuum limit.

Experimental results

Research questions

• RQ1 Does the invariance of Wilson loops under area-preserving diffeomorphisms (APD) survive non-perturbatively in 2D non-commutative U(1) gauge theory?
• RQ2 Can a stable, IR-finite phase of NC QED exist in 4D despite perturbative IR instabilities?
• RQ3 Is the photon a viable physical degree of freedom in a non-commutative spacetime, particularly in the Double Scaling Limit?
• RQ4 Does the Double Scaling Limit lead to a well-defined, continuous NC geometry in both 2D and 4D?
• RQ5 What is the nature of the phase transition in NC QED 4D, and does it correspond to spontaneous breaking of translation symmetry?

Key findings

• In 2D, non-perturbative simulations show that Wilson loop invariance under area-preserving diffeomorphisms is broken, including the subgroup $ SL(2,\mathbb{R}) $, indicating no residual symmetry.
• In 4D, the Double Scaling Limit leads to a phase of intermediate coupling strength with broken translation symmetry, which is IR-stable and does not suffer from the perturbatively known IR instability.
• The open Polyakov line exhibits hysteresis and a first-order phase transition at $ \beta \simeq 0.35 $, with the broken phase extending as $ \beta \propto N^2 $, confirming spontaneous symmetry breaking.
• The photon dispersion relation in the broken phase is linear and massless, consistent with the Nambu-Goldstone mode of spontaneously broken translation symmetry.
• The DSL trajectories for fixed $ \theta $ always terminate in the broken phase, as $ \beta \propto \sqrt{N} $ in the DSL, while the broken phase extends as $ \beta \propto N^2 $, ensuring physical relevance.
• Numerical results confirm that the TEK model correctly reproduces the non-commutative U(1) gauge theory at finite $ N $, validating the matrix model approach for non-perturbative studies.