Oxidation, Reduction and Semi-Classical Limit for Quantum Matrix Geometries

Abstract

Matrix configurations define noncommutative spaces endowed with extra structure including a generalized Laplace operator, and hence a metric structure. Made dynamical via matrix models, they describe rich physical systems including noncommutative gauge theory and emergent gravity. Refining the construction in [1], we construct a semi-classical limit through an immersed submanifold of complex projective space based on quasi-coherent states. We observe the phenomenon of oxidation, where the resulting semi-classical space acquires spurious extra dimensions. We propose to remove this artifact by passing to a leaf of a carefully chosen foliation, which allows to extract the geometrical content of the noncommutative spaces. This is demonstrated numerically via multiple examples.

Figures (5)

• Figure 1: A covering with local coordinates of the hybrid leaf through $(1,2,1)\in\mathbb{R}^3$ for $N=4$, $\alpha=0.9$. Left: quantum manifold $\mathcal{M}$, right: embedded quantum space $\Tilde{\mathcal{M}}$
• Figure 2: Modifiedjointfootnote relative deviation of (\ref{['Hypothesis']}.1) ($d_\mathbb{1}$, blue), (\ref{['Hypothesis']}.2) ($d_X$, orange) and (\ref{['Hypothesis']}.3) ($d^m_{\{\}}$, green). The generic parameter choice is the setup of figure \ref{['fig:sfsSpaces']}. Top left: dependence on $N$ ($m=100$), top right: dependence on $\alpha$ ($m=100$), bottom left: dependence on the choice of leaf ($m=2$) with $\lambda\cdot(1,2,1)\in\mathbb{R}^3$ lying in and hence specifying the leaf, bottom right: dependence on the choice of distribution ($m=100$) with TS being the hybrid leaf, QM being the hybrid leaf using omega and GQM being the same with differently constructed coordinates
• Figure 3: Left: comparison of the classical and quantum space constructed with identical random coefficients, blue: classical space -- image of random points on the sphere, orange: embedded quantum space -- scan of the hybrid leaf through $(1,2,1)\in\mathbb{R}^3$ for $N=10$ and $c=4$, right: plot similar to those of figure \ref{['fig:sfsValidation']} depending on $c$ ($m=2$)
• Figure 4: Left: a covering with local coordinates of the hybrid leaf through $(1,2,1,0)\in\mathbb{R}^3$ for $N=5$ -- a stereographic plot of the embedded quantum space, right: plot similar to those of figure \ref{['fig:sfsValidation']} depending on $N$ ($m=0.02$)
• Figure 5: Left: embedded quantum space for $n=3$ -- image of Cartesian coordinate lines in target space, right: embedded quantum space for $n=3$ (different perspective) -- two directions of local coordinates in the hybrid leaf