{getToc} $title={Table of Contents}
Summary
The paper studies the Moyal sphere, a noncommutative space with a metric similar to the spherical metric of ordinary space. The authors calculate the scalar curvature and area of the Moyal sphere and find that it returns to an ordinary sphere when the noncommutative parameter approaches 0.
Highlights
- The Moyal sphere is a noncommutative space with a metric similar to the spherical metric of ordinary space.
- The scalar curvature of the Moyal sphere is calculated and found to return to the curvature of the ordinary sphere when the noncommutative parameter approaches 0.
- The area of the Moyal sphere decreases as the noncommutative parameter increases and approaches 0 when the parameter approaches infinity.
- The total curvature integral of the two-dimensional Moyal sphere coincides with the Gauss-Bonnet formula of the two-dimensional Euclidean space.
- The conformal metric of constant curvature of the Moyal space is studied and an approximate expression is obtained.
- The area and constant curvature metric expression of a generalized deformed Moyal sphere with two noncommutative parameters are calculated.
- The results have implications for the study of mathematical structures and physical properties of noncommutative spaces.
Key Insights
- The Moyal sphere is a noncommutative space that can be studied using the Moyal product, which is a noncommutative product of functions.
- The scalar curvature of the Moyal sphere is an important property that characterizes the space and is found to return to the curvature of the ordinary sphere when the noncommutative parameter approaches 0.
- The area of the Moyal sphere is another important property that is found to decrease as the noncommutative parameter increases and approaches 0 when the parameter approaches infinity.
- The total curvature integral of the two-dimensional Moyal sphere is a topological invariant that coincides with the Gauss-Bonnet formula of the two-dimensional Euclidean space, indicating that the noncommutative algebraic structure of the smooth function space does not change some topological properties of the two-dimensional Moyal sphere.
- The conformal metric of constant curvature of the Moyal space is an important property that characterizes the space and is found to have an approximate expression that deviates from the classical case for small radii and approaches the classical case for large radii.
- The area and constant curvature metric expression of a generalized deformed Moyal sphere with two noncommutative parameters are important properties that are found to have similar results to the Moyal sphere, indicating that the noncommutative parameters have a similar effect on the space.
- The results have implications for the study of mathematical structures and physical properties of noncommutative spaces, which are important in various areas of physics, including quantum field theory and gravity.
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Citation
Chen, H.-L., & Lin, B.-S. (2024). Curvature, area and Gauss-Bonnet formula of the Moyal sphere (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2412.20483