Noncommutative Geometry and Spacetime: A Historical Reconstruction

Noncommutative geometry (NCG) is a branch of pure mathematics with a wide range of applications to spacetime physics. Stemming from the divergence problem in QFT, modern contributions conjecture that the fundamental structure of spacetime is noncommutative. This seemingly homogeneous picture is the result of almost a century of discontinuous interest in noncommutative spacetime (NCST). In this paper, I present a three-phase division of the development of NCST approaches. The initial phase (1930s-1950s) introduced noncommutativity as a means of addressing the divergence problem while maintaining Lorentz-invariance. The second phase (1950s-early 1990s) initially dismissed noncommutativity and focused on the pressing problem of localisation at high energies. In this context, independent alternative approaches to QG identified the value of a fundamental length scale, which ultimately contributed to a reconsideration of the conjecture of spacetime noncommutativity. This was despite the fact that it undermined the original operationalist methodology. Finally, a third phase (1990s-today) has seen a renewed interest in NCST. I argue that this resurgence can be attributed to the discovery of new mathematics in the 1980s. Furthermore, I demonstrate how the third phase builds upon and continues instances from the previous attempts, but redirects the research question. In light of this, I finally argue that the history of NCG in physics can be understood as an example of "interlaced convergence." This instance of theory convergence may prove useful as a case study for the more general problem of theory building in QG.