Original $\mathbb F_1$ in emergent spacetime

Abstract

The existence of a quantum field theory over the "field with one element" was first addressed in 2012 by Bejleri and Marcolli, where it was shown that wonderful compactifications of the graph configuration spaces that appear in the calculation of Feynman integrals, as well as the moduli spaces of curves, admit an $\mathbb F_1$ structure. Recently, we also examined some advantages of studying finite fields $\mathbb F_q$, wherein $\mathbb F_1$ represents the fundamental string with the Planck length, playing a fundamental role in the model. Such a role was briefly described by comparing the similarity between the collapse of the spacetime concept probing the scales below the Planck length and the mathematical collapse of the 'field' concept at $q=1$. In this letter, we elaborate more on this role by explaining how Kapranov and Smirnov's perspective based on the Iwasawa theory is a perfect mathematical fit for string theory. Particularly, their work suggests that the existence of $\mathbb F_1$ alone is sufficient to create other fields $\mathbb F_q$ emergent as its extensions, reflecting the postulate of string theory, where various vibrational modes of the fundamental string manifest as other fields. As support, a couple of evidence are provided that illustrate the physical importance of the Weyl group (known to be a reductive group over $\mathbb F_1$) and explain why the calculation of the amplitudes in the "amplitudes=combinatorial geometry" program (initiated by Arkani-Hamed et al. in 2013) exhibits a combinatorial nature and simplifies the exponentially growing ($\mathcal{O}(4^n)$) calculations of the Feynman diagrams to a polynomially growing order ($\mathcal{O}(n^2)$) in the kinematic space of scattering data for the scalar theory with cubic interactions.