$p$-adic String Theories Provide Lattice Discretization to the Ordinary String Worldsheet

A class of models called $p$-adic strings is useful in understanding the tachyonic instability of string theory. These are found to be empirically related to the ordinary strings in the $p\to 1$ limit. We propose that these models provide discretization for the string worldsheet and argue that the limit is naturally thought of as a continuum limit in the sense of the renormalization group.

Figure 1

The worldsheet of the 3-adic string ${\mathcal{B}}_3$, $\partial {\mathcal{B}}_3={\mathbf{Q}}_3$. The dotted line is the path from the boundary points 0 to $\infty$.

Figure 2

The lattice with spacing $a^{\prime }$ leads to a coarse-grained one with spacing $a=m{a}^{\prime }$ ($m=3$ here), when the gray branches are integrated out.