Quantum Path Integrals on Non-Differentiable Spaces

In differential geometry, Hermann Weyl showed that defining the Laplace-Beltrami operator on Riemannian manifolds has helped to uncover geometric information from spectral data encoded in functions like the heat kernel and zeta function. This is reflected in the question famously raised in Mark Kac's paper 1966, "Can One Hear the Shape of a Drum?" However, defining geodesics and other geometric information on non-differentiable spaces such as quantum graphs and fractals have been less explored but exhibit exotic properties that have potential to uncover new physics. By extending the study of the heat kernel trace and zeta function to irregular mathematical structures, we aim to compute the quantum path integral beyond Riemannian geometry. As a result, we developed an algorithm to allow for a geometric interpretation of Roth weights on complete graphs, revealing novel insights into parity and symmetry structure by enumerating all closed paths. Additionally, we generalized the short-time equations of the heat kernel trace on a diamond fractal and the Weierstrass function, thereby revealing better agreement with long-time behavior. These findings support the initial hypothesis that quantum path integrals can be generalized beyond Riemannian geometry by leveraging spectral geometry on discrete spaces.