Zenith Grant Awardee
Dr. Jenny Harrison
University of California at Berkeley
Project Title
Axioms of Calculus and Mechanics
Project Summary
We present axioms of calculus that unify the discrete and the smooth continuum with applications to fractals, soap films, charged particles, and smooth manifolds. Properties that follow from the axioms include broad generalizations of the integral theorems of calculus that are equally valid for the above domains. A goal for the period of this grant is to establish similar axioms of mechanics that unify the quantum and classical viewpoints.
Technical Abstract
We present axioms of calculus that unify the discrete and the smooth continuum with applications to fractals, soap films, charged particles, and smooth manifolds. Properties that follow from the axioms include broad generalizations of the integral theorems of calculus that are equally valid for the above domains. Emerging from the axioms is a new derivative and integral called prederivative and preintegral. The first takes on the role of weak derivative of distributions and permits us to multiply singular distributions without a contradiction of mathematics. Preintegral together with prederivative suggest a way to formalize conservation of energy in a curved space. Hodge decomposition in our category leads to rigorous proof of the existence of the Coulomb field and Maxwell's equations. A similar proof shows how 2-dimensional shell fields form a dynamic fabric of space. We find new representations of the CCR with physical realism using prederivative and its adjoint. A goal for the period of this grant is to establish similar axioms of mechanics that unify the quantum and classical viewpoints.
QSpace Latest
PressRelease: Precision experiment puts pressure on quantum collapse theories
Quantum mechanics, the theory governing the microscopic world, is famously counterintuitive. A particle can exist in a superposition of multiple states, such as different positions, until a measurement is performed. At that point, the wavefunction describing that particle appears to ‘collapse’ to a single outcome. This puzzle lies at the heart of the measurement problem, famously illustrated by Schrödinger’s cat, suspended between life and death until observed. The XENONnT detector, which was designed to be sensitive to rare physics events, has tightened constraints on one family of possible solutions to the measurement problem, known as ‘collapse theories.’ The work, which was partially funded by FQxI, was reported in Physical Review Letters in March 2026. Image credit: XENON Collaboration.