The points of space are not points: their wavefunctions remove the ultraviolet divergences of quantum field theory

Abstract

The points $p(x)$ of space are observables that obey quantum mechanics with average values $\langle p(x) \rangle$ that obey general relativity. In empty flat 3-space, a point $p(x)$ may be represented as the sum $ \boldsymbol p(x) = \langle \boldsymbol p(x) \rangle+ \boldsymbol \epsilon(x)$ of its average value $\langle \boldsymbol p(x) \rangle = \boldsymbol x$ and its quantum remainder $ \boldsymbol \epsilon(x)$. The ubiquitous Fourier exponential $\exp( i \boldsymbol k \cdot \boldsymbol p(x))$ is then $\exp( i \boldsymbol k \cdot ( \boldsymbol x + \boldsymbol \epsilon))$. If the probability distribution of the quantum remainders $ \boldsymbol \epsilon(x)$ is normal and of width $\sigma$, then the average of the exponential $\langle \exp(i \boldsymbol k ( \boldsymbol x + \boldsymbol \epsilon ) \rangle$ contains a gaussian factor $ \exp( - \sigma^2 \boldsymbol k^2/2 ) $ which makes self-energies, Feynman diagrams and dark energy ultraviolet finite. This complex of ideas requires new bosons and suggests that supersymmetry is softly broken.