36. The shape of points
This section describes a research topic,∗ and as such is not a compendium of generally
Using only the expressions for the Compton wavelength λ = h/mc and for the Schwarz-
schild radius rs = 2Gm/c2, a number of arguments are presented which lead to the con-
clusion that at Planck energies, space-time points and point particles must be described, in
contrast to their name, by extended entities. These arguments point towards a connection
between microscopic and macroscopic scales, confirming the present results of string the-
ory and quantum gravity. At the same time, they provide a pedagogical summary of this
aspect of present day theoretical physics.
1. It is shown that any experiment trying to measure the size or the shape of an elementary
particle with high precision inevitably leads to the result that at least one dimension of the
particle is of macroscopic size.
2. It is argued that there is no data showing that space-time is continuous, but enough
data showing that it is not. It is then argued that as a consequence, one necessarily needs
extended building blocks to build up an entity, such as the vacuum, which is extended in
3. The existence of minimum measurable distances and time intervals is shown to im-
ply the existence of space-time duality, i.e. a symmetry between very large and very small
distances. Space-time duality in turn implies that the fundamental entities which make up
vacuum and matter are extended.
4. In another, purely logical argument it is argued that the constituents of the universe and
thus of space-time, of matter and of radiation cannot form a set and that as a consequence
any useful description of nature must use extended entities.
5. The Bekenstein–Hawking expression for the entropy of black holes is used to argue
that its surface dependence confirms that both space-time and particles are composed of
6. A similar argument, based on extending statistical properties to Planck scales, shows