[FOM] Could spacetime be discrete?

Giuseppe.Longo@ens.fr Giuseppe.Longo at ens.fr
Tue Jan 17 02:17:22 EST 2006

On Saturday 14 January 2006 21:11, H Z wrote:
> On 1/14/06, Alasdair Urquhart <urquhart at cs.toronto.edu> wrote:
> > Richard Haney raises the question as to whether spacetime
> > could be discrete.  I am not sure what this means, but if
> > it means that there is a minimum length, this appears
> > inconsistent with special relativity.

The only reasonable mathematical definition of discrete, I guess, is:
the discrete topology is natural
whatever natural? may mean in mathematics (but we know what it may mean:
the  discrete topology on Cantor s reals is NOT natural, say, or Friedman
s  combinatorial statements are natural?; you may accept this or not - if
not,  you just miss something).
Now, in Quantum Physics non-locality, non-separability are the opposite of
 discrete in that sense: the discrete topology separates points, one may 
access to them locally, individually.
Of course, from the energy spectrum and Planck s h (which has the
dimension of  an action) one may deduce Planck s time and length. But
these are related to  measure and access to phenomena, in no way they are
intrinsic to space and  time. And this access is subordinated to the
entanglement effect, just  mentioned.
Space-time is not Cantor continuous nor topologically discrete, it is
whatever  it is. We have fantastic mathematics of continua (from the
Geometry of  non-linear systems to Relativity, in modern times), with
their intended  structure of determination (such as: determined does not
need to imply  predictable, since Poincare) and a still hill understood
mathematics for the  interface between microphysics (Feynman integral,
typically) and those other  areas of (mathematical-)Physics. One thing is
sure: the Laplacian determination, which comes  with discrete data bases,
discrete clocks and space, as Turing understands in  his late papers, is
remote from todays microphysics, with entanglement and  its intrinsic
probability correlation, as for measure (yet, Shroedinger  equations are
linear, which greatly helps in computer simulations!).  The major issue is
that there are no space-time trajectories in Quantum  Mechanics (the
mathematical challenge of Feynman integral), after 2400 years  of a
Physics of trajectories (Aristotle, Galileo, Newton, Laplace, Poincare, 
Einstein... even Boltzmann, all worked on trajectories). No naive approach
is allowed in these frames.
More on this may be found in:
Giuseppe Longo.  Laplace, Turing and the "imitation game" impossible
geometry:  randomness, determinism and programs in Turing's test.
(downloadable : http://www.di.ens.fr/users/longo/ )

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