St-Petersburg State University Research Institute of Physics Faculty of Physics High Energy and Elementary Particles Physics
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Discretization of space-time is a necessary
step  in  quantizing  gauge theories, including
gravity,   when,  as  a result of quantization,
the gaugegroup becomes finite and infenetezimal
transformations   sieze   to   make  sense.  We
identify   a   minimal   set  of  variables  of
discretized    gravity   on   which   Newtonian
interaction   between   two  particles  can  be
calculated.   The   space   is  discretized  by
dividing    it    into   quadrants   by   three
three-dimensional  surfaces intersecting at the
worldlines  of  the  particles.  The  variables
needed  for calculating Newtonian potential are
the  holonomies of $SO(4,1)$ connection dual to
the  above  three  surfaces plus the holonomies
around the images of the particle worldlines on
every  surface.  The  equations  of motions for
these   variables   are  ordinary  differential
equations    with   no   spatial   derivatives.
Newtonian  portential  in  the  usual  form  is
reproduced  from  linearized  version  of these
equations, in which the holonomies of $SO(4,1)$
group  are  replaced  by  abelian holonomies of
$so(4,1)$  algebra.  The  full  equations  with
$SO(4,1)$  holonomies indicate  the presence of
a  lower  bound  on  the  distance  between the
particles.  In the limit when  one particle has
much  bigger  mass  than  the other, this bound
corresponds  to the Schwarzschild radius of the
heavy   particle.    We  discuss  the  possible
implications  of this effect for short dictance
physics in quantum theory.
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March 2009