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Klein-Gordon dispersion relation: mass=0.2
3
Upper curve: continuum
2.5
Lower curve: discrete
2
1.5
1
0.5
0
-3
-2
-1
0
1
2
3
kc
Fig. 4.
Continuum and quantum LBE dispersion relations for
m
=0
.
2
phenomena that are typically regarded as numerical artifacts or anomalies in
the continuum, might be credited as potentially trustful physical effects. Differ-
ent lattice formulations of a given continuum quantum wave equation can then
provide a guideline to identify the admissible structure(s) of space-time around
the Planck scale. In this light, the relevant ultraviolet limit is no longer the
continuum one,
ω,k →∞
, but rather
ω,k → ω
p
,k
p
. If the continuum limit is
approached smoothly, say with second order corrections in
k/k
p
, there is clearly
no chance of detecting observable effects of space-time discreteness at any scales
but those in the near vicinity of the Planck scale. This conclusion would however
change completely in the case of a turbulent space-time with violent bursts gener-
ating scales much larger than the original Planck scale. The present work is only
scratching the very surface of a di
4
cult subject: quantitative insights require the
description of quantum wave-motion with many internal (non-Abelian) degrees
of freedom in discrete spacetimes with a self-consistent dynamic curvature. This
is a formidable task, but possibly not a completely hopeless one. Random lat-
tices [12], discrete-kinetic versions of Regge calculus [13], cellular automata in
dynamic geometries [14], are just a few tantalizing possibilities. As to the lattice
Boltzmann approach, a few preliminary efforts in this direction (although con-
fined to classical fluids) have made their appearence in the recent past [15]. Much
more work is required to understand whether the discrete kinetic approach can
shed any new light into this di
4
cult and fascinating frontier of modern physics.
References
1. Witten, E.: The ultimate fate of space-time, Physics Today,
49
(1996) 24-28.
2. t'Hooft, G.: A confrontation withinfinity, Int. J. Mod. Pys. A,
15
, (2000) 4395-
4402
