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problems [10]. Recently, it has also been extended to the case of non-linear rela-
tivistic interactions [11]. Apart from numerical considerations, the message from
QLBE seems to be that violation of causality and locality between matter and
anti-matter at a lattice scale does not translate into a corresponding violation
at larger scales, but it is reabsorbed into an effective mass and group speed of
the quantum wavepacket.
This is shown in Figures 3 and 4.
Klein-Gordon dispersion relation: mass=1
3.5
3
2.5
Continuum
2
Discrete
1.5
1
0.5
0
-3
-2
-1
0
1
2
3
kc
Fig. 3. Continuum and quantum LBE dispersion relations for m =1 . 0
From Figure 3, we see that in the proximity of the lattice scale ( m =1,
kc = π ) the discrete scheme exhibits a strong departure from the continuum
dispersion relation. In particular, the quantum wave packet is basically stopped
by the interaction with the lattice. At just twice larger wavelengths, such an
effect is much less dramatic. And much more so at frequencies smaller than the
lattice cut-off frequency (Planck frequency) ω p , as indicated in Figure 4. From
this Figure, it is apparent that departures from the continuum dispersion relation
remain very negligible up to wavelengths pretty close to the lattice cut-off. As
a result the major effect of lattice discreteness is a sensible renormalization of
the particle mass and a substantial slowing down of the particle speed. In our
opinion, it is by no means obvious that these effects should be regarded as mere
“computational artifacts”.
5 Summary
At a scale at which space-time continuity becomes questionable various finite-
difference formulations of quantum wave equations become actual statements
on the fine-grain structure of space-time. Once this point of view is endorsed,
 
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