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3 Discrete Limit and the Quantum Lattice Boltzmann
Equation
Let us now go down to a scale such that space-time discreteness can no longer
be ignored. For coinciseness, we shall identify the Planck length and time with
the corresponding lattice pitches:
t p = ∆t, l p = ∆x = c∆t,
The question is to write down a fully discrete equation yielding the Klein-Gordon
equation in the large-scale limit. In doing so, we shall require the discrete formu-
lation to comply with fundamental principles of stability, unitarity and space-
time locality. These requirements provide a selection rule for the structure of the
discrete lattice. Once these are secured, the finite size of the lattice pitch, ∆t ,
should simply affect the quantitative values of the measurable observables, such
as the effective mass and group speed of the propagating waves. Along the spirit
of taking the lattice for real, we shall try to interpret these departures from the
continuum limit as potential trustful physical phenomena at the ”Planck“ scale.
Let us now proceed to examine a few simple discretizations.
An explicit time-marching along the trajectories would deliver the exact ana-
logue of the lattice Boltzmann equation for classical fluids [7,8,9]:
r ( x + ∆x,t + ∆t ) − r ( x,t )= ml ( x,t )
(6)
l ( x − ∆x,t + ∆t ) − l ( x,t )= −mr ( x,t )
(7)
where r,l are shortands for right/left-ward propagation. Note that in physical
units m is the ratio of particle Compton frequency ω c = mc 2 /h , to the 'Planck'
frequency ω p 2 π/∆t :
m = ω c ∆t = ω c p
The scheme (6)is immediately shown to be unconditionally unstable for any size
of ∆t , no matter how small. Hence, discrete quantum motion can not take place
on such type of space-time connectivity.
The
typical
remedy
is
the
well-known
Cranck-Nicolson
implicit
time-
marching:
r ( x + ∆x,t + ∆t ) − r ( x,t )= m
2 [ l ( x,t )+ l ( x + ∆x,t + ∆t )](8)
l ( x − ∆x,t + ∆t ) − l ( x,t )= −m
2
[ r ( x,t )+ r ( x − ∆x,t + ∆t )](9)
This corresponds to integrating the right-hand-side (collision term in Boltzmann
language) along the characteristic trajectory of the field at the left-hand side.
It is immediately seen that, due to the simultaneous dependence introduced by
terms at t + ∆t at the right-hand side, the state of the quantum field at ( x,t ) de-
pends on all sites at the previous time step t−∆t . From a strict numerical point
of view, this implies the solution of a linear algebraic system, a rather heavy
task. However, here we shall not deal with numerical issues, but focus instead on
 
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