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m = 0 and of the lattice function F around k = 0 and ω = 0, deliver corrections
to the continuum dispersion relation to all orders in the smallness parameter m .
The lowest order (quadratic) terms must reproduce the Klein-Gordon equation.
Renormalizability requires such type of expansions to remain finite even when
the bare mass m is sent to infinity, a property which is manifestly fulfilled by the
relation (15). More generally, the renormalization condition can be formalized
as follows
M ( ω c p )= F ( ω c p )
(16)
which is:
ω c p = F 1 M ( ω c p )
with the further requirement that the function G ≡ F 1 M be sublinear ( G ( z ) <
z , z denoting a generic argument) and remains finite as z is sent to infinity. In
view of condition (16), the discrete dispersion relation reads as:
F ( ω/ω p )= F ( ω c p ) F ( k/k p )
(17)
from which further composition rules in terms of the particle-antiparticle prop-
agators k/k p ± ω c p can be deduced, depending on the specific form of the
lattice function F .
The claim is that, while continuum properties depend on the behaviour of the
lattice function F ( z ) in the vicinity of z = 0, the relevant physics at the Planck
scale is dictated by the behaviour of the same lattice function in the vicinity of
|z| =1.
In order for general relations such as (15), to be interpreted as more fun-
damental than the corresponding continuum limit, one should show that they
possess at least the same symmetries (and possibly more ) than the correspond-
ing continuum partial differential equations. To date, we have not been able to
either prove or disprove such a property for the quantum LBE.
In operatorial sense, the discrete space-time partial differential equation as-
sociated with (15) reads as follows:
F ( t ψ
ω p
)= M ( ω c
ω p ) F ( x ψ
)
k p
where the functor F (now function of operators) resums an infinite series of
partial derivatives associated with the exact representation of discrete derivatives
in the finite space and time grid. Finally, it is worth noting that the discrete light-
cone condition ∆x = c∆t , namely ω p = k p c , is essential to all of the above, for
it cancels a host of instabilities which would otherwise spoil the entire picture.
4 Numerical Simulations
The scheme (10), named “quantum Lattice Boltzmann scheme” (QLBE) has
been demonstrated by actual simulations of some simple quantum mechanical
 
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