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2 Preliminaries
Let us consider relativistic bosons described by the Klein-Gordon equation (in
1D for simplicity):
( t − c 2 a ) φ = −ω c φ
(1)
where
ω c = mc 2 /h
is the Compton frequency of the material particle. We describe relativistic bosons
by a set of n =2 D complex wavefunctions ψ j ( x,t ), such that the j -th component
can propagate only along the j -th direction according to the light-cone condition:
dx ja = c ja dt
(2)
where j =1 ,n runs over the discrete directions and the latin index a runs over
spatial dimensions 1 ,D . The discrete wavefunctions are postulated to fulfill the
following (complex) Boltzmann equation:
j ψ j = M jk ψ k
(3)
where:
j ≡ c i,µ µ =0 ,D
(4)
is the covariant derivative along direction j . Here the (anti-symmetric) scatter-
ing matrix M jk represents the interaction between the j and k components of
the spinorial wavefunction, the analogue of Boltzmann collisions. More precisely
each wavefunction has a dual partner propagating along the opposite direction:
according to Feynman's picture this dual partner can be identified with an anti-
particle propagating backwards in time. The Klein-Gordon equation is obtained
by acting upon (3) with the unitary matrix U jk :
1 i
1 − i
2
U =
(5)
Simple algebra delivers the Klein-Gordon equation for the partner fields φ j =
U jk ψ k . It is known that that in the adiabatic limit v/c << 1, the fast antisym-
metric mode (index 2) is enslaved to the slow symmetric one (index 1),
|∂ t φ 2 | << 2 m|φ 2 |
so that the Klein-Gordon equation reduces to the non-relativistic Schroedinger
equation. This limit is formally analogue to the adiabatic relaxation of the kinetic
Boltzmann equation to the fluid-dynamic Navier-Stokes equations. This analogy
provided the starting point for the quantum Lattice Boltzmann scheme [6]to be
described in the next section.
 
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