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In-Depth Information
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
,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.