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in the role of a particle-antiparticle mixing angle. This mixing angle can also be
interpreted as the effective mass,
m
=
θ
(
m
), acquired by the particle as a result
of the interaction with the discrete lattice. Taylor expansion of both terms in the
above expression shows that
m
/m
=1+
O
(
m
3
), indicating that the effective
mass connects pretty smoothly to its continuum limit (see Figure 2).
6
5
4
bare mass
3
effective mass
2
1
0
0
1
2
3
4
5
6
omega_c/omega_p
Fig. 2.
Lattice renormalization of the bare mass.
In addition, the effective mass stays finite even when the bare mass goes
to infinity, a typical signature of renormalization. As to propagating modes, we
note that the quantum lattice Boltzmann scheme produces the correct dispersion
relation up to second order in the time-step
∆t
. Standard Fourier analysis of (10)
delivers the following dispersion relation:
cos ω∆t
=
cos
(
θ
)
cos
(
k∆x
)
(14)
It is readily checked that a second order expansion of the cosines of the above
dispersion relation yields the well known continuum dispersion relation for rela-
tivistic bosons
ω
2
=
c
2
k
2
+
ω
c
(encoding Lorentz invariance) up to terms
O
(
m
3
).
It is tempting to conjecture that (14) is just a specific instance of a more general
expression of the form
F
(
ω/ω
p
)=
M
(
ω
c
/ω
p
)
F
(
k/k
p
)
(15)
where the function
F
and the form-factor
M
encode the details of the lattice
texture. Some general requirements are identified. The consistency requirement:
lim
m→
0
M
(
m
)=1
,
guarantees compatibility of the discrete dispersion relation with its continuum
photonic limit
ω
=
kc
. Systematic expansions of the form factor
M
(
m
) around
