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the structure of space-time generated by a given numerical discretization. The
above scheme is easily shown to be unconditionally stable, for any
∆t
. Stability
comes however at the price of wasting locality, the 'revenge' of space-time for
overruling the principle of causality. Thus, again, this type of connectivity can-
not correspond to a realizable quantum motion at the lattice space-time scale.
Interestingly,
causality can be restored without loosing stability.
Consider the
following-apparently minor-variation of the Cranck-Nicolson scheme (see Figure
1):
r
(
x
+
∆x,t
+
∆t
)
− r
(
x,t
)=
m
2
[
l
(
x,t
)+
l
(
x − ∆x,t
+
∆t
)](10)
l
(
x − ∆x,t
+
∆t
)
− l
(
x,t
)=
−m
2
[
r
(
x,t
)+
r
(
x
+
∆x,t
+
∆t
)](11)
l^
r^
t+dt
l r
t
x−dx
x
x+dx
Fig. 1.
Space-time diagram of the modified Cranck-Nicolson scheme. The interaction
takes place along the arrows labeled
r
and
l
In other words, at the right-hand side, quantum fields are evaluated at the
end position of their own flight-path. This is twice non-local, since it adds a non-
locality in space on top of the usual time non-locality of the standard Crank-
Nicolson scheme. To be noted that
the non-causal link only connects matter-
antimatter fields
. These two non-localities annihilate each other in a way to
produce a
local
and yet
stable
scheme for any size of
∆t
. This is easily realized
by noting that (10) is a simple 2
×
2 system which can be solved independently
site by site, to deliver the following explicit scheme:
r
(
x
+
∆x,t
+
∆t
)=
r
(
x,t
)
cosθ − l
(
x,t
)
sinθ
(12)
l
(
x − ∆x,t
+
∆t
)=
r
(
x,t
)
sinθ
+
l
(
x,t
)
cosθ
(13)
with
cosθ
=(1
− m
2
/
4)
/
(1 +
m
2
/
4)
