Note that in the spacetime state
framework, we may replace talk of states of particles relative to elements of a foliation
with direct talk of the states of spacetime regions, such as
R
j
,S
j
and
T
j
.
18
sense in which nonunitary dynamics has consequences going beyond those of
unitary dynamics. To this end we shall now allow ourselves to talk directly of
the states of the pertinent spacetime regions shown in Figure 1; considering first
the unitary case, before seeing how things differ under nonunitary dynamics.
Thus suppose that in Figure 1 the transformations performed at
X
1
and
X
2
were unitary—say,

0
) −→ 
1
)

1
) −→ − 
0
)
;
(5)
a spinflip rather than a measurement. The states of the various relevant regions
are easily calculated:
hatwide
ρ
R
1
=
hatwide
ρ
S
1
=

α

2

0
) (
0

+

β

2

1
) (
1

hatwide
ρ
R
2
=
hatwide
ρ
S
2
=

β

2

0
) (
0

+

α

2

1
) (
1

hatwide
ρ
T
1
=

β

2

0
) (
0

+

α

2

1
) (
1

hatwide
ρ
T
2
=

α

2

0
) (
0

+

β

2

1
) (
1

and
hatwide
ρ
Λ
i
=

ψ
i
) (
ψ
i

where

ψ
1
)
=
α

0
) 
1
)
+
β

1
) 
0
)

ψ
2
)
=
α

1
) 
1
) −
β

0
) 
0
)

ψ
3
)
=
−
α

0
) 
0
)
+
β

1
) 
1
)

ψ
4
)
=
−
α

1
) 
0
) −
β

0
) 
1
)
.
We observe that, as expected, the state on both Λ
2
and Λ
3
is changed from
Λ
1
, since either
X
1
or
X
2
lies in the past light cone of each region. The states
of the individual particles, however, are unambiguously fixed by the dynamics,
and neither particle’s
individual
state is affected by a transformation outside its
light cone —
X
1
lies outside the past light cone of
S
2
, for instance, so
hatwide
ρ
S
2
=
hatwide
ρ
R
2
even though
S
2
is part of Λ
2
and the state of Λ
2
is changed by
X
1
.
Things are otherwise in the event of collapse, where we make a measurement
rather than a spin flip at
X
i
. The (hypothetical) stochastic dynamics produce
perfectly reasonable (nondeterministic) predictions for all the joint states—
assigning a nonzero probability to, say,
hatwide
ρ
Λ
i
=

φ
i
) (
φ
i

where

φ
1
)
=
α

0
) 
1
)
+
β

1
) 
0
)

φ
2
)
=

φ
3
)
=

φ
4
)
=

0
) 
1
)
.
But there is no unique prescription for the oneparticle states.
Take
S
2
, for
instance—if it is regarded as part of Λ
1
then it has state

β

2

0
) (
0

+

α

2

1
) (
1

;
(6)
19
if it is regarded as part of Λ
2
then it has state

1
) (
1

. We might interpret this
(at the risk of confusion with a terminology adopted elsewhere in philosophy of
QM) as a
contextuality
of local states—the state of a region depends on which
larger region it is regarded as a part of. We might also interpret it as denying
the existence of states for many regions of spacetime—regions only have states
at all if they are assigned the same state by all foliations. We might call this
nihilism
about local states, a name which will be particularly apt if, as we are
inclined to suspect (but have not proved), in realistic models virtually no regions
smaller than entire foliations are assigned states.
To conclude this section: there is an “hierarchy of counterintuitiveness” here.
We have
1. Full locality
(classical physics). All spacetime regions have states; the state
of a region supervenes on the states of its subregions.