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and
i
ij
O
j
O
j
/
i
E
ij
Note that the O
j
can be identical but in any case they are considered separate
subspaces for which the observables are totally unlinked. We can now define the
synthetic model as an encoding of M in terms of the direct sum of subspaces O
j
=
j
a
O
j
s
M
(5)
We observe that in the synthetic model an abstract state p
M is encoded by
an ordered tuple of single-valued elements q
1
q
2
q
j
, with q
j
∈
∈
O
j
.
Each q
j
may, however, also be understood as an ordered set or vector
q
j
=
¯
00q
j
00
such that we can write for the p of a synthesised M
p
=¯
q
1
+¯
q
2
+···
This evidently describes an observable on M,
=
j
p
1j
2j
ij
(6)
where
1j
2j
ij
is an element of
i
ij
O
j
. More specifically,
=
j
a
O
j
M
→
s
M
=
j
p
→
p
j
q
j
where we view
j
as an operator, that is a map, with
j
=
1j
ij
The sum of observables on O
j
induces therefore an observable on M.
5. SYNTHETIC VS. ANALYTIC MODELLING
With an observable Eqn 6, defined on M but arising from a synthetic model, we
can now compare this to the analytic approach. Towards this end let us generalise