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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
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