= Tr PS(t)/Tr S(t), where

is the absolute or objective probability that the answer to the question P asked at time t will be Yes. But no individual processor "knows" the absolute or objective state S(t), which could be written S(t)=P(t_n)U(t_n,t_(n-1))P(t_(n-1)...P(t_1)S(0)P(t_1)...P(t_n). where P(t_i) is the question answered Yes at time t_i, and U(t_i,t_j) = exp(-iH(t_i - t_j)). If some processor knows only part of the information needed to construct S(t) then he must in principle ADD the contributions corresponding to the various possibilities compatible with what he knows. Thus if he knows that question P(t_i) was posed at time t_i, but does not know that the answer was Yes, then he must add to the actual S(t) the similar expression with P(t_i) replaced by (1-P(t_i)). This gives a personal S(t) that represents his own knowledge. It can be used in place of the objective S(t) in the formula for

to give the probability that is entailed by
his knowledge.
Of course, no processor knows even all the questions
that have been posed over the course of history.
But one can make prediction about small subsystems
that have been prepared in a known way, and are known to
be isolated from interacions with the rest of nature
between preparation and subsequent measurement. In
such a case the space can be divided into the part built
on the degrees of freedom of the isolated system and
the rest, so that the matrix elements of S(t) are
~~. After preparation the system and its
environment evolve independently up until the measurements.
von Neumann's theory explains how information about
the preparation and the measurements are conveyed into the
brain of the observer. If P_0 is the image in the space of the
system of the knowledge of the preparation, and P is the image
in the space of the system of the knowledge corresponding to
a possible outcome then the probability of P given P_0 and
a state S(t) at the time of the preparation is
~~

= Tr PU(t',t)P_0 S(t) P_0 U(t,t')P/Tr P_0 S(t),
where U(t',t) =exp -iH(t'-t) is the inverse of U(t,t').
The definition of Tr X gives
Tr X = Sum over all s, Sum over all r of =
Tr_(s) PU'(t',t)P_0 S_s P_0 U'(t,t') P/Tr_(s) P_0 S_s
where Tr_(s) means trace over the s variables, and
S_s = Tr_(r) S = Tr_s S,
[according to our notation, which makes Tr_s the trace
over all variables "other" than those associated with
the system, which, in an abuse of language I call s:
I use s also to denote the indices that label the basis
vectors of the system s).
In an ideal preparation the initial state of s is completely
fixed by P_0: S_s(t) = a multiple of the identity. Then
= Eq. 1
Tr_s P U'(t',t) P_0 U(t,t') /Tr_s P_0,
where I have repeatedly used PP=P, U(t',t)U(t,t')=1, and
Tr_(s) AB = Tr_(s) BA.
Eq. 1 is the Copenhagen formula.
Suppose P_1 is measured in region 1, and P_2 is
measure at the same time in the spacelike separated
region 2.
This means P=(P_1)(P_2), and (P_1)(P_2)= (P_2)(P_1).
Then the general formula gives
~~.
The U(t',t) and U(t,t') break into a product of
two independent factors,
~~~~ = ~~~~ ~~