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Observing the observer
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The relation between observer and observed is one of the controversial
issues in quantum mechanics.
Two of the most influential thinkers on the matter, John von Neumann
and Eugene Wigner, used certain physical assumptions to deduce
important consequences with mathematically rigorous arguments.
However, the validity of the consequences depends crucially on the
validity of the assumptions....
Von Neumann discusses the measurement problem in Chapters V and VI of
his famous 1932 book. These two chapters are reprinted on pp. 549-647
of the reprint volume ''Quantum Theory and Measurement'' by Wheeler
and Zurek, from which I take the page numbers (original page numbers
are not given there).
He begins by contrasting process 1 (Measurement as orthogonal
projection to an eigenstate of the operator R measured) and process 2
(the Schroedinger dynamics). His U denotes the density matrix, and is
transformed to P^*UP by a measurement corresponding to the projection
operator P, and by a unitary transform under the Schroedinger dynamics.
The discussion of process 1 assumes that R has discrete spectrum and
that measurements produce exact eigenvalues of R (p.449) and are
instantaneous (p.554), ''i.e., must be carried through in so short a
time that the change of U given 2. is not yet noticeable''.
After a long thermodynamical interlude von Neumann introduces on p.622
the perception of the observer: ''at some time we must say: and this
is perceived by the observer. That is, we must always divide the world
into two parts, the one being the observed system, the other the
observer. [...] The boundary between the two is arbitrary to a large
extent. [...] experience only makes statements of this type: an
observer has made a certain (subjective) observation; and never any
like this: a physical quantity has a certain value.''
To prepare the derivation of the independence of the measuring process
on where precisely this boundary is placed, von Neumann discusses the
quantum description of the combination system+detector (detector is my
short word for his ''measurement instrument''), culminating in the
result on p.639 top characterizing the entanglement of system and
detector (but the word entanglement was invented only a few years
later by Schroedinger).
On p.641 it is assumed that the state of the observer is completely
known (i.e., a pure state), and on p.645 enters the assumption that at
some time before the measurement the density matrix of system+detector
factors. Based on this, the proof of the boundary independence is
completed on p.647.
In conclusion, von Neumann's analysis is based on five questionable
assumptions:
1. The existence of process 1 as a real process.
But why should Nature respond to measurement differently than to
everything else? Was there no state vector reduction before the first
measurement was built, or before the first living being looked at
something?
2. The assumption that measurement results are exact eigenvalues of
the measured operator.
This is appropriate for the measurement of spin or helicity that have
a simple rational spectrum but not for most real measurements, where
the spectrum (though discrete) may consist of irrational numbers,
which one can hardly claim to be exactly measurable.
3. The assumption that measurements are instantaneous.
The questionability of the instantaneity assumption is discussued by
von Neumann himself and found harmless only in case of measurements
that result in the mere emission of a light quantum (p.557).
4. The assumption that the state of the observer is pure.
Von Neumann notes on p.639 that in most cases, the states of two
disjoint subsystems of a bigger system are not pure, but does not see
that this essentially conflicts with his assumption.
5. The assumption that before the measurement, the density matrix of
system+detector factors.
In view of the fact that the multi-particle (or field) Hamiltonian
representing the dynamics of system+observer destroys separable states
very quickly via decoherence, this is reasonable only if one assumes
that the observer state is a thermal mixture in which details are
averaged over, against assumption 4.
In addition, since system and detector are commonly composed of the
same kind of indistinguishable particles, the separability assumption
is in direct conflict with the (anti)symmetrization known to be
necessary for all quantum systems composed of indistinguishable
particles.
In a contribution to a book with the title ''The Scientist Speculates'',
reprinted on pp. 168-181 of the volume cited above, Wigner turns the
cautious remarks of von Neumann about the possible involvement of the
brain in quantum mechanics into a full-blown esoteric interpretation,
complete with
-- the concept of consciousness as the actor in achieving a wave
function collapse (''The preceding argument for the difference in the
roles of inanimate observation tools and observers with a
consciousness - hence for a violation of physical laws where
consciousness plays a role - is entirely cogent so long as one accepts
the tenets of orthodox quantum mechanics in all their consequences.'',
p.178), and
-- a subjective interpretation of the state vector (as if quantum
mechanics had nothing objective to say): ''The wave function is a
convenient summary of that part of the past impressions which remain
relevant for the probabilities of receiving the different impressions
when interacting with the system at later times.'' (p.171)
He pays lip service to objectivity (''The information given by the
wave function is communicable'', p.171) - without explaining why, when
it is based on subjective impressions only. In his caricature of the
real thing, the wave function turns into a separable state of system
and observer already when ''his answer gives me the impression that he
did'' [see the flash], ''the joint wave function of friend+object will
change into one in which they even have separable wave functions''.
Clearly, true to the title of the article collection, the scientist
speculates here - nothing more.
In a much more serious article (reprinted on pp. 260-314 of the above
reprint volume), Wigner recapitulates von Neumann's analysis (in much
easier to read terms), repeating all his assumptions, but discussing
its limitations in a bit more detail.
-- ''One has to admit, on the other hand, that (35) is a highly
idealized description of the measurement. [...] The fact that the
measurement is of finite duration introduces a more serious problem.
[...] To which position at which time does the measurement then refer?
This issue is unclear and is rarely discussed.'' (p.284)
-- ''for many if not most operators, this expression - or any other
expression which might lead to that equation - contradicts some of the
basic principles of quantum theory. What then are the limitations of
measurability? Only quantities which commute with all additive
conserved quantities are precisely measurable'' (p.298)
This leaves very little, since the Hamiltonian is additively conserved
and commutes for most systems with hardly any of the traditionally
measured variables. Moreover, if the Hamiltonian has irrational
eigenvalues (which is the case with probability one), these cannot be
exactly measured either.