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A. Neumaier

Science Advisor

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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 ofr 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''.

True to the title of the article collection, the scientist speculates - nothing more.

In a 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.