Understanding the Von Neumann Chain Concept in Modern Quantum Theory

[Varon]

Is anyone thoroughly familiar what von Neumann was trying to say in his von Neumann Chain concept where one can locate anywhere the observer and the observed? I've been reading the original and analyzing it for hours and can't seem to completely get the context in light of present day concept.

Is he talking about figurative or literal? In Bohr original quantum formulation, classical and quantum is the cut or boundary. In von Neumann approach.. it looks like it's all quantum.. that is.. he treats the classical as quantum too.. hence the cut or boundary can be located anywhere in the all quantum stuff?

It was proposed over 70 years ago. Now in light of decoherence. How does one interpret what von Neumann was saying? Is it part of Copenhagen? Is this the other variant compared to the original Bohr classical-quantum cut. How do you understand it? Can you give an actual setup or experiments to demonstrate this von Neumann chain and the collapse cut that can be move anywhere? Thanks.

 

[malreux]

One application that clarifies the concept a bit is to be found in James Ladyman's:

http://iopscience.iop.org/1742-6596/254/1/012013/pdf/1742-6596_254_1_012013.pdf

The relevant spot is at the bottom of page three. There the author describes a tension within the original formulations of the measurement problem - what about situations where observation is not best modeled as discrete but is in fact continuous, e.g. Sudbery's example of the Geiger counter left with a decaying particle for the period of its half-life. If this kind of observation is modeled with the standard formalism one considers a number of discrete measurements with some small time interval between them and takes the limit as the time interval goes to zero. However, applying the projection postulate to each measurement in this model and then time-evolving the state to obtain the transition probability for the next measurement, leads to the conclusion that the difference in the probabilities of nding the system in a given state at dierent times tends to zero. I.e. the 'watched pot' scenario.

Sudbery proposes a model of continuous observation in which the system and an apparatus
are coupled and evolve in time together, hence "continuous observation is not continual
measurement". He reformulates the projection postulate in terms of density operators. According to his Postulate A: Suppose D is an observable being continuously observed and P_k are the projection operators onto the eigenspaces of D. If ρ₀ is the density matrix of the system at time t = 0 its density matrix at time t is:
T_t(ρ₀) = [itex]\sum_k[/itex] P_k e ^-iHt ρ₀e^iHt P_k.

... which can be seen as a supplement to the projection postulate. As Ladyman points out, however, the measurement problem does not go away. Sudbery demonstrates that although probabilities are independent of exactly where the line between system and observer is drawn, but that nonetheless some such line must be drawn. This result is due to the Chain, and demonstrates its usual consequence. This illustration is indirect, but hopefully gives the flavour.

 

[bhobba]

Von Neumann meant it literally - it can be made anywhere.

But nowadays dechorence solves the issue. And with digital recording equipment in the form of computers these days its really hard to see exactly why you need conciousness to collapse it anyway.

Thanks
Bill