Dear Ulf Klein,

Congratulations on an exceptional essay. You beautifully state that the Copenhagen interpretation "shows several strange features, which have as a common origin the switching forth and back between reality and un-reality of properties as observation begins and ends."

You note that the cornerstones of Copenhagen are the interpretation the uncertainty principle that...

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Dear Ulf Klein,

Congratulations on an exceptional essay. You beautifully state that the Copenhagen interpretation "shows several strange features, which have as a common origin the switching forth and back between reality and un-reality of properties as observation begins and ends."

You note that the cornerstones of Copenhagen are the interpretation the uncertainty principle that conjugate properties "cannot be simultaneously real" and the particle/wave duality that seems to switch between particle *OR* wave (as opposed to de Broglie-type particle *AND* wave). You also note that Copenhagen 'beliefs' are *not* part of the quantum theoretical formalism.

I invite you to read and comment upon my essay, The Nature of the Wave Function, which develops the idea that individual (non-point) particles induce real waves in a field, while the inherent indeterminacy of the associated phase implies that the individual behavior of the local particle is inherently non-deterministic. But the waves *do* relate (through de Broglie's wavelength/momentum realtion) to the energy of the particle, and the energy, through the partition function, to the associated probability. In this perspective the individual particle behavior is unpredictable (p-incomplete) but the statistical behavior is probabilistic, tracing to the wave, but not connected to an individual wave.

At first your reference to Tonomura may seem at odds with this, but a closer reading does not necessarily imply "lack of a wave", but presence of a particle, which is completely compatible with my particle-plus-wave model, so that I agree "single particles are always particles." I hope you retain this point in mind when you read my essay [as I hope you will do.]

As for the classical limit, my model implies that the thermal disruption of 'quantum coherence' and the very real limit on the 'extent' of the physical wave lead me to expect a "classical limit" with potential exceptions for 'macro-clusters' of hundreds of particles and super-cooled or superconducting coherent quantum aspects of behavior.

I am in full agreement with your semantic analysis of 'complete'. My model is not p-complete due to the inherently unknown phase of the wave. But it does address the meta-physical connection between the individual particle and the statistical probability amplitude, and seems to comport with recent experiments thus linking a p-incomplete description of the particle to the statistical nature of QM.

I especially like and agree with your observation that EPR was "rewritten" as a hidden variable theory in which "elements of reality" become hidden variables. Your logical analysis of EPR and Bell is superb, and I fully agree that "an individuality interpretation of quantum theory does not exist".

I hope you find my model fully compatible with your analysis as it appears to me it is. If I had read your paper before finishing my own I would have said a few things differently, but no conclusions would change.

I believe that this FQXi contest is producing some important essays, and yours is one of the very best.

Edwin Eugene Klingman

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Dear Ulf Klein,

I agree with you on that the Individuality Interpretation of Quantum Theory is Wrong and that Einstein has finally won the debate over Bohr.

In the section 3 of my essay I propose a generalization of the Schrödinger equation of motion for a stochastic state vector |\Psi>. It can be shown that if we split this stochastic vector into an average vector |\psi> plus...

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Dear Ulf Klein,

I agree with you on that the Individuality Interpretation of Quantum Theory is Wrong and that Einstein has finally won the debate over Bohr.

In the section 3 of my essay I propose a generalization of the Schrödinger equation of motion for a stochastic state vector |\Psi>. It can be shown that if we split this stochastic vector into an average vector |\psi> plus a fluctuating correction, the average |\psi> verifies the Schrödinger equation. That is, the Schrödinger equation only describes the average behaviour of an ensemble obtained from averaging over the noise. Each individual system satisfies an equation beyond the Schrödinger equation.

A more clever demonstration is found in my recent work Positive Definite Phase Space Quantum Mechanics, finished yesterday, where I offer details on the Liouvillian formulation of quantum mechanics, which I mentioned in my essay. It is shown therein that the Schrödinger equation is obtained when we consider an ensemble (7). For a plane wave, (7) confirms early Ballentine's claims [3] on that the wavefunction describes an ensemble of particles with different positions but the same momentum.

The confirmation of the statistical interpretation of quantum mechanics eliminates the usual paradoxes --Schrödinger cat, collapse, etcetera-- associated to the old Copenhagen interpretation.

I have also confirmed your findings on the \hbar-->0 limit of quantum mechanics (Am J. Phys.). The new formalism shows that there is not wavefunction (neither Wigner function nor density matrix) that can reproduce classical physics!

The problem of ordinary quantum mechanics is found in the Hilbert space structure, which is not general enough to contain classical states. Some authors are considering Rigged Hilbert spaces structures as a way to treat classical states in a rigorous way, but that approach is difficult and ambiguous --there are ambiguities regarding the test space and the proper nature of the extension--. However, the positive phase space quantum mechanics presented above naturally contains classical states beyond the Hilbert space.

Regards.

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