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On Aug 11, 2012, at 1:41 AM, Basil Hiley <This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote:


On 27 Jul 2012, at 07:00, nick herbert wrote:


On Jul 26, 2012, at 9:50 AM, nick herbert <This email address is being protected from spambots. You need JavaScript enabled to view it.> wrote:

1. The oft-cited remark that non-relativistic Bohmian mechanics gives the same result as conventional QM for all conceivable experiments is plain wrong. The two theories possess radically different ontologies which lead to radically different consequences.

BH: How can it be wrong?  It uses exactly the same mathematics, without the addition or subtraction of any new mathematical structure.  Its predicted expectation values found in all experiments are identical to those found from the conventional rules.  If you want to criticise it, why not simply say "It adds no new experimental predictions, so why bother with it?"  Then you can get into arguments about which interpretation is better in your opinion.  Then it is a matter of opinion not experimental science.

JS: However, Antony Valentini's extension does add new predictions consistent with my own independent investigations and also Brian Josephson's which already has observational evidence in its favor (Libet, Radin, Bierman, Puthoff-Targ, Bem)



NH: What exists in QM is a wavefunction, spread out in configuration space (and this wavefunction is "real" according to PBR). For a given quantum state all systems represented by that state have the same ontology.

BH: The ontology gives meaning to the notion of a "quantum state".  What does it mean to say "For a given quantum state  all systems represented by that state have the same ontology"?


NH: What exists in BM is an actual particle which for S-states has the remarkable property that v=0. In BM all systems represented by the same state are different--their difference (in the S-state case) being the differing positions of the static electron. A Bohmian S-state consists of an ensemble of stationary electrons each in a different position whose position pattern is given by psi squared.

It is this v=0 property of BM S-wave electrons that is used to create counterexamples to the contention that BM and QM give the same predictions.

1. Muonic Hydrogen. Like t! he electron the muon in the BM picture is stationary. Hence the muon lifetime in BM is the just the natural lifetime. However in QM the muon has a velocity distribution so the lifetime is lengthened by relativity. BM and QM predict different lifetimes for the muonic atom. One may object
that I have introduced relativity into a non-rel situation. However the QM and BM states are still non-rel.
The lifetime of the muon can be seen as a measuring device probing the ontology of the muonic hydrogen.
The probe uses a relativity effect to measure a non-rel configuration.


BH: I recall having already answered this criticism some time ago.  Time dilation is a relativistic phenomenon so you must use the relativistic Dirac theory in this case. 
JS:: Yes, Nick's error here is obvious. He appeals to the wrong equation for the problem. It's a Red Herring.

BH: In the past there I have been entirely happy with the treatment of the Bohm model of the Dirac equation that we have given.  However Bob Callaghan and myself have now obtained a new complete treatment of the Dirac equation with which I am completely happy. It uses the Clifford algebra in a fundamental way as it must to link with the known successful spinor structure.  See Hiley and Callaghan:  Clifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation.  Foundations of Physics,  42 (2012) 192-208.
 DOI:  10.1007/s10701-011-9558-z and in more detail in The Clifford Algebra Approach to Quantum Mechanics B: The Dirac Particle and its relation to the Bohm Approach,  (2010)    aXriv: 1011.4033.
Our work shows that the Bohm charge velocity of the electron is, in fact, given by v= Psi alpha Psi,  where alpha is the Dirac 4x4 matrix, which is related to the Dirac gamma matrices. (See Bohm and Hiley The Undivided Universe, p. 272 for our original treatment which is confirmed by our latest work.)  If you now look at the wave function of the ground state of the Dirac hydrogen atom which you can find in Bjorken and Drell p. 55 you will find the electron is moving in the ground state.  What is interesting is that when you take this expression and go to the non-relativistic limit you find the velocity is zero, exactly the result that the Schrödinger equation gives.  Remember the energy levels calculated from  the Schrödinger hydrogen atom are only approximations to those calculated using the Dirac hydrogen atom.

Do you have a reference to the paper that measures the lifetime of the muon in muonic hydrogen?  I can't find a good reference to a clean experiment which shows exactly how to measure the time dilation you mention.  I have recently written up the details of the calculation that I have outlined above but I would like to add a better reference to the actual measurement.


2. Electron Capture decay. Certain radioactive elements (Beryllium 7, for instance) possess an excess positive charge and do not have enough energy to decay by positron emission. Instead they capture the S-state electron which transforms a nuclear proton into a neutron and neutrino (inverse beta decay). Electron Capture (EC) is a very delicate probe of the ontology of the S-state electron. QM ontology (all electrons the same) predicts a smooth exponential decay. After many half-lifes all the Be7 is gone.
BM ontology predicts a very different outcome: exponential decay for all electrons located inside the nucleus;
infinite li! fe for stationary Bohmian electrons located outside the nucleus.

BH: You must read past the simple Bohm model introduced in chapter three of our book, "The Undivided Universe".  The first ten chapters contain a discussion of the non-relativistic Bohm model.  There we show that if you want to apply the theory to problems where the particles interact either with other particle or with fields like the electro-magnetic field, you must introduce an appropriate interaction Hamiltonian.  In section 5.3 to 5.5 we show how to deal with a very simple example of two-particle interactions.  These sections were written simply to illustrate how the mathematics work and how you can explain the results using the Bohm interpretation. NB the interpretation is only applied after we have solved the Schrödinger equation containing the interaction Hamiltonian.  You can't solve these equations exactly so you have to use perturbation theory.  Remember the maths is the same as for the standard interpretation.  It is the interpretation that is different.

What happens if the interaction Hamiltonian involves the electromagnetic potentials?  To discuss interaction with the electromagnetic field you must go to a relativistic theory.  This means you must use the Dirac equation.  Chapter 12 of our book begins to show you how to do this.  The work of Bob Callaghan and myself mentioned above takes this further.  What we have done is to discuss the free Dirac electron for simplicity.  We simply wanted to show how it worked without introducing more realistic interaction Hamiltonians. 
Now let me try to answer your question as to how we deal with electron capture.  In order to describe this capture, we have to introduce the appropriate interaction Hamiltonian.  What is the appropriate interaction Hamiltonian in this case?  To find this we have to go to a review article like "Orbital electron capture by the nucleus" [Rev. Mod Phys. 49 (1977) 77-221].  You will see that the interaction Hamiltonian is a weak electron current-hadron current interaction.  You must now put that into the Dirac equation and calculate away.  Well the calculations are all done in the Rev. Mod. Phys. paper and all we need to do is to interpret the results according to the Bohm model.

Where your analysis goes wrong is that you assume (1) the non-relativistic theory and (2)  there is no interaction between the nucleus and the electron.  You can do that to a first approximation to explain the principle of the Bohm model to, say, a first year undergraduate, but you must not say that's all there is.  It is not a true reflection of the processes that are involved!  There is an interaction between the nucleon and the electron and you must take this into account even in the Bohm model if you want to understand the physics.

If your message is simply to say that the naive Bohm model based on the Schrödinger is inadequate to deal with these problems then I totally agree with you.  Bohm and I have always recognised that the '52 work was just a first step.  Let me quote from his Causality and Chance book p. 118

“It must be emphasized, however, that these criticisms are in no way directed at the logical consistency of the model, or at its ability to explain the essential characteristics of the quantum domain.  Rather they are based on broader criteria, which suggest that many features of the model are implausible and, more generally, that the interpretation proposed in section 4 [of the ‘52 paper] does not go deep enough.”

I thought that in our book, "The Undivided Universe", we made it clear that chapter 3 was a first step.  All the remaining chapters were to show how the model was to be developed to meet many different actual situations found in nature.  Finally in chapter 15, we outlined what was going to be developed in a second volume, which would probe a much deeper structure but unfortunately Bohm died just as we were finishing the first book.

NH: If these two counter-examples to the QM/BM experimental identity conjecture have been discussed in the literature,
I! am unaware of it. But they should be.

BH: You are quite right, these points should be discussed in the literature.  Unfortunately I have been too involved in developing the ideas outlined in chapter 15 and that means going deeper into what I think really underlies quantum phenomena.  You will find some of this work in the latest publications of mine which are accessible on the net.  A good place to find a comprehensive review of my latest efforts is in my paper Process, Distinction, Groupoids and Clifford Algebras: an Alternative View of the Quantum Formalism, in New Structures for Physics, ed Coecke, B., Lecture Notes in Physics, vol. 813, pp. 705-750, Springer (2011).  Unfortunately I don't think it is available on the net at present but if you are interested I can send you a copy.

Thank you for your interest in our work.

Basil.


Nick Herbert

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