Nick, that is a Red Herring. The bare Bohm model you cite never claims to be a complete theory. Indeed, I suspect the problem you raise below was one of the reasons Vigier introduced the quantum noise - sub-quantal Brownian motion terms. Decay of unstable real particles is a zero point vacuum fluctuation effect not found in bare non-relativistic quantum mechanics.
For example, the bare Dirac equation does not give the Lamb shift and the magnetic moment of the electron correctly - you need QED radiative correction. Same idea here for bare Bohm theory. It's only the zero order starting point so to speak.
In any case, Basil will correct me if my memory here is mistaken?
On Jul 26, 2012, at 12:42 PM, nick herbert <
If these two BM examples have been treated
in the literature I am not aware of it.
You are more knowledgable than I.
Please cite references.
Note that ordinary QM treats these two problems
quite simply without recourse to zero-point fluctuations.
Jack: No, I don't think so. You must couple the atomic electron Hamiltonian to a random EM field ~ j.Azpf mimicking virtual photons to compute decays if you don't use QED.
In other words, if you simply start with
H0 = p^2/2m + e^2/r
where m is the muon mass & r is the relative coordinate
you cannot calculate the decay of the muon in orthodox QM without adding some Hint to H0. So it comes down to the same thing in essence no matter which interpretation you use.
Nick: I am interested in reading Vigier's BM description
of these two experiments to see if his method
does indeed, as you are apparently claiming,
give the same results as non-relativistic QM.
On Jul 26, 2012, at 11:55 AM, JACK SARFATTI wrote:
On Jul 26, 2012, at 9:50 AM, nick herbert <
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.
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.
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 the 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.
Nick, you have neglected coupling to the zero point vacuum fluctuations that trigger when the unstable particle decays. The unstable real particle gets a kick from a virtual particle giving it a velocity. I think this is done explicitly in Vigier's sub-quantum stochastic Brownian motion addition to the bare NR QM Bohm model you cite. So when you do that everything works. Similarly below.
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 life for stationary Bohmian electrons located outside the nucleus.
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.
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