Stochastic quantum mechanics, cont.

Date: Tue, 19 Dec 1995 14:32:44 -0800
From: Vic Stenger <vjs@uhheph.phys.hawaii.edu>
To: quantum-d list <quantum-d@teleport.com>
Subject: QUANTUM-D: Stochastic quantum mechanics, cont.

On Mon, 18 Dec 1995, Paul Easton wrote:

>     Vic Stenger wrote:
> 
>    > Thus a stochastic interpretation in which the Brownian motion occurs 
>    > in spacetime, so that steps backward in coordinate time are allowed 
>    > (proper time continues to change monotonically) provides a picture of 
>    > definite particle paths that still gives all the results of quantum 
>    > mechanics...
> 
> 1) The Schrodinger equation is not "just" a diffusion equation. The 
>    factor i that multiplies the time derivative makes it a complex 
>    field equation. Given an initial condition whose absolute square 
>    is a guassian function the absolute square solution will look like 
>    diffusion. However the simplest solution is a propagating complex 
>    plane wave. It is not clear how this would be enabled by Brownian 
>    motion.

Well, the derivation includes another step that is similar to what Bohm
did, and Mandelung before him.  You assume a wavefunction psi=Rexp(iS)
where R and S are real.  This can be justified on classical grounds.  See
Goldstein _Classical Mechanics_ 2nd ed p. 484.  If a particle has a
momentum p, the surfaces of constant action S, where p = grad(S), are
wavefronts of wavelength lambda and frequency nu that propagate with a
phase velocity u = lambda*nu where lambda = h/p and nu = h/E, where E is
the energy of the particle and h is a constant.  All we do in QM is assert
the value of h by experiment and let R be less than one, interpreting it as
a probability amplitude, that is probability density rho = R^2. Using the
Fokker-Planck equation to give drho/dt you get the Hamilton-Jacoby
classical equation of motion with an addition potential Q =
-hbar^2/2m*Laplacian(R^2)/R, which is Bohm's "quantum potential." 
>From this, and the equation of continuity, you get the Schrodinger 
equation.

What has taken me a long time to realize is that the stochastic 
interpretation is identical to Bohm's interpretation in the case where 
the hidden variables are indeterministic, and that this gives standard 
quantum mechanics.  The alternate of deterministic hidden variables 
remains possible, but this is a new theory beyond conventional QM that 
is necessarily superluminal (not just nonlocal).
 
>    In any case, if one had a derivation for Schrodinger's equation 
>    consistent with vacuum fluctuations, it would be interesting but 
>    almost as weird as current theory. One still has the unphysical 
>    complex field. And the reasoning may be circular. Could one derive 
>    vacuum fluctuations without using Schrodinger's equation?

Since quantum fluctuations have been a part of physics for forty years, I 
do not know what you call it "weird."  I think we have put the cart before 
the horse, but the horse is not chasing its tail.  We make a case for 
quantum fluctuations by simply arguing that spacetime is not indefinitely 
indivisible.  h > 0. Discrete spacetime -> fluctuations -> quantum 
mechanics.

As I show above, the complex field is not unphysical.  It is an alternate 
way of describing particle motion even classically, where the dynamics 
are determined by a complex wave exp(iS) (R=1).  As Goldstein shows, this 
explains the close link between the most fundamental principle of 
classical mechanics, least action, with Fermat's principle in optics.  
And, I should note, Feynman's path integrals, from which quantum 
mechanics can be derived, give least action in the decoherence limit - 
which is the way quantum becomes classical.

> 2) In any case the weirdest thing about QM is not the wave equation 
>    but measurement theory. Can the Brownian motion approach help us 
>    here?

Yes, this is what I have tried to indicate in 
http://www.phys.hawaii.edu/vjs/www/visual.ps (or .txt).  It explains 
nonlocality when you allow the stochastic motion to include backward steps 
in coordinate time.

...and then Rhett jumped in, asking: 

| he did not say 'nonlocality,' but rather 'measurement.' i think he
| meant the whole conundrum of how reduction can take place in a linear
| theory with unlimited tolerance for the superposition principle, &
| all that. maybe you think of nonlocality as being the central problem
| in theory reconciliation (like Stapp who said that the the biggest
| mystery was 'how information gets around so quick')? but these things
| need careful sketching, eh? ...but as far as paul's question goes, i
| thought that the whole point is that when we begin as Bohm did then
| we have the 'collapse' (i.e. the localized aspect) for free - that is
| one of our basic ingredients! having to explain collapse is a problem
| of people who believe in abstract wave-particle duality, not those who
| chose the stochastic way - so the question needs no answer at all. do
| you agree with this? can you comment on the issue i have raised, vis
| a vie the measurement problem and nonlocality?

I assumed by measurement problem he was referring to the nonlocality
problem, which is really the only problem, isn't it?  How do you get
wavelike behavior from particles? Once you explain that, the rest falls
in place because then you are free to carry on QM in the normal way
without having to justity the wave-particle dualism.

Vic

*******************************************
Vic Stenger
http://www.phys.hawaii.edu/vjs/www/vjs.html
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