Edward Nelson's nice book on brownian motion and physics made online here.

Hi akimon - that's a great link!

See also: Edward Nelson, "Derivation of the Schroedinger Equation from Newtonian Mechanics", Physical Review, Vol. 150, No. 4, 28 Oct 1966.Using Brownian motion, Nelson provides an interesting derivation of the time-dependent Schoedinger equation. Roughly speaking, his work suggested that QM might be derived from ordinary mechanics combined with a hypothesis that elementary particles are subject to a universal jiggling of unspecified cause. He makes a keen observation about diffusion processes and interference:" ... Some people seem to think that in a diffusion process rho [theprobability density] must spread out as time increases, but this is notso. It can spread out, bunch up again, spread out, bunch up at fiveseparated peaks, produce rings of alternating density - in short, thereare no restrictions. The reason we don't observe this kind of behaviorwhen ink diffuses in water is a dynamical one - this diffusion hasdissipative dynamics. But the motion of billiard balls is also lessinteresting if someone has poured molasses all over the table."Imre Feynes, a Hungarian physicist and contemporary of von Neumann, pre-dates Nelson and argued along similar lines. I've always felt that this interpretation of quantum mechanics was a bit underrated.

Yeah, I think the interesting stuff is from chapter 9 onwards.. He introduces a Weiner process "random jiggling" term in the equations of classic vibrating mass and damped oscillation systems.QuoteUsing Brownian motion, Nelson provides an interesting derivation of the time-dependent Schoedinger equation. Roughly speaking, his work suggested that QM might be derived from ordinary mechanics combined with a hypothesis that elementary particles are subject to a universal jiggling of unspecified cause. That's interesting, I always thought that the schrodinger equation comes from the conservation of energy, and the assumption that particles travel in waves (the complex exponential). Nelson mentioned that a few results of quantum mechanics agree with those of stochastic mechanics in his book. My old stat-mech text had a brief mention of this, but that course left me very confused

Tell please, whether there are any attempts of experimental check of stochastic interpretation of the quantum mechanics?

lesha, you might be interested in this paper by Timothy Wallstrom: "On the derivation of the Schrodinger equation from stochastic mechanics," Found. Phys. Lett. 2, 113-126,(1989). I'm curious to know why you're interested in experimental tests? As far as I know, it's just an interpretation of quantum theory -- an alternative to the Copenhagen interpretation. Have you heard differently? I'm not aware of experimental predictions that would differ from those of standard quantum mechanics. There may have been some doubt about Nelson's calculations on the hydrogen atom, but I don't recall the details.

In my opinion the main difference from the standard quantum mechanics is, that the stochastic mechanics gives probability of transition. Shredinger equation does not contain such information. For confirmation of this or that version of the stochastic mechanics it is necessary to measure probability of transition experimentally. Me interests whether somebody tried to make it? Or whether thought any body about it? I have some reasons as in experiment to measure probability of transition. But I afraid "to invent a bicycle ". And consequently I want to know all that is known on experimental checks the stochastic mechanics.

[]

Omar mentions Prof Ray Streater's web page on 'Lost causes in physics'. Here you do find someting on Nelson's stochastic interpretation of QM and other correct and slanted views of Prof Streater. I wonder why John Bell took up Bohm's work and developed the Bell's inequalities, if as Streater says, it is a lost cause? Science has no causes, won or lost! There are still people looking at the stochastic way like Francesco Guerra. For further information on this interpretation, look at the Los Alamos web site at http://uk.arxiv.org and the book 'Jean Pierre and the Stochastic Interpretation of Quantum Mechanics' ISBN 0-968389-5-6. All this is just food for thought and should not be taken as dogma!

I very much liked your last statement: I think, that only experiment will show what of versions of stochastic interpretation is true.

For some reason on a homepage (http: // www.mth.kcl.ac.uk/staff/rf_streater.htm l) I have not met the unit ' Lost causes in physics'. How long time you saw this section?

Prof. Omar,Is it necessary for students to read "Dynamical Theories of Brownian Motion" to learn higher level mathematical finance?

Try www.mth.kcl.ac.uk/~streater/lostcauses.html to find Prof. Ray Streater's article on 'Lost Causes'

I think I can answer mglendza's question, which was, "I wonder why John Bell took up Bohm'swork and developed the Bell's [sic] inequalities, if as Streater says, it is a lost cause?"Bell did not have, in 1960, the advantage of knowing the result of the Aspect experiments, which were in fact, done later as a response to Bell's own work. All Bell had was von Neumann's result, which showed that in any quantum theory, there do not exist anydispersion-free states. This had, up till then, been used to argue that no classical theory couldreproduce all the results of quantum theory. In fact, it is very difficult to test von Neumann'scriterion experimentally. To do so would require the experimentallist to look at ALL states, and show that, for each one, there was an observable with a non-zero standard deviation. The brilliant work of Bell (which was, incidently, quite original, and not anticipated by Bohm)involved showing that in any quantum theory (in a Hilbert space of dimension greater than three)there were predictions that could not be true in ANY classical theory, whether or not the theory contained hidden variables. More, he, Bell, suggested an experiment in which the prediction of quantum theory could be tested: if quantum theory can be explained by a classical model, Bell's inequalities could NOT be violated.It is well known that Aspect's results violated Bell's inequalities; so the data cannot be described by ANY classical theory, and this includes Nelson's stochastic mechanics, or any (classical)extension of it that may be claimed to describe spinning particles. The non-local theory of Bohm(the one with the quantum potential) also cannot describe the data of Aspect. What is needed isa generalisation of probability theory, and this is what quantum theory is. It is rather likethe history of non-euclidean geometry: Kant tried to argue that Euclidean geometry, and indeed, Newtonian physics, MUST be true, as it can be derived from pure logic. The existenceof non-euclidean geometries was, first, proved mathematically, and then later, with Einstein,used as a better theory of gravity. We cannot fit experiment to the assumption of a flatspace-time. In the same way, atomic data cannot fit a classical model of probability.

I think that knowledge of stochastic differential geometry is very valuable in finance for two reasons:1. Change of measure is no problem. Remember that Girsanov is just an exponential map, like all exp martingales.2. You can avoid Karatzas and Shreve since definitions for stochastic processes on a manifold need to be precise.And yes, stochastic processes mirror the wave function (remember Feynman), but who needs another derivation of QM even if you get spin for free.