## Re: How probable are realizations of stochastic processes? (was:

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nNick Maclaren wrote:\n&gt; In article &lt;4125E601.5070604@univie.ac.at&gt;,\n&gt; Arnold Neumaier &lt;Arnold.Neumaier@univie.ac.at&gt; writes:\n\n&gt; |&gt; The case of spin is more difficult to analyze because as stated, it is\n&gt; |&gt; not a well-defined stochastic process. If it is taken as a continuous\n&gt; |&gt; measurement, the flip is at random times, and so even a single flip at\n&gt; |&gt; a definite time has probability zero.\n&gt;\n&gt; Continuous stochastic processes are a perfectly good mathematical\n&gt; model - Kolmogorov and other Russian probabilists have worked on\n&gt; them. They are, however, so unspeakably evil to handle that most\n&gt; people who know about them recoil in horror at the thought of\n&gt; touching them and discretise the problem ....\n&gt;\n&gt; No, I don\'t know anything more about them than that :-)\n\nBut I know much more. There is an extended theory of stochastic\ndifferential equations, starting with Brownian motion.\nAll these are continuous-time stochastic processes.\nTheir proper formal understanding requires a bit more measure\ntheory and the theory of Ito integrals; but the results are\n(with careful interpretation) mostly analogous to the discrete case.\n\n\n&gt; This is, however, relevant to quantum mechanics, where analysing\n&gt; interactions over time is very close to continuous Markov theory\n&gt; (which is the model I am referring to).\n\nIndeed, dissipative quantum systems are usually described by\nquantum versions of continuous-time stochastic processes.\nThe Lindblad form of dissipative QM is the quantum version of\nFokker-Planck equations, which describe the evolution of an initial\nprobability distribution by means of a stochastic process.\nAnd there are several versions of turning this into processes in which\nindividual realizations make sense: Quantum diffusion processes\nand quantum jump processes.\n\nEEQM, which started off this discussion of probability, is an example\nin case.\n\n\nArnold Neumaier\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Nick Maclaren wrote:
> In article <4125E601.5070604@univie.ac.at>,
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> writes:

> |> The case of spin is more difficult to analyze because as stated, it is
> |> not a well-defined stochastic process. If it is taken as a continuous
> |> measurement, the flip is at random times, and so even a single flip at
> $|> a$ definite time has probability zero.
>
> Continuous stochastic processes are a perfectly good mathematical
> model - Kolmogorov and other Russian probabilists have worked on
> them. They are, however, so unspeakably evil to handle that most
> people who know about them recoil in horror at the thought of
> touching them and discretise the problem ....
>
> No, I don't know anything more about them than that :-)

But I know much more. There is an extended theory of stochastic
differential equations, starting with Brownian motion.
All these are continuous-time stochastic processes.
Their proper formal understanding requires a bit more measure
theory and the theory of Ito integrals; but the results are
(with careful interpretation) mostly analogous to the discrete case.

> This is, however, relevant to quantum mechanics, where analysing
> interactions over time is very close to continuous Markov theory
> (which is the model I am referring to).

Indeed, dissipative quantum systems are usually described by
quantum versions of continuous-time stochastic processes.
The Lindblad form of dissipative QM is the quantum version of
Fokker-Planck equations, which describe the evolution of an initial
probability distribution by means of a stochastic process.
And there are several versions of turning this into processes in which
individual realizations make sense: Quantum diffusion processes
and quantum jump processes.

EEQM, which started off this discussion of probability, is an example
in case.

Arnold Neumaier

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In article <412CE73E.4090201@univie.ac.at>, Arnold Neumaier writes: |> $|> >$ Continuous stochastic processes are a perfectly good mathematical $|> >$ model - Kolmogorov and other Russian probabilists have worked on $|> >$ them. They are, however, so unspeakably evil to handle that most $|> >$ people who know about them recoil in horror at the thought of $|> >$ touching them and discretise the problem .... $|> >|> >$ No, I don't know anything more about them than that :-) |> |> But I know much more. There is an extended theory of stochastic |> differential equations, starting with Brownian motion. |> All these are continuous-time stochastic processes. |> Their proper formal understanding requires a bit more measure |> theory and the theory of Ito integrals; but the results are |> (with careful interpretation) mostly analogous to the discrete case. It isn't what you know that causes the trouble; it's what you know that isn't so. If you want to find out the issues to which I am referring, search a good pure mathematics and statistics library for the term 'continuous Markov'. But be aware that the theory is too hard for most excellent pure mathematicians, and was developed by Kolmogorov and people of close to that calibre. The summary is that you need MUCH more advanced mathematics than you state, and the problem is with the phrase "with careful interpretation". Probability and statistics have a lot of "gotchas" where you really do need to understand the subject to know when the obvious interpretation is likely to be wrong. And this area is far, far worse than most of those subjects. This is analogous to the related areas of chaos theory (including turbulent flow), where the accepted approaches for many decades turned out to be completely misguided. It wasn't until people realised that that the areas started progressing. Regards, Nick Maclaren.



"Nick Maclaren" schrieb im Newsbeitrag news:cgkb2i$cdo$1@pegasus.csx.cam.ac.uk... > $|> >$ Continuous stochastic processes are a perfectly good mathematical > $|> >$ model - Kolmogorov and other Russian probabilists have worked on > $|> >$ them. They are, however, so unspeakably evil to handle that most > $|> >$ people who know about them recoil in horror at the thought of > $|> >$ touching them and discretise the problem .... > In article <412CE73E.4090201@univie.ac.at>, > Arnold Neumaier writes: > |> There is an extended theory of stochastic > |> differential equations, starting with Brownian motion. Indeed, the Wiener process and closely related diffusion processes are hardly unspeakably evil to handle. Undergraduates do that all the time. I think that Arnold Neumaier has a point in his comments on the relation between quantum mechanics and statistics, and that it is not too helpful to avoid thinking about this point by declaring that such thinking is unspeakably hard, just because about any topic that there is advanced mathematical literature on can be made to look extremely detached. The question if and how quantum mechanics can be modeled by (continuous) stochastic processes has been studied, of course. One interesting example that I happen to be a little familiar with is Edward Nelson's work. For a summary (which can be understood by undergraduates) as well as further literature see here: http://www-stud.uni-essen.de/~sb0264/stochastic.html I think Nelson's construction of a certain class of continuous stochastic processes whose equations of motion are equivalent to the Schroedinger equation might supplement the present discussion by some interesting results. For some fun, those interested should have a look at the java-applet which illustrates Nelson's stochastic process in the example of a double-sit configuration: http://www-2.cs.cmu.edu/~lafferty/QF/ .

## Re: How probable are realizations of stochastic processes? (was:

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nI was hoping to stop posting on this, but one of the things that I\nfind unacceptable is being quoted out of context.\n\nI shall not respond further.\n\n\nIn article &lt;412db4c2\\$1@news.sentex.net&gt;,\n"Urs Schreiber" &lt;Urs.Schreiber@uni-essen.de&gt; writes:\n|&gt;\n|&gt; &gt; |&gt; There is an extended theory of stochastic\n|&gt; &gt; |&gt; differential equations, starting with Brownian motion.\n|&gt;\n|&gt; Indeed, the Wiener process and closely related diffusion processes are\n|&gt; hardly unspeakably evil to handle. Undergraduates do that all the time.\n\nThe point about such processes is that they are a simplification of\nthe general case and, as with many such simplifications, the full\ncomplications do not arise. In fact, that is precisely WHY many\nsuch simplifications are used - not just because they are simpler,\nbut because they restrict the analysis to where it is reliable.\n\nThe context of this is that I was saying "be VERY careful extending\nyour understanding of the simplified case to the general case,\nbecause doing that is KNOWN to be unreliable." That is not exactly\na rare situation in physics!\n\n|&gt; I think that Arnold Neumaier has a point in his comments on the relation\n|&gt; between quantum mechanics and statistics, and that it is not too helpful to\n|&gt; avoid thinking about this point by declaring that such thinking is\n|&gt; unspeakably hard, just because about any topic that there is advanced\n|&gt; mathematical literature on can be made to look extremely detached.\n\nEh? Precisely WHO is denying that there is a close relationship\nbetween quantum mechanics and statistics?\n\nThe context of this is that I was pointing out to Arnold Neumaier\nthat his understanding of the bases of probability is non-standard\nand (as I understand what he is saying) mathematically inconsistent,\nthough it is reliable enough when applied to simple cases. I was\nthen saying that it is therefore NOT a good idea to extend it to\nadvanced analyses in quantum mechanics.\n\nFurthermore, I was saying that I (as someone who was once a fairly\ngood specialist in this area) do not and never did know enough to\nknow where the "gotchas" arise in this particularly difficult\nsituation, and that therefore it is an area where great care is\nindicated. In particular, using an inconsistent mathematical model\nis very likely to lead to erroneous conclusions.\n\n\nRegards,\nNick Maclaren.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>I was hoping to stop posting on this, but one of the things that I find unacceptable is being quoted out of context. I shall not respond further. In article <412db4c2$1@news.sentex.net>,
"Urs Schreiber" <Urs.Schreiber@uni-essen.de> writes:
|>
$|> > |>$ There is an extended theory of stochastic
$|> > |>$ differential equations, starting with Brownian motion.
|>
|> Indeed, the Wiener process and closely related diffusion processes are
|> hardly unspeakably evil to handle. Undergraduates do that all the time.

The point about such processes is that they are a simplification of
the general case and, as with many such simplifications, the full
complications do not arise. In fact, that is precisely WHY many
such simplifications are used - not just because they are simpler,
but because they restrict the analysis to where it is reliable.

The context of this is that I was saying "be VERY careful extending
your understanding of the simplified case to the general case,
because doing that is KNOWN to be unreliable." That is not exactly
a rare situation in physics!

$|> I$ think that Arnold Neumaier has a point in his comments on the relation
|> between quantum mechanics and statistics, and that it is not too helpful to
|> unspeakably hard, just because about any topic that there is advanced
|> mathematical literature on can be made to look extremely detached.

Eh? Precisely WHO is denying that there is a close relationship
between quantum mechanics and statistics?

The context of this is that I was pointing out to Arnold Neumaier
that his understanding of the bases of probability is non-standard
and (as I understand what he is saying) mathematically inconsistent,
though it is reliable enough when applied to simple cases. I was
then saying that it is therefore NOT a good idea to extend it to

Furthermore, I was saying that I (as someone who was once a fairly
good specialist in this area) do not and never did know enough to
know where the "gotchas" arise in this particularly difficult
situation, and that therefore it is an area where great care is
indicated. In particular, using an inconsistent mathematical model
is very likely to lead to erroneous conclusions.

Regards,
Nick Maclaren.



Nick Maclaren wrote: > If you want to find out the issues to which I am referring, > search a good pure mathematics and statistics library for the > term 'continuous Markov'. But be aware that the theory is too > hard for most excellent pure mathematicians, and was developed > by Kolmogorov and people of close to that calibre. Kolmogorov worked on it many, many years ago. Would you call general relativity too hard for most excellent pure mathematicians, just because it was developed by Einstein and people of close to that calibre? What was advanced many years ago has usually been streamlined to an extend that it can be understood by anyone putting the effort in. In particular, today continuous-time stochastic processes are a well-understood part of probability theory. Arnold Neumaier



"Urs Schreiber" wrote in message news:412db4c2\$1@news.sentex.net... [snip] > I think that Arnold Neumaier has a point in his comments on the relation > between quantum mechanics and statistics, and that it is not too helpful > to avoid thinking about this point by declaring that such thinking is > unspeakably hard, just because about any topic that there is advanced > mathematical literature on can be made to look extremely detached. It maybe of interest to note here what Brian Greene in his over simplified book "The Fabric of the Cosmos" wrote, page 209: "...Thus, the telltale difference between the quantum and the classical notions of probabilities is that the former is subject to interference and the latter is not. Decoherence is a widespread phenomenon that forms a bridge between the quantum physics of the small and the classical physics of the not-so-small by suppressing quantum interference - that is, by diminishing sharply the core difference between quantum and classical probabilities." However, on page 212, Brian goes on to explain that he is not "yet fully convinced" that "decohence (histories)" exist but others claim that they have developed decoherence into a complete frame- work that solves the measurement problem. Regards Joe --- Outgoing mail is certified Virus Free. Checked by AVG anti-virus system (http://www.grisoft.com). Version: 6..744 / Virus Database: 496 - Release Date: $8/24/04$



Urs Schreiber wrote: > The question if and how quantum mechanics can be modeled by (continuous) > stochastic processes has been studied, of course. One interesting example > that I happen to be a little familiar with is Edward Nelson's work. For a > summary (which can be understood by undergraduates) as well as further > literature see here: > http://www-stud.uni-essen.de/~sb0264/stochastic.html > > I think Nelson's construction of a certain class of continuous stochastic > processes whose equations of motion are equivalent to the Schroedinger > equation might supplement the present discussion by some interesting > results. While it gives an interesting aspect to quantum mechanics and its classical limit, Nelson's description has a severe deficiency in that it cannot handle the situation when the wave function vanishes at some point. At all such points, R has a singularity, and S is entirely undefined. (This happens, e.g., for excited states of hydrogen, hence is an integral part of standard QM.) Also, it is awkward to do scattering calculations in Nelson's framework. Moreover, as you quote nelson on your p. 16, ''Quantum mechanics can treat much more general Hamiltonians for which there is no stochastic theory.'' Thus it is unlikely to be useful as a 'fundamental' description of nature. Instead, natural stochastic forms of quantum mechanics are those of quantum diffusion processes and quantum jump processes, in which the wave function itself is regarded as a classical random object. For their use in an experimental context, see, e.g., http://www.arxiv.org/abs/quant-ph/9805027. Arnold Neumaier



"Arnold Neumaier" schrieb im Newsbeitrag news:4131A59C.6050706@univie.ac.at... > Urs Schreiber wrote: > > > The question if and how quantum mechanics can be modeled by (continuous) > > stochastic processes has been studied, of course. One interesting example > > that I happen to be a little familiar with is Edward Nelson's work. For a > > summary (which can be understood by undergraduates) as well as further > > literature see here: > > http://www-stud.uni-essen.de/~sb0264/stochastic.html > > > > I think Nelson's construction of a certain class of continuous stochastic > > processes whose equations of motion are equivalent to the Schroedinger > > equation might supplement the present discussion by some interesting > > results. > > While it gives an interesting aspect to quantum mechanics and its > classical limit, Nelson's description has a severe deficiency > in that it cannot handle the situation when the wave function vanishes > at some point. At all such points, R has a singularity, and S is entirely > undefined. (This happens, e.g., for excited states of hydrogen, > hence is an integral part of standard QM.) True. As far as I recall the claim was that the respective stochastic process is still well defined away from these 0s. This is referred to as 'trapping' in the respective literature, which I think is mentioned in these notes at some point. > Also, it is awkward to do scattering calculations in Nelson's > framework. Sure! Nobody should use this weird stochastic process to do any QM calculations. The whole interest of this stochastic process is that it exists and has dynamical laws equivalent to the Schroedinger equation (away from these 0s, anyway). Moreover, as you quote nelson on your p. 16, > ''Quantum mechanics can treat much more general Hamiltonians > for which there is no stochastic theory.'' > > Thus it is unlikely to be useful as a 'fundamental' description > of nature. Yes, for this and other reasons one must be very careful with any interpretation. I just mention it for curiosity reasons. Did you ever have a look at the very similar ideas by Stephen Adler: http://golem.ph.utexas.edu/string/ar...0.html#c001495 ?



"Urs Schreiber" schrieb > "Arnold Neumaier" schrieb > > Urs Schreiber wrote: > Moreover, as you quote nelson on your p. 16, > > ''Quantum mechanics can treat much more general Hamiltonians > > for which there is no stochastic theory.'' > > > > Thus it is unlikely to be useful as a 'fundamental' description > > of nature. > Yes, No. If QM can treat much more general Hamiltonians, but for the Hamiltonian of our universe a stochastic theory exists, this makes it is more likely that stochastic theory is useful as a fundamental description. It gives stochastic theory explanatory power - stochastic theory explains why the Hamiltonian is of such a special type. Compare with the following reasoning: Religious thinking can treat much more general worlds for which there is no quantum theory. Thus it is unlikely for QM to be useful as a fundamental description of nature. Ilja

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