http://arxiv.org/pdf/math.PR/0203249 ]]>

Harry,

it would be desireable to have such an entry, but I am afraid I won’t be able to do it justice without going back and reminding myself of lots of literature.

One apparently well thought-through approach is that by Adler. There is this book review which gives a useful impression.

]]>@Urs: Could you at some point write up an article for the cafe with an account of some attempts at making QM deterministic and what went wrong in those attempts (written for mathematicians who aren't trained in physics)? I'm really interested in the subject, but I don't ever plan to be a "physics guy", and I assume that it would be interesting to a lot of the readers at the cafe who are pure mathematicians and would get pretty lost actually trying to read through the literature. Just an idea for a post at some point in the future.

]]>@Harry #65: Thanks for the references. Very interesting.

]]>Concerning the t’Hooft-article:

there are of course many attempts in the literature to realize standard quantum mechanics as the coarse-grained version of a non-probabilistic theory. Great minds have tried themselves on this, not the least t’Hooft. But it’s one of those dangerous questions to follow…

]]>Harry wrote:

Gerard ’t Hooft (not a crackpot) put a paper up on the ArXiv a few years ago talking about a mathematical basis for a deterministic theory of QM

Oh, interesting, I did not know that.

My last comment may seem to imply that I think that quantum mechanics *enforces* an interpretation including some sort of randomness, but the situation I had in mind was a little bit different: My friends did not know that any physical theory existed with *any* non-deterministic interpretations around, that was completly new to them.

Probability theory is concerned with mathematical models of phenomena that exhibit randomness , or more generally phenomena about which one has incomplete information.

Perfect, thanks! ]]>

http://arxiv.org/abs/0909.5075

Rob Spekkens, one of the co-authors, has developed an epistemic theory of QM that has gained some traction and the underlying processes are deterministic in his theory to some extent. ]]>

Okay, I see. I reworked the first sentence at probability theory to

]]>Probability theory is concerned with mathematical models of phenomena that exhibit

randomness, or more generally phenomena about which one has incomplete information.

probability theory is not attempting to model the real world

Sure it is. What do the think the word “frequency” in “frequentist interpretation” comes from?

But I agree with your other point: in as far as probability theory is about the observable world, it is different from most other theory of physics, much more universal, yes. That’s why I tried to be careful, wrote “similar status” and put that funny “other(?!)”.

]]>Ian, since your query box is not at all about what probability theory describes but whether or not quantum mechanics is a probabilistic theory, I have removed the query box and instead reproduce it here, for whatever discussion may here come out of it:

]]>Ian Durham: At the risk of starting another argument, I would just like to point out that there are theoretical physicists for whom the phenomena described by probability theory are not random. Specifically, both the epistemic and consistent histories approaches to quantum theory do not view the quantum processes that are described by probability theory as being random. Their view is that we use probability theory simply due to a lack of knowledge on our part, i.e. some things, which are perfectly non-random, are nevertheless unknowable. Just my two cents.

@TvB: Gerard 't Hooft (not a crackpot) put a paper up on the ArXiv a few years ago talking about a mathematical basis for a deterministic theory of QM http://arxiv.org/abs/quant-ph/0604008

I don't know what became of it, but since he's a nobel prize winner, it can't be that bad, right?

Edit: Here's a more recent paper of his with similar ideas: http://xxx.lanl.gov/abs/0908.3408

Edit 2: Oh wait, Urs probably knows him!

]]>@Ian: No problem, but that is a question that I would like to “outsource” to a different page, we can do that if we take “random” as given on the probability theory page and discuss its different interpretations somewhere else (like is it “insufficient knowledge” or “god playing dice” :-)

I know some mathematicians specializing in mathematical statistics who firmly believe that our physical world is deterministic (I think they still do, even after I told them about quantum mechanics). To them it’s really only about mathematical models that help you decide what to do in certain situations, without any epistemologic relevance.

]]>From the idea section (could somebody put the code for a query box on the sidebar? I always have to go to another page to check it, and I can never remember it.):

Notice that in this respect probability theory has a similar status as (other(?!)) theories of physics: there is a mathematical model (measure theory here as the model for probability theory, or for instance symplectic geometry as a model for classical mechanics) which can be studied all in itself, and then there is in addition a more or less concrete idea of how from that model one may deduce statements about the observable world (the average outcome of a dice role using probability theory, or the observability of the next solar eclipse using Hamiltonian mechanics)...

This is not true. Unlike the "(other?!) theories of physics", probability theory is not attempting to model the real world (it is an a priori, not an a posteriori field of discourse, which I thought we all agreed on (at least earlier in this thread)). The analogy is much closer with higher category theory, where there are different models of (oo,1)-categories that all describe fundamentally the same objects.

Probability differs from physics in the following way: results derived in probability theory are independent from the universe in which we live. They are mathematical truths, rather than sketches of physical laws that can vary from time to time and place to place and can never be judged to be completely accurate (this is why mathematics is the "queen of the sciences". It is not subject to the same philosophical issues that one encounters in the empirical sciences).

Anyway, in my opinion, the "Idea" section currently detracts from the value of the rest of the page.

]]>Thanks David!

Very nice. Finally some constructive progress here.

I have further expanded the Idea section at probability theory a bit, and then I added plenty of hyperlinks to the nPOV material that you had added.

]]>(Although the last few comments have gotten back to discussing actual nLab pages, this discussion has veered sufficiently away from actual nLab pages that I’ve moved it into the Atrium.)

]]>I had a go at it :-) Is the book “stochastic relations” by Doberkat of relevance? I saw that David mentioned it on the nCafé and skimmed the table of contents, but all I have learned yet is that it obviously is about stochastics and uses the language of category theory.

]]>OK. So I’ve made a small start there, pasting in some cafe material. We need an introduction, which could be category theory free.

]]>Unless the nPOV has something particular to say on the subject, I don’t see why nLab should wade into this controversy.

I don’t think there is need to wade into any controversy. I don’t see a deep controversy anyway.

But, David, I am hoping one day we’ll have a decent page probability theory that, among other things, describes the category-theoretic approaches to it, which you have been disucssing so often on the nCafe.

]]>