In QM, local action does make sense

We have been too busy with our accreditation-related work, but I still had to squeeze in a comment at Scott Aaronson’s blog.

In case you don’t know, Scott Aarsonson is a tenured Associate Professor in CS at MIT (I mean the one in Cambridge, MA, USA). Scott’s opinions count—at least, they are very widely read (and often, also very extensively commented on and discussed).

This year, Scott was invited to respond to the Edge’s annual question [^]. In the latest post on his blog covering his and others’ responses to the question [^], Scott singled out three answers by others (at the Edge forum) which he thought were heading in the wrong direction. In Scott’s own words:

Then there were three answers for which the “progress” being celebrated, seemed to me to be progress racing faster into WrongVille

In particular, the following residents of the so-called “WrongVille” were of immediate interest to me; let me continue quoting Scott’s words:

Ross Anderson on an exciting conference whose participants aim to replace quantum mechanics with local realistic theories.  (Anderson, in particular, is totally wrong that you can get Bell inequality violation from “a combination of local action and global correlation,” unless the global correlation goes as far as a ‘t-Hooft-like superdeterministic conspiracy.) [Emphasis in bold is mine.]

The minimum implications here are these two: (i) quantum mechanics—not this interpretation or that interpretation of its existing mathematics, but the entire mechanics of the quanta itself—cannot ever be local, and (ii) therefore, any attempts to build a local theory to explain the quantum phenomena must be seen as a replacement for QM [a lock, stock and barrel replacement, I suppose].

One further implicit idea here seems to be that any local theory, if it yields the necessary global correlation, must also imply superdeterminism. In case you don’t know, “superdeterminism” here is primarily a technical term, not philosophical; it is about a certain idea put forth by the Nobel laureate ‘t Hooft.

As you know, my theorization has been, and will always remain, local in nature. Naturally, I had to intervene! As fast as I could!!

So I wrote a comment at Scott’s blog, right on the fly. (Literally. By the time I finished typing it and hit the Submit Comment button, I was already in the middle of some informal discussions in my cabin with my colleagues, regarding arrangements to be made for the accreditation-related work.)

Naturally, my comment isn’t as clear as it should be.

It so happens that our accreditation-related activities would be over on the upcoming Sunday, and so, I should be able to find the time to come back and post an expanded and edited version early next week. Until then, please make do with my original reply at Scott’s blog [^]; I am copy-pasting the relevant portion “as is” below:

Anderson’s (or others’) particular theory (or theories) might not be right, but the very idea that there can be this combination of a local action + a global correlation, isn’t. It is in fact easy to show how:

The system evolution in QM is governed by the TDSE, and it involves a first derivative in time and a second in space. TDSE thus has a remarkable formal similarity to the (linear) diffusion equation (DE for short).

It is easy to show that a local solution to the DE can be constructed. Indeed, any random walks-based solution involves only a local action. More broadly, starting with any sub-domain method and using a limiting argument, a deterministic solution that is local, can always be constructed.

Of course, there *are* differences between DE and TDSE. TDSE has the imaginary $i$ multiplying the time derivative term (I here assume TDSE in exactly that form as given on the first page of Griffith’s text), an imaginary “diffusion coefficient,” and a complex-valued \Psi. The last two differences are relatively insignificant; they only make the equation consistent with the requirement that the measurements-related eigenvalues be real. The “real” difference arises due to the first factor, i.e. the existence of the i multiplying the $\partial \Psi/\partial t$ term. Its presence makes the solution oscillatory in time (in TDSE) rather than exponentially decaying (as in DE).

However, notice, in the classical DE too, a similar situation exists. “Waves” do exist in the space part of the solution to DE; they arise due to the separation hypothesis and the nature of the Fourier method. OTOH, a sub domain-based or random walks-based solution (see Einstein’s 1905 derivation of the diffusion equation) remains local even if eigenwaves exist in the Fourier modeling of the problem.

Therefore, as far as the local vs. global debate is concerned, the oscillatory nature of the time-dependence in TDSE is of no fundamental relevance.

The Fourier-theoretical solution isn’t unique in DE; hence local solutions to TDSE are possible. Local and propagating processes can “derive” diffusion, and therefore, must be capable of producing the TDSE.

Note, my point is very broad. Here, I am not endorsing any particular local-action + global-correlation theory. In fact, I don’t have to.

All that I am saying is (and it is enough to say only this much) that (i) the mathematics involved is such that it allows building of a local theory (primarily because Fourier theoretical solutions can be shown not to be unique), and (ii) the best experiments done so far are still so “gross” that existence of such fine differences in the time-evolution cannot be ruled out.

One final point. I don’t know how the attendees of that conference think like, but at least as far as I am concerned, I am (also) informally convinced that it will be impossible to give a thoroughly classical mechanics-based mechanism for the quantum phenomena. The QM is supposed to give rise to CM (Classical Mechanics) in the “grossing out” limit, not the other way around. Here, by CM, I mean: Newton’s postulates (and subsequent reformulations of his mechanics by Lagrange and Hamilton). If there are folks who think that they could preserve all the laws of Newton’s, and still work out a QM as an end product, I think, they are likely to fail. (I use “likely” simply because I cannot prove it. However, I *have* thought about building a local theory for QM, and also do have some definite ideas for a local theory of QM. One aspect of this theory is that it can’t preserve a certain aspect of Newton’s postulates, even if my theorization remains local and propagational in nature (with a compact support throughout).)

OK. So think about it in the meanwhile, and bye for now.

[BTW, though I believe that QM theory must be local, I don’t agree that something such as superdeterminism is really necessary.]


A Song I Like:

(Hindi) “aaj un se pehli mulaaqaat hogi…”
Singer: Kishore Kumar
Music: R. D. Burman
Lyrics: Anand Bakshi

[E&OE]

 

 

 

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