Bohmian Mechanics (2017) | Hacker News Reader

Bohmian Mechanics (2017)

plato.stanford.edu

41 pointsdoctor_eval10 comments

I've got a PhD in theoretical particle physics. During my undergrad years I was very interested in interpretations of quantum mechanics. I was swayed by Bohmian interpretation and decided to try to "convert" problems and solutions of a standard QM-101 course into that Bohmian language. The first several chapters went well, but I got stuck on particle indistinguishably - I couldn't find any adequate explanation as to how to get, say, fermions with Bohm trajectories. Almost 20 years have passed since - I've been returning to discussions of Bohmian interpretation here and there - but no one so far have given me what I'm looking for. The indistinguishably question is just getting ignored or brushed off as insignificant (yeah Pauli exclusion principle is "insignificant").

The key to identical particles is understanding the configuration space properly. Basically, instead of ordered n-tuples (R^3N), one can use sets of size n whose elements are points in R^3. This immediately eliminates everything except the "symmetric" wave functions. To get the anti-symmetric functions needed for fermions, one needs to extend the theories under consideration a bit. One could use a covering space approach, demanding a suitable projection of the trajectories downwards or one can approach it from the perspective of having a twisted bundle for the value space of the wave function, one in which it looks locally like complex-valued wave functions, but globally it has a parallel transport action that leads to sign change around a closed path that suitably permutes the set.

The nice thing about the parallel transport point of view is the extension to spin. Again, it will look locally like the usual tensor product, but by using an index set for the tensor product being just the set of positions itself, one can again get a non-trivial parallel transport that does just what one would expect.

This is what restricts the permissible wave functions.

One reference for this is my own thesis from about 20 years ago:

http://jostylr.com/thesis.pdf

There were some papers that were published from that, but they mainly focused on the general topological story rather than having very much in the way of identical particles.

One thing we never did accomplish from this point of view was explaining why fermions went with half spins and bosons integer spins. But at least we did get it reduced to just the two choices.

Great article. Bohmian mechanics is often dismissed with reference to the impossibility of hidden variables due to Bell. So I enjoyed reading the following bit:

> Here’s how Bell himself reacted to Bohm’s discovery:

But in 1952 I saw the impossible done. It was in papers by David Bohm. Bohm showed explicitly how parameters could indeed be introduced, into nonrelativistic wave mechanics, with the help of which the indeterministic description could be transformed into a deterministic one. More importantly, in my opinion, the subjectivity of the orthodox version, the necessary reference to the “observer”, could be eliminated. …

The way I've always understood it is that Bell's theorem eliminates theories with local hidden variables, but Bohmian mechanics has explicitly non-local hidden variables.

I recently read a paper [1] by the author of asymetric numerical systems fame, Jarek Duda, which I found a fun dive into what I realized was Bohmian mechanics. It was chuck full of typos and hand waiving, but it was pretty approachable as a short dive through how one might get from basic mechanics of ellipsoid potentials to standard model particles. It posits a view of maximum entropy among all potential paths in space-time in a Lagrangian framework determines physics. Fun to think about, but I'm not sure if anyone has explored enough to see if it really can get to some of the tricky corners of phenomenology.

[1] "Four-dimensional understanding of quantum mechanics" https://arxiv.org/abs/0910.2724

Speaking of typos: "chuck full" should be chock-full.

You're welcome

When I was a student, I was struck by the lack of focus on the weirdness of QM in the QM courses. It hit me super hard when it switched from courses where you could spend extra time and feel like you really understood it to instead having a course where that extra time is a counterproductive rabbit hole and “shut up and calculate” is the most useful school of thought.

I absolutely love anything that tries to make progress on the foundational questions.

Stephen Wolfram is trying to make progress in this area.

https://www.youtube.com/watch?v=-t1_ffaFXao&t=3955s

Despite what ShamelessC says about the skepticism of other physicists, I found Stephen Wolfram's talk absolutely fascinating. For the first time, some disparately related notions in my mind ceased to be so and now make sense to me in that they're integrally related in ways I'd not given much thought to previously.

Irrespective of whether he's on the correct track or not, his explanation of certain phenomena, energy for instance, made complete sense. His way of using analogies to describe abstract processes is superb. By that I mean he is able to present one with conceptually understandable mental pictures or images of difficult physics that one would normally only understand through mathematical equations.

It's a shame more physicists don't take a leaf [or do I mean branch - pun intended ;-)] from his way of presenting these concepts.

Which has received a good deal of skepticism from the scientific community.

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