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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

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").

jostylr 6y ago
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 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.

jostylr 6y ago

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 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.