Wednesday, May 22, 2024

What happens to particles after they have been interacting according to Bohm?

 Once more, I am trying to better understand the Bohmian or pilot wave approach to quantum mechanics. And I came across this technical question, which I have not been able to successfully answer from the literature:

Consider a particle, described by a wave function \(\psi(x)\) and a Bohmian position \(q\) that both happily evolve in time according to the Schrödinger equation and the Bohmian equation of motion along the flow field. Now, at some point in time, the (actual) position of that particle gets recorded, either using a photographic plate oder by flying through a bubble chamber or similar. 

Unless I am not mistaken, following the "having a position is the defining property of a particle"-mantra, what is getting recorded is \(q\). After all, the fact, that there is exactly one place on a photographic place that gets dark was the the original motivation of introducing the particle position denoted by \(q\). So far, so good (I hope).

My question, however, is: What happens next? What value of \(q\) am I supposed to take for the further time evolution? I see three possibilities:

  1. I use the \(q\) that was recorded.
  2. Thanks to the recording, the wave function collapses to an appropriate eigenstate (possibly my measurement was not exact, I just inferred that the particle is inside some interval, then the wave function only gets projected to that interval) and thanks to the interaction all I can know is that \(q\) is then randomly distributed according to \(|P\psi|^2\) (where \(P\) is the projector) ("new equilibrium").
  3. Anything can happen, depending on the detailed inner workings and degrees of freedom of the recording device, after all the Bohmian flow equation is non-local and involves all degrees of freedom in the universe.
  4. Something else
All three sound somewhat reasonable, but upon further inspection, all of them have drawbacks: If option 1 were the case, that would have just prepared the position \(q\) for the further evolution. Allowing this to happen, opens the door to faster than light signalling as I explained before in this paper. Option 2 gives up the deterministic nature of the theory and allows for random jumps of the "true" position of the particle. This is even worse for option 3: Of course, you can always say this and think you are safe. If there are other particles beyond the one recorded and their wave functions are entangled, option 3 completely gives up on making any prediction about the future also of those other particles. Note that more orthodox interpretations of quantum mechanics (like Copenhagen, whatever you understand under this name) does make very precise predictions about those other particles after an entangled one has been measured. So that would be a shortcoming of the Bohmian approach.

I am honestly interested in the answer to this question. So please comment if you know or have an opinion!

Wednesday, January 24, 2024

How do magnets work?

I came across this excerpt from a a christian home schooling book:

which is of course funny in so many ways not at least as the whole process of "seeing" is electromagnetic at its very core and of course most people will have felt electricity at some point in their life. Even historically, this is pretty much how it was discovered by Galvani (using forge' legs) at a time when electricity was about cat skins and amber.

It also brings to mind this quite famous Youtube video that shows Feynman being interviewed by the BBC and first getting somewhat angry about the question how magnets work and then actually goes into a quite deep explanation of what it means to explain something

But how do magnets work? When I look at what my kids are taught in school, it basically boils down to "a magnet is made up of tiny magnets that all align" which if you think about it is actually a non-explanation. Can we do better (using more than layman's physics)? What is it exactly that makes magnets behave like magnets?

I would define magnetism as the force that moving charges feel in an electromagnetic field (the part proportional to the velocity) or said the other way round: The magnetic field is the field that is caused by moving charges. Using this definition, my interpretation of the question about magnets is then why permanent magnets feel this force.  For the permanent magnets, I want to use the "they are made of tiny magnets" line of thought but remove the circularity of the argument by replacing it by "they are made of tiny spins". 

This transforms the question to "Why do the elementary particles that make up matter feel the same force as moving charges even if they are not moving?".

And this question has an answer: Because they are Dirac particles! At small energies, the Dirac equation reduces to the Pauli equation which involves the term (thanks to minimal coupling)
$$(\vec\sigma\cdot(\vec p+q\vec A)^2$$
and when you expand the square that contains (in Coulomb gauge)
$$(\vec\sigma\cdot \vec p)(\vec\sigma\cdot q\vec A)= q\vec A\cdot\vec p + (\vec p\times q\vec A)\cdot\vec\sigma$$
Here, the first term is the one responsible for the interaction of the magnetic field and moving charges while the second one couples $$\nabla\times\vec A$$ to the operator $$\vec\sigma$$, i.e. the spin. And since you need to have both terms, this links the force on moving charges to this property we call spin. If you like, the fact that the g-factor is not vanishing is the core of the explanation how magnets work.

And if you want, you can add spin-statistics which then implies the full "stability of matter" story in the end is responsible that you can from macroscopic objects out of Dirac particles that can be magnets.