Interesting analogy but I'm not sure if it holds. Infinitesimals are an alternative to the limit-based formalism typically seen. It changes the way you do calculus.
However Bohmian Mechanics is a different interpretation, not a different formalism. You can accept the interpretation, but you still end up solving the Schrödinger equation in the same way, using the same mathematical methods.
The analogy works better for matrix mechanics as compared to Schrödinger's wave mechanics. These are mathematically quite different even though they give the same results.
You can use the same formalism for most things, but Bohmian mechanics is a different formalism in some cases. For instance, you can derive the Born rule for systems in equilibrium, but that leaves open the possibility of non-equilibrium regimes, which simply don't exist in orthodox QM.
An introduction for the interested [1]. It would be neat if someone could come up with an experimental test. The very existence of non-equilibrium would immediately falsify Copenhagen.
The Heisenberg picture and the Schrodinger picture the same, sometimes using one basis is more useful and more intuitive than using the other.
Pilot Waves, on the other hand, are pretty much always superfluous. Please link me, if you have seen a situation where using Pilot Waves simplifies a calculation.
I think you mean - you can always reparameterize things to have a 'pilot wave' around. That doesn't mean they're 'there', really.
I would think that the 'default' interpretation would be excluding them, since (clearly) they're not necessary (evidence being: interpretations without them exist). That would be the definition of superfluous, right? Unless they give different experimental results somewhere, which I'm pretty sure in't the case.
> I think you mean - you can always reparameterize things to have a 'pilot wave' around. That doesn't mean they're 'there', really.
The wave equation is always there, or you wouldn't get quantum behaviour. There exist no-go theorems demonstrating that the wave function can't just be a reflection of our ignorance, it must be "ontic", ie. exist, in some real way. What's disputed is whether the particles need to be given separate existence from the waves that are necessarily there.
> I would think that the 'default' interpretation would be excluding them, since (clearly) they're not necessary (evidence being: interpretations without them exist). That would be the definition of superfluous, right?
No, because the interpretations you speak of can't actually explain the measurement problem without positing additional axioms which then make them less plausible. It's well known that pilot wave theories are more axiomatically parsimonious, meaning that they require fewer assumptions overall to explain all of our observations.
For example, the Born rule must be assumed by most interpretations of QM, with little rhyme or reason other than we know it's empirically valid. But because pilot wave theories posit real physical entities with well understood properties, we can actually derive the Born rule. This is just one example that demonstrates how pilot wave theories positing additional real entities can make for an overall simpler set of assumptions.
Pilot Wave posits a bunch of extra crap and derives the Born rule. Regular / MWI QM posits the born rule and skips the extra crap. They're both positing something - but the Born rule is a far simpler claim (just in, like, mathematical complexity, to me).
Also: aren't we to the point where measurement makes perfect sense (besides the values of the probabilities, as given by the Born rule), via entanglement between experiment+lab frames, and decoherence of unrelated Degrees of Freedom? Cause I thought we were. (see, for example, http://www.preposterousuniverse.com/blog/2014/06/30/why-the-...)
Also, pilot wave fails badly in relativistic extensions, for the obvious reason that a universe-wide pilot wave function is hard to make covariant. There are attempts at fixing this but last I heard none of them are doing a good job. So that's another strike against it, in my book.
> Pilot Wave posits a bunch of extra crap and derives the Born rule. Regular / MWI QM posits the born rule and skips the extra crap. They're both positing something - but the Born rule is a far simpler claim
Except it's not, because you also have to posit the measurement postulates in orthodox QM. This so-called "extra crap" reproduces the measurement postulates and the Born rule, thus replacing a large set of assumptions with a much smaller set.
Many-Worlds is indeed much simpler than orthodox QM, but it's still not simpler than pilot waves. They are roughly comparable, with many-worlds still having unresolved conceptual difficulties surrounding probabilities, among other issues [1]. Which is more parsimonious between many-worlds and pilot waves is hotly debated among philosophers of science.
> Also: aren't we to the point where measurement makes perfect sense (besides the values of the probabilities, as given by the Born rule), via entanglement between experiment+lab frames, and decoherence of unrelated Degrees of Freedom?
"Measurement now makes perfect sense" is an interpretation-specific claim. Measurement still doesn't make sense in Copenhagen, measurement mostly makes sense in Many-Worlds, modulo some of the difficulties I mentioned earlier [1].
> Also, pilot wave fails badly in relativistic extensions, for the obvious reason that a universe-wide pilot wave function is hard to make covariant.
It's actually pretty trivial if you're willing accept a preferred foliation of space-time, as long as the preferred frame is unobservable. This seems aesthetically unappealing, hence why people perpetuate this myth of "difficulty", but it's not a priori wrong.
Fortunately, a preferred foliation can actually be derived from the wave function itself, which means this foliation exists in every interpretation of QM [2].
This is the kind of surprising result that probably no one would have even bothered looking for, and I think it proves John Bell's position that non-locality is the unresolved problem of QM [3]. Other interpretations just let you paper over it, to our detriment IMO.
However Bohmian Mechanics is a different interpretation, not a different formalism. You can accept the interpretation, but you still end up solving the Schrödinger equation in the same way, using the same mathematical methods.
The analogy works better for matrix mechanics as compared to Schrödinger's wave mechanics. These are mathematically quite different even though they give the same results.