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.
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 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.
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.
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.
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.
[1] https://plato.stanford.edu/entries/qm-manyworlds/#6
[2] https://arxiv.org/abs/1307.1714
[3] John Bell, who was a big fan of pilot waves by the way.
A more appropriate analogy to the SA v NSA would be GRW formulation vs Bohmian or even a well-defined many worlds (that does exist, but it does not involve splitting of worlds which is as problematic as collapse). These are different theories and they can lead to different generalizations.
In particular, GRW seems more amenable to being relativistic without adding a foliation or some other structure.
Bohmiam mechanics, on the other hand, does requires using a foliation though there are possibilities to use within the existing structures: https://arxiv.org/abs/1307.1714
I also think NSA has a naming problem. By its very name, how can it ever be standard?
I think the issue is also that the limit fits in with numerical analysis in that it talks about errors. NSA sounds like it skips the error analysis and is only concerned with stuff at the limit which is certainly useful in many instances, but not always.
"Nonstandard" isn't referring to how people view the field, but to additional elements you add to the model. Just to throw out a weird example, here's Edward Nelson's text on the subject:
https://web.math.princeton.edu/~nelson/books/1.pdf
"Theorem 4. There is a finite set that contains every standard object."
That is, in Nelson's system, the set of standard natural numbers is finite. That finite number is a nonstandard natural number, larger than any standard natural number.
Mixing "nonstandard"(in the sense of an undefined predicate added to the base logic) with "nonstandard"(in the sense of some professors thinking it's weird) is a type error.
Someone correct me if I'm wrong here, but logically, standard theory and pilot theory can't BOTH be right, can they?
Newton's fluxions and Leibniz's differentials provided great guidance, and in a lot of cases they even worked and gave correct results, but they were not really rigorously defined, and later mathematicians started running into serious problems with that approach. Those were resolved with the approach that we all now known and love (loath?): limits and the epsilon/delta approach to them.
NSA extends the real field to contain a new kind of number that is positive but is less than 1/N for any integer N, giving a new field called the hyperreals. You can then do things like define the derivative of f at x to be (f(x+h)-f(x))/h where h is an infinitesimal, work out that using ordinary algebraic manipulations, and then just drop any terms that are infinitesimal.
NSA goes far beyond just calculus. You can use these ideas all through analysis.
Seems pretty cool, and a lot clearer than a limits approach. So why has it not taken over?
I believe that a big part of it is that it has been shown that NSA and standard analysis are equivalent. Anything that you can prove in one can be proven in the other. Perhaps more important, one doesn't seem any better than the other at leading to deep insights. If you can't solve a problem in standard analysis, you probably aren't going to figure out the solution if you use NSA, and vice versa.
So while it might be overall nicer if we were starting from scratch to use NSA instead of standard analysis everywhere, so much is already done in standard analysis that switching now would be too painful.