The h Bar

Semiclassical

18:00

The orthodox account states that the wavefunction is nothing more than a computational tool for expressing our knowledge of the system.

It only exists in a conceptual sense.

With dBB there's a difference of opinion.

The more traditional account, tied to people like de Broglie and Bohm, is that the wavefunction is a real, genuine wave in configuration space

vzn

@bolbteppa de broglie (initially) thought the wavefn was real, was he "insane"? maybe he (mistakenly!) gave up on the idea because his cohorts told him something similar :P ... the idea that something real is really a ghost is the anti-hallucination (delusion) going on for ~1century...

Semiclassical

We only interact with it through the particles, but it still serves to pilot the particles through configuration space

bolbteppa

Expected value of say $n$ measurements of velocity, $E(v) = p_1 v_1 + ... + p_n v_n = <p_1,\dots,p_n> \cdot <v_1,\dots,v_n> = (\sqrt{p_1}e^{-i\theta_1},\dots,\sqrt{p_n}e^{-i\theta_n}) \begin{bmatrix} v_1 & 0 & \dots & 0 \\ 0 & v_2 & \dots & 0 \\ \dots & \dots & \dots \\ 0 & 0 & \dots & v_n \end{bmatrix} \begin{bmatrix} \sqrt{p_1} e^{i\theta_1} \\ \vdots \\ \sqrt{p_n} e^{i\theta_n} \end{bmatrix} = \Psi^* \hat{v} \Psi$

Semiclassical

The more recent account, coming from those of a more strictly philosophical bent (e.g. Goldstein Durr etc) is that the wavefunction is real in roughly the same way as the Hamiltonian in classical mechanics is real

Namely, that it expresses/encodes the laws of motion for the particles

bolbteppa

Note the last part is $\Psi^* \hat{v} \Psi$, we ended up with quantum wave functions from the probabilistic idea of 'expected value' i.e. 'expected measurement', and velocity as a linear operator, the idea of wave functions being real is simply bizarre

Semiclassical

18:05

@bolbteppa Keep in mind that, to the extent that Bohmians talk about 'measurement' in a simple way, they mean 'position measurements'

so $\Psi^*(x) x\Psi(x)=x\rho(x)$

For momentum measurements, by contrast, they therefore necessarily have to say that one never measures the momentum of a particle directly

Bohmian measurement of velocity is a tricky thing, frankly

vzn

@bolbteppa any molecular wave can only be measured in individual particles trajectories do we then claim no such wave exists, that it is imaginary? there is a similar principle happening at the subquantum scale, now proven by experiment...

bolbteppa

That single line above is all you need to see why wave functions arise in QM, it hinges on even thinking one needs 'expected measurement/value', which of course goes back to paths not existing, it's just the Born rule before you define a wave function basically

The idea of a wave function being real is crazy, the way Bohm in his paper gets to this conclusion is simply assuming Schrodinger axiomatically, I am actually shocked it's that ridiculous a justification

You can get path integrals or density matrices from that line by re-writing it, have a reference for it etc

Semiclassical

I mean, the Bohmian standpoint is basically one where you go from taking points on phase space as essential, i.e. position and momentum as independent, to one where points on configuration space is essential

That's how it conceives about the uncertainty principle: If you can't measure position and momentum simultaneously, then it means that momentum should not longer be regarded as independent from position

It needs some object to provide that relation, and that's the point of the wavefunction

I'd also point out that, if one is merely trying to show the consistency of such a formulation, then the origin of the Schrodinger equation is irrelevant

That's how Bell viewed the 1952 paper, as I understand it: Not to motivate the entire story of dBB, but to show that trajectories of deterministic particles could be introduced in a logically consistent way

So I don't think it's fair to repeatedly criticize Bohm's paper for not providing a motivation of the Schrodinger equation, when that wasn't his intention

that said, it is a fair question to ask what people have done since the 1952 paper

and while I would be shocked if nobody has asked 'where does the Schrodinger equation come from' in the dBB community, I'll admit that I'm disheartened by how blithely people accept that point

(seems like the story would have to be: If you were going to have a story of particles in configuration space, guided by a pilot wave, then what law would have to govern the evolution of that wave?)

(and then argue, by symmetry and/or by the need for the correct classical limit to emerge, that the Schrodinger equation is the only one possible)

@bolbteppa going back once more to what I was saying earlier, I think part of the dBB reply would be to ask: "Where do the measurements postulates come from?"

vzn

18:30

@bolbteppa there is apparently no other wave in physics that is not regarded as real! o_O

@Semiclassical it appears Sch eqn can be derived from classical wave eqn + de Broglie matter waves eqn. simple enough!

Semiclassical

You need one more ingredient for that to work, namely the classical Hamiltonian $H=p^2/2m+V$

once you have that, then at least heuristically one can infer the schrodinger equation by considering a wave $\psi=Ae^{ikx-\omega t}$ and applying the relations $E=\hbar\omega $, $p=\hbar k$

Captain Bohemian

@Blue How could you reply a message so long ago?

Semiclassical

Wikipedia (on the dBB page) glosses that as "Schrödinger's equation can be derived by using Einstein's light quanta hypothesis $E=\hbar \omega$ and de Broglie's hypothesis $p=\hbar k$." and then goes on to give a similar motivation for the guidance equation

vzn

@Semiclassical there seem to be multiple derivations, think saw that in a paper once... would like to see a survey... this maybe comes close, Styer, 9 formulations of QM aapt.scitation.org/doi/10.1119/1.1445404

Semiclassical

Same. Wikipedia surveys a few, but it's hardly exhaustive.

vzn

18:40

@Semiclassical (idea) maybe would make a good Physics question for anyone who has the nerve

Semiclassical

And even the same formulation usually contains differences of opinion across different authors as far as how to do so

vzn

@Semiclassical actually maybe Bohm seems to have addressed this himself in the 1982 paper pointing to madelung fluid need to look at that further, Sch eqn seems to have been connected to fluid dynamics long ago, am very partial to that direction myself

bolbteppa

@Semiclassical no idea why the position and momentum can't be measured simultaneously in dBB, no idea why that happens just out of 'practical necessity', no idea why the Schrodinger equation has absolutely anything to do with anything, why not the Heat equation? This is where it gets into psychology: 'because it works' is going to be the answer...

He absolutely had to justify using the Schrodinger equation and why it doesn't egregiously conflict with his assumption about paths existing (just being hidden). The point I'm basically being led to by thinking about his paper is what he was doing just isn't consistent, I'm fine for it to make sense, but even worse he then goes off the axiomatic Schrodinger equation and then calls the wave function a real thing, a force!

vzn

@bolbteppa your criticisms in a way ask more of Bohmian theory than of QM! in some/ many ways an impossible demand :(

Semiclassical

He doesn't call the wavefunction a force.

He does call it an objective field, but he doesn't call it a force

bolbteppa

18:45

Well, 'analogous to a force'

vzn

fields ←→ forces

Semiclassical

(given how non-newtonian dBB is, the word 'force' is problematic)

bolbteppa

"This field exerts a force on the particle in a way that is analogous to, but not identical with, the way in which an electromagnetic field exerts a force on a charge, and a meson field exerts a force on a nucleon. In the last analysis, there is, of course, no reason why a particle should not be acted on by a $\Psi$-field, as well as by an electromagnetic field," cqi.inf.usi.ch/qic/bohm1.pdf (P170)

vzn

...the psi-field is the electromagnetic field. o_O

Semiclassical

it's really not.

@bolbteppa Again, that language strongly reflects Bohm's pseudo-Newtonian presentation

bolbteppa

18:49

I know it's going to get hairier understanding why dB called it a real field

Semiclassical

Part of what makes this tricky is that there's been a few eras of activity

there was de Broglie's initial work in the 1920's

there was a resurgence of interest after the 1952 paper

there was another return to the subject when Bell looked at it

and then finally there's been more recent development by those of a more philosophical bent by people like Goldstein

There's continuity across all four of those eras, of course, but there's also a lot of divergence

bolbteppa

I'm willing to grant that the first 3 sections of Bohm's paper could make sense, for argument's sake at the very least, like his set-up seems like the best way you could make a case like this, to me section 4 starting from Schrodinger blindly then examining it to end up re-interpreting the wave function as a real thing, considering how weighty the claims of the first 3 sections are, it's almost burst-out-laughing ridiculous, I'm sure people did better since the 50's

Semiclassical

I tend to focus on the more recent stuff, if only because it's in a position to take advantage of all of the prior history

bolbteppa

I think I understand what you mean abot his pseudo-Newtonian presentation, I think that's because he breaks it up into $Re^{iS/\hbar}$ and then derives things off this special form and ends up with this $U(x)$ potential

Semiclassical

Right.

'the quantum potential'

formally it works, but it feels artificial as all heck

I like the 'wave characteristics = particle trajectories' idea better

for one because it emphasizes the role of the density in configuration space

bolbteppa

18:57

Here's a potential problem with the whole characteristics argument, it may be my misunderstanding of the method of characteristics

Semiclassical

and also parallels how phase space density evolves in Hamiltonian mechanics, with the spirit of the uncertainty principle being that one can only talk about a configuration space density anymore

bolbteppa

My understanding is that the method of characteristics is a method of solving PDE's by re-expressing the PDE in a coordinate system such that the PDE breaks up into 3 ODE's for 3 curvilinear-orthogonal curves that form a coordinate system on the surface, you are exploiting the fact that any curve on a surface can be interpreted as a characteristic

So calling particles characteristics of the Schrodinger equation is like conflating a simply math method of solution by exploiting special curves on the solution surfaces with actual real particle trajectories as if the 'characteristic' aspect of it means anything, it's really just 'particle trajectories', the 'characteristics' is just a name for the curves they live on anyway

I'm not sure how much the idea of characteristics matters

Semiclassical

it doesn't help that there's a 'method of quantum characteristics' which is different than dBB stuff

bolbteppa

haha

Semiclassical

and of course searching for "dbb characteristics" is going to give so many false hits as to be worthless

Captain Bohemian

19:03

excuse me, what does dBB refer to? I have rolled back much to look for the clue of what is dBB but still couldn't be sure what it is.

bolbteppa

There are a few perimeter videos on dBB I should look at

Secret

Method of quantum characteristics

Quantum characteristics are phase-space trajectories that arise in the phase space formulation of quantum mechanics through the Wigner transform of Heisenberg operators of canonical coordinates and momenta. These trajectories obey the Hamilton equations in quantum form and play the role of characteristics in terms of which time-dependent Weyl's symbols of quantum operators can be expressed. In the classical limit, quantum characteristics reduce to classical trajectories. The knowledge of quantum characteristics is equivalent to the knowledge of quantum dynamics. == Weyl-Wigner association rule... ==

bolbteppa

dBB = deBroglie-Bohm theory

Secret

the hell I am reading

Semiclassical

Looking on google, what I'm seeing makes me wonder if 'particle trajectories = characteristic curves' is the wrong slogan

characteristic curves are related, but I'm having a hard time finding something definitive

bolbteppa

19:04

Characteristic curves := curves tbh

But for a PDE the PDE splits when you get into a coordinate system using those curves, so you can characterize the PDE I guess :p

Captain Bohemian

@bolbteppa is it that theory which competes against Copenhagen's interpretation and supports Einstein's idea?

Semiclassical

In the sense of arguing that the wavefunction alone is not a complete specification of a quantum state, yes.

bolbteppa

@CaptainBohemian it claims to anyway :p

Semiclassical

But it also has the feature of being an explicitly nonlocal theory

Which uh

Not something Einstein would have liked

bolbteppa

I would like to know what Einstein was trying to do and why it was so nuts

Semiclassical

19:07

(I think you could take Bell's theorem as proof that no account of QM would have pleased Einstein, and since dBB is an account of QM it therefore must displease Einstein)

Captain Bohemian

@bolbteppa it claims to be but is not agreed generally in scientific community?

Semiclassical

To the extent that dBB is characterized as a reformulation of the QM formalism, it's not something that can be 'disproven.' The math is what it is.

vzn

@Semiclassical the latest development in dBB is Adler + Vinante late 2017

Semiclassical

It's really not.

vzn

@Semiclassical would like to see a proof of that o_O

@Semiclassical sigh had a feeling youd say that (again) @#%& gotta do everything myself around here >:( :P

Semiclassical

19:13

To the extent that AV is evidence, it'd be evidence of QM's empirical limitations

and therefore of dBB's empirical limitations as well

Regardless of whether you consider it a development, it's a development from outside of dBB rather than within it

bolbteppa

@CaptainBohemian the 'standard' view I guess would be that Copenhagen is QM, and you're not supposed to think about interpretations because it's too hard, dBB is probably mainly ignored because of that, MWI is the second most common view probably, with dBB probably after it, I'd say these other interpretations would gain more traction if the warnings about 'interpretations' weren't so stark, that doesn't mean these other approaches are legitimate

Semiclassical

Again, I think part of the response would be: In the standard interpretation, where do the measurement postulates come from?

Captain Bohemian

@bolbteppa excuse me, what is MWI? I also saw this acronym in your previous conversation, but don't know what it is.

mercio

multi world interpretation ?

bolbteppa

Many-Worlds Interpretation

Semiclassical

19:19

Without the measurement postulates, you've got no way of connecting the wavefunction to actual experiments.

vzn

@Semiclassical reviewing Adler. cant find index of his book online yet, maybe not available. full disclosure his 2 papers dont credit bohm at all. he seems to be fond of citing himself. however the book says its based on lower-level (subquantum) thermodynamics and brownian motion giving the Sch eqn as an emergent property. think the connections will become more apparent over time.

Semiclassical

Yes, and that's not how the Schrodinger equation comes out in the dBB story. Hence why I maintain that it's not a development from within that

Captain Bohemian

@bolbteppa I seem not to have heard of this interpretation. Is it the interpretation underlying multiverse?

bolbteppa

Yeah

Semiclassical

If anything, the Adler et al position would be that dBB is simply not correct. At most, the dBB account would be emergent itself.

vzn

19:21

@Semiclassical again, bohm proposed brownian motion and emergence as a direction to research. etc

Semiclassical

Again, that's Bohm.

dBB, in its mathematical content, is contained in Bohm's 1952 paper. Anything past that is something else

@bolbteppa closest thing I'm finding right now is this:

"Also the negative metaphysical (“surrealistic”) connotations associated with Bohm’s trajectories started changing, since, in the end, they are properly defined from a formal viewpoint. For those familiarized with the treatment and solution of partial differential equations, such trajectories are just the curves arising from the method of characteristics, well-known in mathematical physics to solve finite-order partial differential equations"

however, the fact that I've had such trouble finding stuff like that makes me suspicious that I'm misunderstanding something

bolbteppa

I think that's what I was referring to earlier

Semiclassical

hmm

bolbteppa

People trying to interpret characteristics as trajectories as if a particular solution method generating curves (only by a special choice of coordinate system making the solution easier) has anything to do with physics

Semiclassical

That doesn't sound right, if only because the curves you get in dBB are determined by the wavefunction

bolbteppa

19:28

It seems analogous to saying integrating $\int_a^b \frac{d}{dt} f(t) dt = f(b) - f(a)$ by the FToC is different to integrating it by breaking it into Riemann sums

Semiclassical

They're not just a choice of coordinate system

bolbteppa

Like, the 'characteristics' are just an artifice of a particular solution method

Semiclassical

There is an issue in that vein which I haven't really seen addressed

namely, suppose we at least take for granted the continuity equation $\partial_t \rho+\nabla \cdot \vec{j}=0$

In order to make the dBB story run, we take $\vec{v}=\vec{j}/\rho$

But to what extent is $\vec{j}$ a well-defined object?

I mean, if you replace $\vec{j}\to \vec{j}+\nabla f$, the continuity equation still holds

The implication being that you'd get the same empirical agreement, but the curves would be modified to $\vec{v}\to \vec{j}/\rho+\nabla f/\rho$

and therefore not unique

vzn

AHA Vinante has worked with Bassi on the new subquantum findings, Bassi cited heavily by Adler, Bassi reviewed Adlers book. Bassi has written several papers partial to Bohmian theory Bohmian Mechanics, Collapse Models and the emergence of Classicality arxiv.org/abs/1603.02541 Information and the foundations of quantum theory arxiv.org/abs/1310.8600 Quantum Theory: Exact or Approximate? arxiv.org/abs/0912.2211 ...

Book Review: "Quantum Theory as an Emergent Phenomenon", by Stephen L. Adler arxiv.org/abs/quant-ph/0504214

Semiclassical

that's an objection I've seen mentioned in the Towler slides, for instance, and the only response given is that apparently the choice of $\vec{j}=\frac{\hbar}{m}\text{Im}(\psi^*\nabla \psi)$, as done in standard QM, is unique determined by the fact that it should be the non-relativistic limit of a properly relativistic current

i.e. start with the Dirac theory, write down the (unique!) probability 4-current there, and take the non-relativistic limit

I dunno how satisfying I find that, though

I mean, at best that's applicable to electrons.

For a spin-zero particle...?

mercio

19:46

that review makes me want to read the book

Semiclassical

(in fact, come to think of it: How would dBB even work for a spin-zero particle, since the appropriate relativistic wave equation would be the K-G equation and that doesn't have a continuity equation)

vzn

the paper by Adler + Basi arxiv.org/abs/0912.2211 considers Bohmian mechanics as mathematically equivalent to QM. while not disagreeing, feels like too narrow an interpretation to me & that more credit is due to Bohm for prediction of subtle subquantum effects, ref to brownian motion etc. :( Although their underlying mathematical formulations differ, empirically they are indistinguishable, since they predict the same experimental results as does standard quantum theory.

Semiclassical

It looks like even the dBB people don't try to give a trajectory interpretation to a spin-zero particle, with the ontology in that case being based on fields rather than particles

e.g. "After summarizing the salient points of the nonrelativistic theory, it is explained why a trajectory interpetation of the Klein-Gordon equation is in general untenable" shows up in an abstract

bolbteppa

@Semiclassical what does that mean

Semiclassical

the last line?

bolbteppa

19:55

Yeah

KG

Semiclassical

Well, the comparison is this. If you do the Dirac equation and run the dBB story, what you get works albeit with some features that go against the spirit of relativity

With K-G, by contrast, you can't even do that much

The reason being that, in the Dirac theory, you can write down a genuine continuity equation for a 4-current $(\rho,\vec{j})$ with positive definite density

By contrast: While K-G does have an equation that looks like a continuity equation, the density isn't positive definite

and therefore doing something like $\vec{v}=\vec{j}/\rho$ is right out, since the velocity should be well-defined

so that strategy for producing a trajectory interpretation of a relativistic spin-zero particle will fail

bolbteppa

I see

Semiclassical

i think their response is that, when doing bosons, they treat the fields themselves as basically real and not the particles. by contrast, the opposite seems to work for fermions

...which is sorta goofy

bolbteppa

20:15

I think I made some headway into dBB anyway, more than I had up to now, I'm really disappointed with what I've seen

Think it was useful to figure out the jist of what they were saying

danielunderwood

20:52

Any of you guys that mess with ML ever tried to do a classification problem with a variable output space? Say you want to classify the type of object in an image, but at one point it could be a selection of different species of tree, in another one of a selection of species of shrub, and so on

Or classification with abnormally large output spaces. Although I suspect you'd just use regression and some sort of picking to do that

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