An argument against Bohmian mechanics?

[A. Neumaier]

If one wanted to include fields operators defined at sharp space-time points, one would need a Hilbert space of uncountable dimension.

But one then gets into trouble with locality.

 

[rubi]

But one then gets into trouble with locality.

Possible, but why do you think so? One could require $[\phi(x),\phi(y)]=\delta_{xy}$ instead of $[\phi(x),\phi(y)]=\delta(x-y)$.
I think the motivation for using Schwartz distributions is to have a well-defined notion of Fourier transform, while maintaining as much generality as possible.

 

[A. Neumaier]

Possible, but why do you think so? One could require $[\phi(x),\phi(y)]=\delta_{xy}$ instead of $[\phi(x),\phi(y)]=\delta(x-y)$.

Your suggestion plays havoc with everything important in QFT, since it essentially equips space-time with the discrete topology. Try to base a QFT on your suggestion and you'll see.

 

[rubi]

Your suggestion plays havoc with everything important in QFT, since it essentially equips space-time with the discrete topology. Try to base a QFT on your suggestion and you'll see.

I can see many things going wrong and I agree that one shouldn't do it. It's just an example to clarify things. I was just wondering, why you were specifically thinking about locality, because that seems pretty easy to maintain.

 

[vanhees71]

Well, our lattice-QCD colleagues are pretty successful with such an approach, however only in imaginary time ;-).

 

[A. Neumaier]

Well, our lattice-QCD colleagues are pretty successful with such an approach, however only in imaginary time ;-).

But locality isn't conventionally defined in imaginary time ;-(

 

[vanhees71]

Well, yes. Lattice theory is an approximation as well, but tell this the lattice guys and see their reaction (which is very interesting ;-)).

 

[Demystifier]

Well, yes. Lattice theory is an approximation as well, but tell this the lattice guys and see their reaction (which is very interesting ;-)).

The right question is this: Is it only an approximation in the limit $a\rightarrow 0$?

 

[vanhees71]

Yes, that's indeed the question, but on the other hand computers are finite, and thus also $a$ stays always finite too in practical calculations.

 

[Demystifier]

Well, apparently, this theorem is not established rigorously, so finding a gap is not necessary to criticize it.

I don't understand this statement. You are right, this theorem is not a rigorous theorem; it is a FAPP (For ALL Practical Purposes) theorem*. If you like, you may even call it an argument, rather then a theorem. But to criticize an argument, you must find a gap in the argument. There is no other way.

*To explain what I mean by a FAPP theorem, let me give an example: the law of large numbers in probability theory. In the limit $N\rightarrow\infty$ it is a rigorous theorem. But as such, it is pretty useless. It is only useful for a big but finite $N$, sometimes as small as $N=1000$. For finite $N$, the law of large numbers is only a FAPP theorem.
In fact, the FAPP theorem of Bohmian mechanics could also be translated into a mathematically rigorous theorem, but in such a form it would be physically irrelevant. Yes, it would probably make some mathematical physicists happy, but still it would not be much useful for practical purposes.

 

[rubi]

Well, our lattice-QCD colleagues are pretty successful with such an approach, however only in imaginary time ;-).

Right. You can also view LQG as lattice quantum gravity with the continuum limit taken.

But localityisn't conventionally defined in imaginary time ;-(

Well, locality in the Lorentzian (Wightman) setting translates to the symmetry of the Schwinger functions in the Euclidean setting.

I don't understand this statement. You are right, this theorem is not a rigorous theorem; it is a FAPP (For ALL Practical Purposes) theorem*. If you like, you may even call it an argument, rather then a theorem. But to criticize an argument, you mist find a gap in the argument. There is no other way.

A rigorous argument is an argument that doesn't have gaps. So if your argument is not rigorous, then it has gaps by definition.

In fact, the FAPP theorem of Bohmian mechanics could also be translated into a mathematically rigorous theorem, but in such a form it would be physically irrelevant. Yes, it would probably make some mathematical physicists happy, but still it would not be much useful for practical purposes.

Mathematical rigor is just a different name for having high standards with respect to ones arguments. This is generally a good thing. Whether Bohmian mechanics reproduces QM depends critically on a specific argument, so one ought to have high standards with respect to it. Since interpretations of QM all (claim to) make the same physical predictions, mathematical rigor is the only possible way to exclude some of them. So requiring arguments to be rigorous is the least we should expect in the interpretations business.

 

[vanhees71]

Well, my only quibble is, what's "practical" about Bohmian mechanics. I've never been able to make sense of the claimed trajectories, which cannot be verified empirically. So what's physics wise the merit of Bohmian mechanics compared to minimally interpreted quantum theory? At best it's a nice academic mathematical exercise to calculate the unobservable trajectories, right?

 

[rubi]

To explain what I mean by a FAPP theorem, let me give an example: the law of large numbers in probability theory. In the limit $N\rightarrow\infty$ it is a rigorous theorem. But as such, it is pretty useless. It is only useful for a big but finite $N$, sometimes as small as $N=1000$. For finite $N$, the law of large numbers is only a FAPP theorem.

Of course, mathematicians also study, how big $N$ must be in order to have the probability of deviating from the $N\rightarrow\infty$ limit to be sufficiently small.
(See https://en.wikipedia.org/wiki/Large_deviations_theory )

Well, my only quibble is, what's "practical" about Bohmian mechanics.

Well that seems to be Arnold's point also. Standard quantum mechanics makes correct predictions about the hydrogen atom even without including extra baggage and ending up with a complicated model (of which we don't even rigorously know whether it works, even if we include all the extra baggage).

 

[A. Neumaier]

So what's physics wise the merit of Bohmian mechanics compared to minimally interpreted quantum theory?

It adds to quantum theory a host of unobservable (and hence unverifiable and unfalsifiable) degrees of freedom to give its believers the illusion that particle have infinitely accurate positions at all times. This comes at the cost of lots of other counterintuitive features:

• The nonlocal dynamical equations are one of them. (Even Newton considered the nonlocality of his gravitational forces as a defect of the theory.)
• That in a universe consisting of a single hydrogen atom, the electron stands still is another one.
• Worst of all, it cannot make any measurable prediction without taking the whole universe into account. Once this is done and its effect is eliminated by statistical mechanics, only ordinary quantum mechanics is left.

For those like me who believe that infinitely accurate positions are an artifact of idealization beyond the reasonable, it has no merit at all.

 

[vanhees71]

Hm, QT without Bohm's "extra baggage" describes in great detail how the hydrogen atom works. Its spectrum is among the most precisely described phenomena of relativistic QFT. There's no need for "extra baggage", let alone that's not clear how in the relativistic case one might define its content ;-).

 

[Demystifier]

So if your argument is not rigorous, then it has gaps by definition.

Which does not mean that one does not need to criticize the specific gaps in order to criticize the final conclusion of the argument.

 

[Demystifier]

Well, my only quibble is, what's "practical" about Bohmian mechanics. I've never been able to make sense of the claimed trajectories, which cannot be verified empirically. So what's physics wise the merit of Bohmian mechanics compared to minimally interpreted quantum theory? At best it's a nice academic mathematical exercise to calculate the unobservable trajectories, right?

To paraphrase Feynman, Bohmian mechanics is like sex. Sure, it may have practical applications
https://www.amazon.com/dp/9814316393/?tag=pfamazon01-20
but that's not why we do it.

So why do I do it? Well, for me interpreting QM is like interpreting a magic trick. When a magician pulls out a rabbit from the hat, what do the spectators do? Unfortunately, they cannot come to the stage to explore how the magician really does it. So for a poor spectator there are only a few options:
1) Watch and enjoy the show. (The analog of shut up and calculate for QM.)
2) Accept the minimal interpretation; the magician somehow pulls out the rabbit, and that's all what I can and need to know. (The analog of minimal interpretation for QM.)
3) Interpret it as a true magic. The rabbit was not there from the beginning, but in some moment it was created from nothing. (The analog of true collapse interpretation for QM.)
4) Accept that the magician is really a hypnotist who used hypnosis to make spectators believe they see a rabbit. The rabbit doesn't exist objectively, but only as a spectator's observation. (The analog of qubism for QM.)
5) Try to devise a rational mechanism which could explain it. For instance, perhaps the rabbit was hidden inside the table from the beginning, and perhaps the table on which the hat was sitting has a removable cover from which a rabbit can pass, and perhaps the top of the hat can be removed to allow passing of the rabbit from the table to the hat. Yes, this interpretation involves a lot if hidden variables neither of which can be proved by the spectator. Nevertheless, for a person with scientific instinct who seeks rational explanations, such an interpretation makes much more sense than the other four. (The analog of the Bohmian interpretation for QM.)

The question for everybody: What do you do when you see a magician trick?

 

[A. Neumaier]

The question for everybody: What do you do when you see a magician trick?

for a poor spectator there are only a few options:
1) Watch and enjoy the show. (The analog of shut up and calculate for QM.)

For a poor spectator the entry fee is worth the show only under option 1.

 

[Demystifier]

For a poor spectator the entry fee is worth the show only under option 1.

So you never try to figure out how did he did it?

 

[A. Neumaier]

So you never try to figure out how did he did it?

I am not a poor spactator. I have the thermal interpretation. It is enough to show that particles have an approximate position whenever the state is such that the position can be observed. No hidden variables are needed for that.

 

[Demystifier]

I am not a poor spactator. I have the thermal interpretation.

Fair enough. And is there an analog of thermal interpretation for the rabbit-from-the-hat phenomenon?

 

[Spinnor]

The question for everybody: What do you do when you see a magician trick?

Accept and enjoy that the magician did a great trick and then try and figure out how he did it.

 

[vanhees71]

Where is a abbit-from-the-hat phenomenon in QM that is not already there in classical physics? In a way it's indeed a miracle that we can describe nature quite well with mathematical tools. Someone (Einstein?) said the most incomprehensible about nature is its comprehensibility.

 

[A. Neumaier]

Fair enough. And is there an analog of thermal interpretation for the rabbit-from-the-hat phenomenon?

Yes:

6) Study physics and extrapolate from what the physicists really do when they compare experiments with theory.

This is how I discovered the thermal interpretation.

 

[Demystifier]

Accept and enjoy that the magician did a great trick and then try and figure out how he did it.

Does your attempted explanation involve some hidden variables, like those in 5) above?

 

[Demystifier]

Yes:

6) Study physics and extrapolate from what the physicists really do when they compare experiments with theory.

This is how I discovered the thermal interpretation.

I still don't understand how that helps to explain the rabbit-from-the-hat phenomenon.

 

[Mentz114]

Accept and enjoy that the magician did a great trick and then try and figure out how he did it.

Does the rabbit have a definite position before it is pulled out of the hat ? Or is that interpretation dependent ?

 

[A. Neumaier]

I still don't understand how that helps to explain the rabbit-from-the-hat phenomenon.

Without physics, how can you explain the appearance of the rabbit? Physics tells you that mass is conserved (and much more). So you know that the rabbit must have been there before.

 

[Demystifier]

Without physics, how can you explain the appearance of the rabbit? Physics tells you that mass is conserved. So you know that the rabbit must have been there before.

That still doesn't explain the trick.

 

[A. Neumaier]

That still doesn't explain the trick.

It also tells you that brains are not very reliable detectors and may fail when their attention is somewhere else.

 

[vanhees71]

I still don't know, what you consider as a "trick" in QT. It's a mathematical description of objective empirical facts of nature, no more no less, and it's I am principle not different from classical mechanics, it's just more comprehensible (while still not complete as long as there is not a consistent quantum theory of the gravitational interaction).

 

[Demystifier]

Where is a abbit-from-the-hat phenomenon in QM that is not already there in classical physics? In a way it's indeed a miracle that we can describe nature quite well with mathematical tools. Someone (Einstein?) said the most incomprehensible about nature is its comprehensibility.

The rabbit-from-the-hat can be explained even without mathematics. To a great extent, Bohmian interpretation also can be understood without mathematics, just by visualizing localized wave packets and particle trajectories within them.

 

[Demystifier]

It also tells you that brains are not very reliable detectors and may fail when their attention is somewhere else.

Which still doesn't explain the trick.

 

[vanhees71]

I don't need particle trajectories. The wave packets are enough. If there's a position observable (for all massive particles there is) and if I've prepared a particle with a pretty well determined position, then it is described by a position probality distribution which is quite narrow. That's it. No rabbit in sight ;-).

 

[Demystifier]

I still don't know, what you consider as a "trick" in QT. It's a mathematical description of objective empirical facts of nature, no more no less, and it's I am principle not different from classical mechanics, it's just more comprehensible (while still not complete as long as there is not a consistent quantum theory of the gravitational interaction).

Sure, there is no trick in quantum theory. But it looks as if there is a trick in quantum phenomena. I want to know how Nature (the magician) does it, not how a poor spectator describes what he sees.