An argument against Bohmian mechanics?

[A. Neumaier]

As a mathematician, you certainly know that I can pull back any structure between isomorphic objects

Yes, but $L^2(R^3)$ doesn't have this structure but must be equipped with it in order to have it. Quantum physics is always about Hilbert spaces already equipped with the right group action.

Ignoring that is like saying I can do functional analysis in the set $R_+$ of positive real numbers, since I can equip any set with continuum cardinality with the structure of $L^2(R^3)$. No mathematician in his right mind would talk like this.

 

[A. Neumaier]

There's no state with infinitely many particles in the sense that the expectation value of the number of particles should be finite for the state to make sense.

But a state must only be square integrable. Therefore there are valid states (such as $\sum_{N>0} N^{-1}|N\rangle$) where the expected number of particles is infinite. Nevertheless, upon each Born-style measurement, the number of particles would come out finite.

 

[vanhees71]

Hm, good point. This would be a state without a vacuum contribution. Interesting. I'm a bit in doubt, whether it can be prepared for any type of particles.

 

[A. Neumaier]

I'm a bit in doubt, whether it can be prepared for any type of particles.

It would take an infinite amount of energy to create it, I believe.

This would be a state without a vacuum contribution.

The state $\sum_{N\ge 0} (N+1)^{-1}|N\rangle$ has the same properties but with a vacuum contribution. Thus that's not the source of the difficulties to create such a state. You can add as much vacuum as you like without changing the infinite expected particle number.

 

[rubi]

Yes, but $L^2(R^3)$ doesn't have this structure but must be equipped with it in order to have it. Quantum physics is always about Hilbert spaces already equipped with the right group action.

Ignoring that is like saying I can do functional analysis in the set $R_+$ of positive real numbers, since I can equip any set with continuum cardinality with the structure of $L^2(R^3)$. No mathematician in his right mind would talk like this.

An object must always be equipped with a structure in order to have it. It's just easier to write it down in some cases. I'm of course not suggesting that one should do QFT on $L^2(\mathbb R^3)$, but that one could do it, i.e. a state vector in QM needs as much information to be fully specified as a state vector in QFT. You cannot do QFT in the Hilbert space of a spin-1/2 system. And you cannot do LQG in the Hilbert space of a QFT. The dimension of the Hilbert space is of course a measure for the information content of a quantum state.

Anyway, I have explained what I meant already in post #47 and what I said was correct, so this discussion is a bit pointless.

 

[A. Neumaier]

The dimension of the Hilbert space is of course a measure for the information content of a quantum state.

No. The conventional, and the only reasonable measure of information applicable a quantum state is the entropy. What you talk about does not deserve the name information in the traditional, scientifically established sense.

 

[vanhees71]

You cannot do QFT in the Hilbert space of a spin-1/2 system. And you cannot do LQG in the Hilbert space of a QFT. The dimension of the Hilbert space is of course a measure for the information content of a quantum state.

First I don't understand what you mean by the first quoted sentence, since a "Pauli particle" (spin-1/2 particle in non-relativistic QT) is described by the usual separable Hilbert space and usually reallized as $\mathrm{L}^2(\mathbb{R}^3,\mathbb{C}^2)$ (position-spin representation or "wave mechanics").

For curiosity, what's LQG?

I also don't know what I should make of the third sentence. The information content of a quantum state, or rather our ignorance about it, is given by the (von Neumann) entropy. For a pure state it's 0 (i.e., we have full possible knowledge about the system's preparation).

 

[rubi]

No. The conventional, and the only reasonable measure of information applicable a quantum state is the entropy. What you talk about does not deserve the name information in the traditional, scientifically established sense.

Then please just assign whatever name you like to it, so we don't need to have this discussion about pure semantics. The dimension of the Hilbert space is a classification criterion for quantum systems.

First I don't understand what you mean by the first quoted sentence, since a "Pauli particle" (spin-1/2 particle in non-relativistic QT) is described by the usual separable Hilbert space and usually reallized as $\mathrm{L}^2(\mathbb{R}^3,\mathbb{C}^2)$ (position-spin representation or "wave mechanics").

I said "spin-1/2 system" and not "(Pauli) particle". A spin-1/2 system is conventionally understood as $\mathcal H=\mathbb C^2$ with the standard spin-1/2 representation of $SU(2)$. People study such quantum systems without referencing the coordinates of particles. Especially in quantum information, people rarely use infinite-dimensional Hilbert spaces.

For curiosity, what's LQG?

Loop quantum gravity.

I also don't know what I should make of the third sentence. The information content of a quantum state, or rather our ignorance about it, is given by the (von Neumann) entropy. For a pure state it's 0 (i.e., we have full possible knowledge about the system's preparation).

See above.

 

[A. Neumaier]

The dimension of the Hilbert space is a classification criterion for quantum systems.

A very, very weak one, since the vast majority of quantum systems of experimental interest (except for finite spin systems) have a separable infinite-dimensional Hilbert space, all of which are isomorphic if you don't have additional structure.

 

[rubi]

A very, very weak one, since the vast majority of quantum systems of experimental interest (except for finite spin systems) have a separable infinite-dimensional Hilbert space, all of which are isomorphic if you don't have additional structure.

I agree, but it seemed to me like Demystifier wasn't aware of it, which is why I explained it in my post. At first, it seems unintuitive that a Hilbert space of separable dimension is enough for field variables, because one might think that a field can have values at uncountably many space-time points. A separable Hilbert space suffices, because only smeared fields (smeared with Schwartz functions) are considered. If one wanted to include fields operators defined at sharp space-time points, one would need a Hilbert space of uncountable dimension.

 

[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.