An argument against Bohmian mechanics?

[Demystifier]

Problems only arise if the axioms contradict each other, which doesn't happen in Copenhagen, but might happen in BM.

Can you name two axioms in BM which potentially might contradict each other? I have no idea what those could be.
On the level of rigorous mathematics, it is proved rigorously that the axiom
$$\rho(x,t_0)=|\psi(x,t_0)|^2$$
for an arbitrary initial $t_0$, implies the theorem
$$\rho(x,t)=|\psi(x,t)|^2$$
for all $t$. It is also quite clear that the axiom above does not contradict any of the other axioms. The only issue is whether that axiom is independent on other axioms, or can be derived from the other axioms. There are various arguments that it can be derived, but these arguments are not absolutely rigorous proofs.

 

[rubi]

As I already explained
https://www.physicsforums.com/threa...l-issue-with-pilot-waves.893734/#post-5622848
he makes the same mistake as the other 99% I refer to in post #3.

Well, I find Arnold's objection very reasonable. In a universe with only one hydrogen atom, there is nothing that could perturb the atom, so if Kleinert is right, then there seem to be systems with non-matching predictions between QM and BM.

There is no mathematical contradiction simply because the statement is mathematically so much vague that it is not even wrong. I mean, from that statement it is not clear at all what physical processes do and which physical processes don't count as measurements. By a vague statement it's very easy to avoid mathematical (or any other) contradiction.

But the situation is still much different. The axioms of Copenhagen are mathematically consistent. In order to match a mathematical theory to experiment, one of course always needs to specify how it relates to reality. In the case of Copenhagen, one needs to specify what a measurement is. However, this concerns only the physical side of the theory. For BM, it is already possible for the mathematical side of the theory to be inconsistent, i.e. proving a contradiction.

the axiom
$$\rho(x,t_0)=|\psi(x,t_0)|^2$$

It is also quite clear that the axiom above does not contradict any of the other axioms. The only issue is whether that axiom is independent on other axioms, or can be derived from the other axioms. There are various arguments that it can be derived, but these arguments are not absolutely rigorous proofs.

No, those aren't the only two possibilities. There is also a third possibility: An initial distribution can be derived from the theory (without using the above axiom), but it is not the one specified by the above axiom. In that case, the above axiom would contradict the remaining axioms and the theory would be mathematically inconsistent. If it is in principle possible to derive the above axiom, then it is also in principle possible that a different initial distribution could be derived.

 

[Demystifier]

Well, I find Arnold's objection very reasonable. In a universe with only one hydrogen atom, there is nothing that could perturb the atom, so if Kleinert is right, then there seem to be systems with non-matching predictions between QM and BM.

If by predictions you mean measurable predictions, then your statement is nonsense because QM makes no measurable predictions at all for a universe with only one hydrogen atom. In such a universe there are no measurements, and without measurements there are no measurable predictions.

Or perhaps by predictions you mean also non-measurable predictions? In that case I fully agree that BM and standard QM make different non-measurable predictions, but I don't see that as a problem for any of the two theories.

The axioms of Copenhagen are mathematically consistent.

Except those which are not mathematical at all, such as those that refer to measurement. Non-mathematical statements are mathematically neither consistent nor inconsistent.

In the case of Copenhagen, one needs to specify what a measurement is. However, this concerns only the physical side of the theory. For BM, it is already possible for the mathematical side of the theory to be inconsistent, i.e. proving a contradiction.

By Popper, a potential to prove or disprove a statement is a virtue, not a drawback.

There is also a third possibility: An initial distribution can be derived from the theory (without using the above axiom), but it is not the one specified by the above axiom. In that case, the above axiom would contradict the remaining axioms and the theory would be mathematically inconsistent. If it is in principle possible to derive the above axiom, then it is also in principle possible that a different initial distribution could be derived.

If I understood you correctly, you suggest that we should better use vague theories which cannot be disproved even in principle, because a mathematically precise theory can always be subject to a rigorous analysis, and rigorous analysis always offers a possibility to disprove the theory. Is that what you are suggesting? If it is, then your view is rather anti-Popperian and anti-scientific.

One additional comment. It is known that quantum field theory has mathematical contradictions due to the infinite number of degrees of freedom in the UV and IR limits. On the other hand, classical Newtonian mechanics does not have such contradictions. Should we then reject quantum field theory and use Newtonian mechanics instead?

 

[rubi]

If by predictions you mean measurable predictions, then your statement is nonsense because QM makes no measurable predictions at all for a universe with only one hydrogen atom. In such a universe there are no measurements, and without measurements there are no measurable predictions.

Or perhaps by predictions you mean also non-measurable predictions? In that case I fully agree that BM and standard QM make different non-measurable predictions, but I don't see that as a problem for any of the two theories.

Well, it's already an interesting result if the theories aren't mathematically equivalent, because that's what lots of people believe.

Except those which are not mathematical at all, such as those that refer to measurement. Non-mathematical statements are mathematically neither consistent nor inconsistent.

You can also formulate the axioms without mentioning physics at all. For example, you could say: Let $(P_{t_i})_{t_i}$ be a set of prejectors indexed by real values $t_i$. The evolution law of a state $[\Psi]$ is given by $[\Psi(t)]= [U(t,t_n)P_{t_n}U(t_n,t_{n-1})\cdots\Psi]$. Of course, it would be pointless to do this, but it is in principle possible.

By Popper, a potential to prove or disprove a statement is a virtue, not a drawback.

Popper's principle concerns experimental falsification of otherwise mathematically consistent physical theories. A mathematically inconsistent physical theory is already falsified before any experiment is made. There is no advantage of having a potential mathematical inconsistency in a physical theory.

If I understood you correctly, you suggest that we should better use vague theories which cannot be disproved even in principle, because a mathematically precise theory can always be subject to a rigorous analysis, and rigorous analysis always offers a possibility to disprove the theory. Is that what you are suggesting? If it is, then your view is rather anti-Popperian and anti-scientific.

No, that's not what I am suggesting. I don't like Copenhagen either and I too find it vague, which is why I don't usually advocate it. But you claimed that it contained unproved statements, which isn't true. I can assure you that I am a big fan of Popper. I was talking about the possibility of a mathematical inconsistency in BM, which is swept under the carpet by the BM community. The thread is about potential problems in BM and I think it is not helpful to ignore these problems by directing the attention to other interpretations. That would really be anti-scientific.

 

[Demystifier]

Well, it's already an interesting result if the theories aren't mathematically equivalent, because that's what lots of people believe.

Then let me repeat once again. BM and standard QM are mathematically not equivalent. But it does not mean that they are not observationally equivalent. Moreover, there is a theorem which proves that they are observationally equivalent. In principle it is not impossible that there is a gap in the proof, but then any critique of their observational equivalence should concentrate on finding the gap within that proof itself. Nobody so far have found such a gap within the proof. About 99% of the existing critiques of the equivalence do not refer to that proof at all, which makes such critiques worthless.

I was talking about the possibility of a mathematical inconsistency in BM, which is swept under the carpet by the BM community.

As Godel has shown in his second theorem, not even the standard axioms for arithmetic of integer numbers can prove their own consistency. It is therefore illusory to expect that axioms of any relevant physical theory (including Bohmian mechanics and standard QM) could prove it's own consistency. But even 99% mathematicians are not worried by the fact that they can't prove consistency of their favored axioms. Why should Bohmians be worried then? In practice, for 99% mathematics and for 99.99% physics, it is enough that the axioms look consistent intuitively. And so far, you haven't presented any heuristic argument (let alone proof) that some of the axioms for BM seem inconsistent.

I was talking about the possibility of a mathematical inconsistency in BM, which is swept under the carpet by the BM community.

As I said, nobody so far found any evidence (let alone proof) for such inconsistency. Concerning the proof of consistency, can you prove that standard QM is consistent? If you think you can, then you probably contradict the second Godel theorem.

 

[Demystifier]

For example, you could say: Let $(P_{t_i})_{t_i}$ be a set of prejectors indexed by real values $t_i$. The evolution law of a state $[\Psi]$ is given by $[\Psi(t)]= [U(t,t_n)P_{t_n}U(t_n,t_{n-1})\cdots\Psi]$. Of course, it would be pointless to do this, but it is in principle possible.

It is not pointless at all. Indeed, it is the basis for the consistent histories interpretation, which some think of as Copenhagen interpretation done right. I don't like consistent histories interpretation for other reasons, but the formal part above is, in my opinion, a great progress over the traditional Copenhagen interpretation.

 

[A. Neumaier]

it is not clear at all what physical processes do and which physical processes don't count as measurements.

Is that any different in Bohmian mechanics?

 

[Demystifier]

Is that any different in Bohmian mechanics?

Yes. Those processes that lead to decoherence, i.e. wave-function splitting into macroscopic approximately non-overlapping branches, can be counted as "measurement". There are some ambiguities concerning the meaning of "macroscopic" and "approximately", but that's not a serious problem because the theory of Bohmian measurements is not a part of the axioms for Bohmian theory. When measurements are part of the axioms, like in some versions of Copenhagen, then measurement ambiguities are much more problematic.

 

[A. Neumaier]

the theory of Bohmian measurements is not a part of the axioms for Bohmian theory.

But it must be part of the Bohmian interpretation since otherwise the connection to quantum mechanics is lost. When making this connection you inherit all the ambiguities of the traditional interpretations! In the ensemble interpretation, one usually also subscribes to your statement that

Those processes that lead to decoherence, i.e. wave-function splitting into macroscopic approximately non-overlapping branches, can be counted as "measurement"

since decoherence is a consequence of the ensemble view.

 

[ShayanJ]

decoherence is a consequence of the ensemble view

That doesn't make sense to me. Can you elaborate?

 

[Strilanc]

By measuring the correlator, one usually means measuring $A$ and $B$ separately and then combining the measurement results. However, if
$$[A,B]\neq 0$$
then measurement of $AB$ is not equivalent to measurement of $A$ and $B$ and subsequent multiplication of the measurement results. In this sense, by measuring $A$ and $B$ separately, one cannot measure the correlator (1).

If you were given a "measure A" black box and a "measure B" black box, and only allowed to use one of them on any prepared state, you could estimate the commutator by searching for input vectors that produced noisier outputs in one observable than the other.

Actually, with just the ability to pick inputs and sample the measurement result, you can do full process tomography of a black-box-measurement. That gives (a good approximation of) the matrix representing A, and separately B, at which point you can in principle just directly compute their commutator with a computer.

(Note that the tomography and the computation could take a very long time. It's not tractable to do this to two unknown 50-qubit observables. Still, you might get some quick and dirty estimate of their commutator by sampling output entropies of random inputs.)

 

[A. Neumaier]

That doesn't make sense to me. Can you elaborate?

I don't understand.

The standard derivation of decoherence properties makes use of standard statistical mechanics arguments, which have always been coached in the ensemble language.

C.A. Fuchs and A. Peres,
Quantum theory needs no interpretation,
Physics Today 53 (2000), 70-71.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.77.8442&rep=rep1&type=pdf

J.P. Paz, S. Habib and W.H. Zurek,
Reduction of the wave packet: Preferred observable and decoherence
time scale,
Physical Review D 47 (1993), 488.

W.G. Unruh and W.H. Zurek,
Reduction of a wave packet in quantum Brownian motion,
Physical Review D 40 (1989), 1071.

 

[ShayanJ]

I don't understand.

Those papers are behind paywalls but from their abstracts I can see that they agree with what I know from decoherence. We can formulate decoherence for an individual system that is interacting with an environment. But ensemble view assumes that QM can only be applied to an ensemble of systems. So clearly decoherence doesn't follow from the ensemble view. Actually its part of the formalism of QM and not any particular interpretation.

 

[A. Neumaier]

We can formulate decoherence for an individual system [...] Actually its part of the formalism of QM and not any particular interpretation.

This is inconsistent since the formalism of QM doesn't tell without an interpretation whether or not it applies to an individual system.

Those papers are behind paywalls

The first paper is freely accessible. Note that Peres holds the ensemble view, neatly presented in his book.

We can formulate decoherence for an individual system

But the interpretation of the formulas assumes a notion of probability, which makes no sense for an individual system - unless you give up the objectivity of physics and allow subjective probabilities.

 

[rubi]

Then let me repeat once again. BM and standard QM are mathematically not equivalent. But it does not mean that they are not observationally equivalent. Moreover, there is a theorem which proves that they are observationally equivalent. In principle it is not impossible that there is a gap in the proof, but then any critique of their observational equivalence should concentrate on finding the gap within that proof itself. Nobody so far have found such a gap within the proof. About 99% of the existing critiques of the equivalence do not refer to that proof at all, which makes such critiques worthless.

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

As Godel has shown in his second theorem, not even the standard axioms for arithmetic of integer numbers can prove their own consistency. It is therefore illusory to expect that axioms of any relevant physical theory (including Bohmian mechanics and standard QM) could prove it's own consistency. But even 99% mathematicians are not worried by the fact that they can't prove consistency of their favored axioms. Why should Bohmians be worried then? In practice, for 99% mathematics and for 99.99% physics, it is enough that the axioms look consistent intuitively. And so far, you haven't presented any heuristic argument (let alone proof) that some of the axioms for BM seem inconsistent.

This is something completely different. Consistency proofs in mathematics are of course always relative to some axiom system like ZFC or even some weak subsystem of second order arithmetic (which suffices for separable Hilbert spaces). If a proof of the consistency of BM relative to ZFC cannot be given, it would be a big problem. Copenhagen is of course consistent relative to ZFC. The incompleteness theorem isn't relevant here.

As I said, nobody so far found any evidence (let alone proof) for such inconsistency. Concerning the proof of consistency, can you prove that standard QM is consistent? If you think you can, then you probably contradict the second Godel theorem.

It is of course the task of the BM community to provide a proof for their claims. Mathematical theorems aren't true until they are disproved. It's the other way around. The consistency of Copenhagen (relative to ZFC) can of course be proved easily, since mathematically, it's just the application of the theory of operators in Hilbert spaces, which is of course derived from ZFC. There is no contradiction with Gödel's theorem.

It is not pointless at all. Indeed, it is the basis for the consistent histories interpretation, which some think of as Copenhagen interpretation done right. I don't like consistent histories interpretation for other reasons, but the formal part above is, in my opinion, a great progress over the traditional Copenhagen interpretation.

Yes, I like the consistent histories interpretation much more than Copenhagen and I agree that it's Copenhagen done right. However, mathematical precise language is not the only difference to Copenhagen.

 

[FeynmanFtw]

One additional comment. It is known that quantum field theory has mathematical contradictions due to the infinite number of degrees of freedom in the UV and IR limits. On the other hand, classical Newtonian mechanics does not have such contradictions. Should we then reject quantum field theory and use Newtonian mechanics instead?

I want to convey my own opinion (albeit based on rather limited knowledge of what current research has shown regarding BM or other interpretations), but I'll do that after Christmas :)

However, the above post caught my attention more than most. I assumed that QFT was one of our strongest quantum theories to date, and I cannot find anything about any UV and IR limit problems on the wiki article. What exactly are these contradictions?

 

[rubi]

One additional comment. It is known that quantum field theory has mathematical contradictions due to the infinite number of degrees of freedom in the UV and IR limits. On the other hand, classical Newtonian mechanics does not have such contradictions. Should we then reject quantum field theory and use Newtonian mechanics instead?

I didn't notice this comment earlier. It's not QFT itself, which has problems. Free QFT's are well defined in any dimension and there exist well-defined interacting models in lower dimensions as well. It's not the number of degrees of freedom that causes problems. QFT's have the same number of degrees of freedom as ordinary quantum mechanics, even if the UV and IR regulators are removed (both theories live in separable Hilbert spaces). It's true that known 4-dimensional interacting models are only well-defined if UV and IR regulators are in place, but that's different from being mathematically contradictory. We have a good understanding for why the perturbation expansions still yield physically correct predictions. Of course, the removal of the regulators is still desirable, but that's of course being researched.

 

[A. Neumaier]

I assumed that QFT was one of our strongest quantum theories to date, and I cannot find anything about any UV and IR limit problems on the wiki article. What exactly are these contradictions?

There are no contradiction in the common, renormalized, Poincare invariant setting actually used for predictions. Only the introductory bare treatment is contradictory, but no prediction depends on this purely motivational part. The bare UV divergences are cured by renormalization, while the IR divergences are cured by using appropriate coherent states to describe the scattering.

All this is well understood. There is, however, an unsolved problem - how to phrase QED in a nonperturbative way. The current, perturbative setting only gives asymptotic series for the predictions, though these give excellent approximations when truncated at order up to six.

QFT's have the same number of degrees of freedom as ordinary quantum mechanics

Not in the traditional counting, where each independent harmonic oscillator operator counts as one degree of freedom. With this counting, quantum mechanics is the quantum physics of finitely many d.o.f., and quantum field theory that of infinitely many.

 

[stevendaryl]

Not in the traditional counting, where each independent harmonic oscillator operator counts as one degree of freedom. With this counting, quantum mechnaics is the quantum physics of finitely many d.o.f., and quantum field theory that of infinitely many.

There is a subtle mathematical difference between the quantum mechanics of an unbounded number of degrees of freedom and an infinite number. The Hilbert space used in QFT describes a general state as a pure as a superposition of:

• A state with zero particles
• A state with one particle
• A state with two particles
• etc.

So it describes a situation that involves any finite number of particles. In perturbation theory, the states are described by starting with a vacuum state and applying a finite number of creation operators. But what it doesn't describe is a situation involving an infinite number of particles. Does infinitely many particles make sense? Sure. If the universe is infinite, then it makes sense to have infinitely many particles; there might be finitely many in any finite volume of space, but the volume of space is infinite. As I understand it, trying to describe the quantum mechanics of a truly infinite number of particles leads to a non-separable Hilbert space, and the mathematics is very different.

In practice, it doesn't make any difference, because we can artificially assume that there are finitely many particles at any given time, even if the universe is infinite.

 

[A. Neumaier]

describes a situation that involves any finite number of particles.

But arbitrarily many, which makes the number of degrees of freedom infinite. An $N$-particle system has $3N$ degrees of freedom. Already a free quantum field theory has an indeterminate number of particles. The number of degrees of freedom therefore exceeds $3N$ for any $N$. Thus there are infinitely many degrees of freedom.

Note that we cannot artificially limit the number of particles to be bounded by some large $N$. This destroys the locality of quantum field theory, which is responsible for the cluster decomposition property, without which it would be impossible to do physics. Also, coherent photon states (such as they are produced routinely by lasers) contain an arbitrarily large number of photons with a nonzero probability. These photons are needed on the formal level to give the coherent states the nice quasi-classical properties that are universally exploited when using them.

The main reason for the mathematical difficulties in quantum field theory is precisely that! Many of the techniques from functional analysis valid for quantum mechanics of finitely many degrees of freedom are no longer applicable to quantum field theory since the number of degrees of freedom is infinite.

Does infinitely many particles make sense?

Nothing I said has anything to do with whether the possibility of an infinite number of particles is or is not realized. It isn't, in standard quantum field theory. It might be in quantum gravity, but unlike QED and the standard model, this is largely unexplored territory.

 

[stevendaryl]

But arbitrarily many, which makes the number of degrees of freedom infinite. An $N$-particle system has $3N$ degrees of freedom. Already a free quantum field theory has an indeterminate number of particles. The number of degrees of freedom therefore exceeds $3N$ for any $N$. Thus there are infinitely many degrees of freedom.

Well, I'm uncertain about the terminology, but the distinction that I'm making is between:

1. An unbounded number of particles, and
2. an infinite number of particles.

The distinction is that in the case of an unbounded number of particles, if |\psi\rangle is a state with an unbounded number of particles, then there is at least one nonnegative number n and a state |\psi_n\rangle with exactly n particles such that \langle \psi|\psi_n \rangle is nonzero.

With a truly infinite number of particles, there would be no overlap with any state with just finitely many particles.

 

[A. Neumaier]

Well, I'm uncertain about the terminology, but the distinction that I'm making is between:

1. An unbounded number of particles, and
2. an infinite number of particles.

This distinction is valid, but doesn't affect the fact that - in contrast to the claim of Rubi above - QFT has infinitely many degrees of freedom while QM has only finitely many. This differences is extremely important.

 

[rubi]

This distinction is valid, but doesn't affect the fact that - in contrast to the claim of Rubi above - QFT has infinitely many degrees of freedom while QM has only finitely many. This differences is extremely important.

Well, that's not what I said. I explained in my post, what I meant by d.o.f., i.e. a state in QFT contains as much information as a state in QM. Of course, in the case of QFT, you need to embed a field algebra into the algebra of operators on the separable Hilbert space, instead of a particle algebra. However, this is not what is problematic. You can easily write down a Fock space and field operators on it and you can also write down well-defined interacting Hamiltonians in that Hilbert space. However, you won't be able to write down an interacting Hamiltonian that satisfies the Poincare algebra relations, i.e. produces a Lorentz-invariant theory. Writing down interacting theories that satisfy the Wightman axioms is of course very difficult.

 

[A. Neumaier]

a state in QFT contains as much information as a state in QM.

But this is wrong. In QM, one cannot represent the information that a system is in a superposition of arbitrarily many particles!

 

[rubi]

But this is wrong. In QM, one cannot represent the information that a system is in a superposition of arbitrarily many particles!

That is not relevant to measuring how much information can be encoded into a state. Of course, different theories encode different information in the state, but the amount of information is exactly the same. I can store either Pulp Fiction or Full Metal Jacket on my hard drive, but if I choose to store Full Metal Jacket, the information about what a quarter pounder with cheese is called in France won't be represented.

 

[A. Neumaier]

That is not relevant to measuring how much information can be encoded into a state. Of course, different theories encode different information in the state, but the amount of information is exactly the same. I can store either Pulp Fiction or Full Metal Jacket on my hard drive, but if I choose to store Full Metal Jacket, the information about what a quarter pounder with cheese is called in France won't be represented.

But measuring information in this way is completely irrelevant. Every real number contains in your sense more information than all the physics books in the world. A classical field theory contains in your sense as much information as a quantum field theory.

In any case, it is highly misleading to equate this strange measure of information with degrees of freedom, as you did in your post #47 (without being precise enough there).

 

[rubi]

But measuring information in this way is completely irrelevant. Every real number contains in your sense more information than all the physics books in the world. A classical field theory contains in your sense as much information as a quantum field theory.

In any case, it is highly misleading to equate this strange measure of information with degrees of freedom, as you did in your post #47 (without being precise enough there).

I stated precisely what I meant by it in my post #47, so all of this is just a semantic discussion. Of course, the dimension of the Hilbert space is relevant to what quantum theories can be formulated on it. You cannot formulate QFT in $\mathbb C^2$, but you can (in principle) formulate it in the usual $L^2(\mathbb R^3)$ space of ordinary quantum mechanics. As I explained, the fact that you are working with a field algebra instead of a particle algebra is not a priori problematic.

 

[A. Neumaier]

You cannot formulate QFT in C2C2\mathbb C^2, but you can (in principle) formulate it in the usual $L^2(\mathbb R^3)$ space of ordinary quantum mechanics.

No, you cannot in any sensible way since there is no natural isomorphism between a Fock space for QFT and $L^2(\mathbb R^3)$. The structure imposed on the Hilbert space makes all the difference for its information content! Quantum mechanics does not happen in arbitrary Hilbert spaces but in Hilbert spaces with distinguished group actions!

 

[rubi]

At this point, it seems like you just don't want to admit that you didn't read my post carefully. As a mathematician, you certainly know that I can pull back any structure between isomorphic objects, so of course I can pull back the group action from a Fock space onto $L^2$. I don't need a natural isomorphism for that. (Moreover, it's exactly the other way around. The irreducible representations of the Poincare group that you obtain from Wigner's analysis are naturally defined on $L^2$ spaces. Fock spaces are built from these $L^2$ spaces and only inherit this group action.)

 

[vanhees71]

Well, I'm uncertain about the terminology, but the distinction that I'm making is between:

1. An unbounded number of particles, and
2. an infinite number of particles.

The distinction is that in the case of an unbounded number of particles, if |\psi\rangle is a state with an unbounded number of particles, then there is at least one nonnegative number n and a state |\psi_n\rangle with exactly n particles such that \langle \psi|\psi_n \rangle is nonzero.

With a truly infinite number of particles, there would be no overlap with any state with just finitely many particles.

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. However, you are right in saying that there are states which have a non-zero overlap with any Fock state of a definite finite particle number. An example are the coherent states of the electromagnetic field, which are as close to classical em. waves as you can get. The photon-number probabilities are given by the Poisson statistics.