The Refutation of Bohmian Mechanics

[Demystifier]

One measures a single mean position of the pointer, not the positions of any of the pointer particles.

No, that's not exactly the idea of BM. One observes all these pointer particles collectively, which due to a low resolution appears as a single mean position. But if there was only one pointer particle, due to the low resolution one could not observe it at all.

But this brings me back to my question about the nerves: Where in the nerves is the measured particle whose position indicates whether or not I see a star, and which color it has?

I believe the remark above answers it as well.

 

[A. Neumaier]

No, that's not exactly the idea of BM. One observes all these pointer particles collectively, which due to a low resolution appears as a single mean position. But if there was only one pointer particle, due to the low resolution one could not observe it at all.


I believe the remark above answers it as well.

Not yet. How is the difference between seeing and not seeing the star encoded in the mean position of the nerve particles? Since according to BM the latter is the only thing observable, and we can tell the difference empirically, this difference must be somehow encoded.

 

[Demystifier]

But in the standard QM picture, the wave function of a spin system (e.g. the Ising ferromagnet) has no position or momentum variables, and hence also no velocities associated with it. Given my specific description of the spin system, what is the BM dynamics? Please be as specific in your formal description, rather than using vague words that leave many things unsaid.

Ah, now I see your point. Well, a spin system without position or momentum variables is a toy model that doesn't describe anything in the real world. You are right, for such a system there is no BM dynamics. But BM does not claim to be applicable to every conceivable quantum theory. Instead, it claims to be applicable to the real world.

 

[Demystifier]

OK. But measuring the particle at rest always gives the same measurement result. Thus the positions of the nonmoving nerves in the eye can hardly be used to tell the difference between seeing and not seeing a star.

Not yet. How is the difference between seeing and not seeing the star encoded in the mean position of the nerve particles? Since according to BM the latter is the only thing observable, and we can tell the difference empirically, this difference must be somehow encoded.

What matters is which nerve (with a well-defined position) is excited. But what it means to be excited? It means that there is an electric current in it. Now you will say that the current is nothing but some microscopic ions moving. Sure, but you don't observe one ion. You observe a bunch of them, which makes the current a macroscopic phenomenon. In fact, it seems that all neuro-physics can be well approximated by classical physics:
http://xxx.lanl.gov/abs/quant-ph/9907009 [Phys.Rev.E61:4194-4206,2000]

 

[A. Neumaier]

Ah, now I see your point. Well, a spin system without position or momentum variables is a toy model that doesn't describe anything in the real world. You are right, for such a system there is no BM dynamics. But BM does not claim to be applicable to every conceivable quantum theory. Instead, it claims to be applicable to the real world.

Many real world problems are in fact posed in terms of a Hilbert space which doesn't contain a representation of the Euclidean group and hence has no position and momentum operators.
In particular, all quantum computing is done in such Hilbert spaces. You won't find a position or momentum operator figuring in quantum computation papers.

And precisely this was my point about BM and quantum computing.

BM only adds unnecessary baggage to the standard quantum machinery and has a few tricks to pretend that this gives more reality to QM than the standard view.

Not only that: BM also subtracts a lot from QM, denigrating important features of QM - which to me represent major insights into the nature of physics - to mere calculational tools:

BM sacrifices all important structure in QM - no covariance under canonical transformations, no Heisenberg picture, no natural procedure for forming subsystems, no natural QFT, too many possible variants for the dynamics without any natural means of distinguishing between them. For any serious computation it must resort to the standard QM formulation. The only exceptions are some dynamical simulations, which produce nice pictures and (sometimes) also practically useful results, But the latter can be obtained more efficiently through traditional means, in all cases I know of.

Thus nothing inviting is left.

 

[Demystifier]

Many real world problems are in fact posed in terms of a Hilbert space which doesn't contain a representation of the Euclidean group and hence has no position and momentum operators.
In particular, all quantum computing is done in such Hilbert spaces. You won't find a position or momentum operator figuring in quantum computation papers.

That's all true in a theory, because the particle positions are not essential to understand some aspects of physics, including quantum computations. Yet, an actual quantum computER operating in such a Hilbert space will NEVER be constructed in a laboratory. A true quantum computer can only be made of particles, such as photons, atoms, etc.

 

[Demystifier]

BM sacrifices all important structure in QM - no covariance under canonical transformations,

BM sacrifices the covariance under canonical transformations no more than classical mechanics does. After all, what we really observe in classical mechanics are particle positions, not some bizarre canonical combinations of position and momentum variables.

no Heisenberg picture,

BM can be formulated in the Heisenberg picture as well, but looks more complicated in that picture.

no natural procedure for forming subsystems,

I have no idea why do you think so?

no natural QFT,

I find
http://xxx.lanl.gov/abs/0904.2287 [Int. J. Mod. Phys. A25:1477-1505, 2010]
quite natural.

too many possible variants for the dynamics without any natural means of distinguishing between them.

Even though there are many possible variants, the standard (de Broglie-Bohm) variant is very natural and can be derived in many ways. For a recent derivation based on weak MEASUREMENTS see
http://xxx.lanl.gov/abs/0706.2522
http://xxx.lanl.gov/abs/0808.3324

 

[A. Neumaier]

That's all true in a theory, because the particle positions are not essential to understand some aspects of physics, including quantum computations. Yet, an actual quantum computER operating in such a Hilbert space will NEVER be constructed in a laboratory. A true quantum computer can only be made of particles, such as photons, atoms, etc.

The strength of standard QM is that
-- it can safely ignore all irrelevant variables,
-- it can transform to arbitrary symplectic coordinate systems in phase space,
-- it can work on arbitrary Lie groups adapted to the problem,
without leaving the framework of the theory.

BM has no such option, hence is strictly inferior to the standard view.

Thus it is fully justified that the main stream ignores BM.

The presentation ''Not even wrong. Why does nobody like pilot-wave theory?'' at http://www.tcm.phy.cam.ac.uk/~mdt26/PWT/lectures/bohm7.pdf diagnoses the disease but only has a historical view rather than an answer to that question. The real answer is that the need for BM is marginal compared to the need for QM. BM subtracts from QM too much without giving anything relevant in return.

Though through lip service it encompasses all of QM, in practice it excludes many systems of practical interest because they are not formulated with enough pointer degrees of freedom (and often cannot be
formulated with few enough pointer degrees of freedom to be tractable by BM means). Simulating quantum computing via BM would be a nightmare.

 

[A. Neumaier]

BM sacrifices the covariance under canonical transformations no more than classical mechanics does. After all, what we really observe in classical mechanics are particle positions, not some bizarre canonical combinations of position and momentum variables.

In classical mechanics, a canonical transformation transforms a system in canonical variables into another one in canonical variables. In many systems the observables are not canonical. E.g., distances and angles in molecules - one _cannot_ observe positions, only distances and angles.
(In any Galilei or Poincare invariant theory, positions are unobservable gauge-dependent quantities.)

BM can be formulated in the Heisenberg picture as well, but looks more complicated in that picture.

It looks useless in that formulation.

I have no idea why do you think so?

I wanted to form a subsystem consisting of N spins only, and since the position variables were gone, the BM description was gone.

I find
http://xxx.lanl.gov/abs/0904.2287 [Int. J. Mod. Phys. A25:1477-1505, 2010]
quite natural.

An author usually finds his own work natural. I find it unnatural that this view doesn't reduce to the standard Bohmian view when you translate in the usual way the field theory back into a multiparticle theory. (There is work by Horwitz and Piron on 4D quantum mechanics along similar lines as yours, it never found much resonance, for very good reasons.)

Even though there are many possible variants, the standard (de Broglie-Bohm) variant is very natural and can be derived in many ways. For a recent derivation based on weak MEASUREMENTS see
http://xxx.lanl.gov/abs/0706.2522
http://xxx.lanl.gov/abs/0808.3324

The quantum field variant and the multiparticle variant, which are equivalent in standard QM, have completely different ontologies in the BM setting. What is natural about that?

 

[Demystifier]

The strength of standard QM is that
-- it can safely ignore all irrelevant variables,
-- it can transform to arbitrary symplectic coordinate systems in phase space,
-- it can work on arbitrary Lie groups adapted to the problem,
without leaving the framework of the theory.

BM has no such option, hence is strictly inferior to the standard view.

Thus it is fully justified that the main stream ignores BM.

The presentation ''Not even wrong. Why does nobody like pilot-wave theory?'' at http://www.tcm.phy.cam.ac.uk/~mdt26/PWT/lectures/bohm7.pdf diagnoses the disease but only has a historical view rather than an answer to that question. The real answer is that the need for BM is marginal compared to the need for QM. BM subtracts from QM too much without giving anything relevant in return.

Though through lip service it encompasses all of QM, in practice it excludes many systems of practical interest because they are not formulated with enough pointer degrees of freedom (and often cannot be
formulated with few enough pointer degrees of freedom to be tractable by BM means). Simulating quantum computing via BM would be a nightmare.

All this points to the conclusion that standard QM is more convenient for PRACTICAL applications, with which I agree. But BM is not developed for practical applications (even though sometimes it has practical applications as well). It is developed with an intention to resolve some FOUNDATIONAL issues. As most physicists are more interested in practical issues than in foundational ones, which is fine and even desirable, it is no surprise that most physicists do not care much about BM and other interpretations of QM. But it does not mean that BM (or some other interpretation) is not right, and that it will not became more useful one day when it becomes better developed.

 

[A. Neumaier]

All this points to the conclusion that standard QM is more convenient for PRACTICAL applications, with which I agree. But BM is not developed for practical applications (even though sometimes it has practical applications as well). It is developed with an intention to resolve some FOUNDATIONAL issues. As most physicists are more interested in practical issues than in foundational ones, which is fine and even desirable, it is no surprise that most physicists do not care much about BM and other interpretations of QM. But it does not mean that BM (or some other interpretation) is not right, and that it will not became more useful one day when it becomes better developed.

In practice, practice decides the interpretation.

 

[Demystifier]

I find it unnatural that this view doesn't reduce to the standard Bohmian view when you translate in the usual way the field theory back into a multiparticle theory. (There is work by Horwitz and Piron on 4D quantum mechanics along similar lines as yours, it never found much resonance, for very good reasons.)

Why do you think that it doesn't reduce to the standard Bohmian view when you translate in the usual way the field theory back into a multiparticle theory? It does. Besides, even though it is partially inspired by the work of Horwitz and Piron, it is different from that, precisely in a manner that avoids the problems of their approach.

The quantum field variant and the multiparticle variant, which are equivalent in standard QM, have completely different ontologies in the BM setting. What is natural about that?

Why do you think that the ontologies are different? Both ontologies are in terms of particle positions. Moreover, for the same states the same particle trajectories appear, except for the fact that QFT contains some additional "dead" particles that exist for an infinitesimally short time.

 

[Demystifier]

In practice, practice decides the interpretation.

In practice, a very small number of people cares about BM, a larger but still relatively small number cares about standard QM, and a much much bigger number of people cares about certain non-scientific religious books. Can we conclude anything relevant from that?

 

[A. Neumaier]

Why do you think that it doesn't reduce to the standard Bohmian view when you translate in the usual way the field theory back into a multiparticle theory? It does. Besides, even though it is partially inspired by the work of Horwitz and Piron, it is different from that, precisely in a manner that avoids the problems of their approach.

Why do you think that the ontologies are different? Both ontologies are in terms of particle positions. Moreover, for the same states the same particle trajectories appear, except for the fact that QFT contains some additional "dead" particles that exist for an infinitesimally short time.

Because the interpretation of the probabilisitic meaning of psi(x,t) is completely different in the two forms.
In the Schroedinger picture and in standard BM, the density of x at fixed t is given by |psi(x,t_0)|^2, while in Horwitz/Piron and in your relativistic BM, it is given by |psi(x,t)|^2delta(t-t_0). You cannot assert both simultaneously.

 

[A. Neumaier]

In practice, a very small number of people cares about BM, a larger but still relatively small number cares about standard QM, and a much much bigger number of people cares about certain non-scientific religious books. Can we conclude anything relevant from that?

Yes: Religion is for everyone, quantum mechanics for the general scientist, and BM for the determinsitic scientist only. I am trying to address the first two groups only, though I know about the practices of the third one.

 

[Demystifier]

Because the interpretation of the probabilisitic meaning of psi(x,t) is completely different in the two forms.
In the Schroedinger picture and in standard BM, the density of x at fixed t is given by |psi(x,t_0)|^2, while in Horwitz/Piron and in your relativistic BM, it is given by |psi(x,t)|^2delta(t-t_0). You cannot assert both simultaneously.

First, this is a difference in the probabilistic interpretation, not in the ontology. Second, in
http://xxx.lanl.gov/abs/0811.1905
I explain how both probabilistic interpretations may be right (but not simultaneously). One (Horwitz/Piron) is a fundamental a priori probability, while the other is a conditional probability. Which one is to be applied is context dependent.

 

[A. Neumaier]

First, this is a difference in the probabilistic interpretation, not in the ontology. Second, in
http://xxx.lanl.gov/abs/0811.1905
I explain how both probabilistic interpretations may be right (but not simultaneously). One (Horwitz/Piron) is a fundamental a priori probability, while the other is a conditional probability. Which one is to be applied is context dependent.

But psi(x,t) can have only _one_ meaning consistent with the Schroedinger equation, which is _not_ context dependent. And it must be the one consistent with standard QM.

Swapping meanings as convenient for a particular argument is another of the trickeries of BM.

A fundamental theory must have a unique interpretation.

 

[Demystifier]

But psi(x,t) can have only _one_ meaning consistent with the Schroedinger equation, which is _not_ context dependent. And it must be the one consistent with standard QM.

Swapping meanings as convenient for a particular argument is another of the trickeries of BM.

A fundamental theory must have a unique interpretation.

But there is only one FUNDAMENTAL probabilistic interpretation in my approach - the Horwitz/Piron one. The other interpretation is DERIVED from the fundamental one - by using the standard theory of probability, which includes the concept of conditional probability. You should know that, irrespective of physics, probability is strongly context dependent, depending on what one already knows about the system. Changing knowledge changes the probability, even if physics is the same.

Besides, even though such a fundamental Horwitz/Piron probability is not identical with the standard probabilistic interpretation, I show that the former is compatible with the latter. The former is a generalization of the latter, not merely a replacement of it.

 

[akhmeteli]

Dear A. Neumaier,

thank you for your reply.

If you think giving a reference to an unpublished arXiv paper without discussing it is a serious sin against the rules, you should report it to the PF management, quoting the present post for context.

Why would I do that? I clearly don't want you to be banned. Even if sometimes I use harsh words, I am not your enemy. I just respectfully asked you to voluntarily follow the rules, because otherwise you create very awkward situations: while what you say is just your personal theory, those members of the forum who are not very familiar with the issue tend to rely on your word, as you have their well-deserved respect, and they think that what you said is a well-established fact. As a result, they are misled at least with respect to the status of your statement. On the other hand, those of us who for some reason happen to know more about the specific problem, sometimes just don't want to silently swallow your statement and are forced to confront you and discuss your personal theory. I think what you do is not quite right, but I am not sure I will be able to explain that to you for a reason outlined at the end of this post.

I fully respect the rules as I understand them.

With all due respect, not that I don't believe you, but I don't, for a reason outlined at the end of this post.

But I cannot discuss my claim further because of the PF rules. So your objection standas like my assertion, and readers must make up their own mind.

Yes, we disagree, and no, I cannot be sure I am right, but my main point is your statement just does not belong here, no matter how correct or wrong your statement is.

I asked about the status of your claim "no quantum computing in the Bohm interpretation."

First, I qualified my statement with ''probably'' since I wasn't sure,

You said: “For example, you cannot do quantum computing in Bohmian mechanics” in post 18 in this thread. I looked for word “probably” in that post. That was a long search… You did use the word in your post 24, but there it related to a somewhat different statement: “Bohmians are not aware of many things; they probably never tried to bring quantum computing into their focus.”; furthermore, the damage was already done earlier, when you told us about quantum computing and the Bohm interpretation without qualifying or “caveating” your statement in any way. The same problem arises: it is not easy to tell a personal theory from the ultimate truth.

and indeed, there was a very recent (2010) thesis that tackled it, as was pointed out by others. I immediately acknowledged the article, studied it, and found that it didn't treat spin systems by themselves but only spin systems coupled to an external pointer variable, thus justifying my remark ''The observables used there do not include a position variable, hence the Bohmian trickery is inapplicable.''. However, I learned that the author invented (or got from somewhere else) a new Bohmian trick - namely that one silently changes the system under study to a bigger one, in order to give it the appearance of fitting into the BM philosophy. This lead to a still ongoing discussion.

I truly respect you for taking your opponent’s argument seriously. But I had no intention to criticize you for not having read something “latest and greatest”. My problem was that, even when asked directly about the status of your statement about quantum computing in the Bohm interpretation, you chose to avoid a direct answer. You could say: “This was proven in such and such article”, or “Well, this is my personal opinion/theory”. You did not. This is unfortunate.

If everyone were banned who made more than 10 claims that do not appear in a peer-reviewed article, PF would be nearly empty.

This phrase of yours makes me think that the chance to convince you is very slim and makes it difficult to believe that you fully respect the rules as you understand them. I tend to make the following two conclusions based on this phrase:
1. You read and understand the rules exactly as they are written, and
2. Then you do exactly what you want.

 

[akhmeteli]

For example, you cannot do quantum computing in Bohmian mechanics.

What is the status of this statement?

If you don't agree, then please tell me how to do quantum computing in Bohmian mechanics..

I am under no obligation to prove that your claim is wrong. Furthermore, I have no idea if it is indeed wrong or right.

It is wrong. I supplied him with an appropriate reference demonstrating how to do deBB quantum computing in #22

Dear camboy,

Thank you for your response.

With all due respect, I am not sure the reference you supplied is indeed appropriate. I admit that I don't know much about quantum computing and don't have time to study your lengthy reference, so I can only judge it by formal criteria. It may well be that this is a paper of the century (at least it seems A. Neumaier wrote about it with some respect), but, as far as I am concerned, this is just an unpublished Master's thesis, so, under the forum's rules, it is not appropriate for discussion here. I am aware that the author's supervisor is well-known for his publications on the Bohm interpretation, but let us wait until Mr. Roser and Dr. Valentini publish this work properly.

As for my personal opinion on A. Neumaier's claim, I don't want to start a flame here, so maybe I'll PM you in a couple of days.

 

[A. Neumaier]

But there is only one FUNDAMENTAL probabilistic interpretation in my approach - the Horwitz/Piron one. The other interpretation is DERIVED from the fundamental one - by using the standard theory of probability, which includes the concept of conditional probability. You should know that, irrespective of physics, probability is strongly context dependent, depending on what one already knows about the system. Changing knowledge changes the probability, even if physics is the same.

It is a myth believed (only) by the Bayesian school that probability is dependent on knowledge.

You cannot change the objective probabiltiies of a mechanism by forgetting about the knowledge you have.

Lack of knowledge results in lack of predictivity, not in different probabilities.

Besides, even though such a fundamental Horwitz/Piron probability is not identical with the standard probabilistic interpretation, I show that the former is compatible with the latter. The former is a generalization of the latter, not merely a replacement of it.

Then please tell me how the probability theory of the ground state of the 1-dimensional quantum harmonic oscillator with H= p^2/2 + q^2/2 - 1/2, where hbar=1 and p,q acting on psi(x,t) (x in R) in the standard way, which in standard QM is modeled by psi(x,t)=e^{-x^2/2} independent of t, is generalized to your fundamental view.

And how the standard view is obtained by taking conditional probabilites.

 

[Demystifier]

It is a myth believed (only) by the Bayesian school that probability is dependent on knowledge.

You cannot change the objective probabiltiies of a mechanism by forgetting about the knowledge you have.

Lack of knowledge results in lack of predictivity, not in different probabilities.

I strongly disagree, but elaboration would be an off topic.

Then please tell me how the probability theory of the ground state of the 1-dimensional quantum harmonic oscillator with H= p^2/2 + q^2/2 - 1/2, where hbar=1 and p,q acting on psi(x,t) (x in R) in the standard way, which in standard QM is modeled by psi(x,t)=e^{-x^2/2} independent of t, is generalized to your fundamental view.

And how the standard view is obtained by taking conditional probabilites.

If you disagree that probability may depend on knowledge, then there is no point in explaining it (which, by the way, I have already explained in a paper I mentioned several times on this thread).

 

[A. Neumaier]

I strongly disagree, but elaboration would be an off topic.

It is not off-topic here:
https://www.physicsforums.com/showthread.php?p=3278689#post3278689

If you disagree that probability may depend on knowledge, then there is no point in explaining it.

This is strange, since the concept of conditional probability exists also in the frequentist school of objective probability and also in the interpretation-less Kolmogorov probability theory.

(which, by the way, I have already explained in a paper I mentioned several times on this thread).

Did you really discuss there, as requested, the ground state of the harmonic oscillator?

 

[A. Neumaier]

I just respectfully asked you to voluntarily follow the rules,

I do follow the rules, of which I quote here the relevant part:

Physicsforums.com strives to maintain high standards of academic integrity. There are many open questions in physics, and we welcome discussion on those subjects provided the discussion remains intellectually sound. It is against our Posting Guidelines to discuss, in most of the PF forums or in blogs, new or non-mainstream theories or ideas that have not been published in professional peer-reviewed journals or are not part of current professional mainstream scientific discussion. Personal theories/Independent Research may be submitted to our Independent Research Forum, provided they meet our Independent Research Guidelines; Personal theories posted elsewhere will be deleted.

_Everything_ I say is my personal opinion (though it often agrees with established scientific fact), and when appropriate I give references to what I believe is a valid source. It is neither against the rules to voice a personal opinion (most contributors do that regularly) nor to refer to unpublished articles if they are ''part of current professional mainstream scientific discussion'' (Streater's book shows that my remarks on wrong signs in time correlations in BM is part of that).

You said: “For example, you cannot do quantum computing in Bohmian mechanics” in post 18 in this thread. I looked for word “probably” in that post. That was a long search… You did use the word in your post 24, but there it related to a somewhat different statement: “Bohmians are not aware of many things; they probably never tried to bring quantum computing into their focus.”; furthermore, the damage was already done earlier, when you told us about quantum computing and the Bohm interpretation without qualifying or “caveating” your statement in any way. The same problem arises: it is not easy to tell a personal theory from the ultimate truth.

The remainder of the discussion has shown in which sense my statement was a fact.

when asked directly about the status of your statement about quantum computing in the Bohm interpretation, you chose to avoid a direct answer. You could say: “This was proven in such and such article”, or “Well, this is my personal opinion/theory”. You did not. This is unfortunate.

That a fact has no convenient reference doesn't make it a personal theory in the sense that it would belong only to the IR section of PF. It just takes more space to provide the evidence, and the discussion with Demystifier has provided it.

This phrase of yours makes me think that the chance to convince you is very slim and makes it difficult to believe that you fully respect the rules as you understand them.

Our understanding of the rules is different, and your arguments did not convince me that your interpretation is better than mine. Only superior arguments than my own are suitable to convince me of something different from what I am already convinced of.

 

[A. Neumaier]

which, by the way, I have already explained in a paper I mentioned several times on this thread).

I assume you meant your paper http://xxx.lanl.gov/abs/0811.1905 . It explains the connection to conditional probability in (6) to (9). But this doesn't apply to the ground state of the harmonic oscillator since there psi(x,t)=e^{-x^2/2} (independent of t) considered as a function of (x,t) in R^4 is not normalizable.

Thus the allegedly more fundamental 4D description has serious normalization problems already in the simple example of the harmonic oscillator.

 

[Demystifier]

I assume you meant your paper http://xxx.lanl.gov/abs/0811.1905 .

Yes.

It explains the connection to conditional probability in (6) to (9). But this doesn't apply to the ground state of the harmonic oscillator since there psi(x,t)=e^{-x^2/2} (independent of t) considered as a function of (x,t) in R^4 is not normalizable.

Thus the allegedly more fundamental 4D description has serious normalization problems already in the simple example of the harmonic oscillator.

See page 5, the paragraph that begins with "Before discussing ...".

 

[A. Neumaier]

See page 5, the paragraph that begins with "Before discussing ...".

I find it strange that you refer to the divergence of the integral over time as ''they cannot be localized in time'', since what you are trying to do is globalizing the state rather than localizing it.

But let me follow your recipe by taking finite time integration, and taking the limit at the end of the calculation. Assuming the already normalized 3D eigenstate psi_0, I normalize the state psi(x,t)=psi_0(x) over the interval [0,T]. This gives me the normalized state phi=psi/sqrt{T}. Now the probability of finding the particle anywhere in a time interval of length Delta is
\int dx \int_0^\Delta dt |\phi(x,t)|^2 =\Delta/T.
Taking T to infinity tells me that there is a zero probability for finding the particle in any given time interval of length Delta.

What did i do wrong to get this very strange result?

 

[Demystifier]

But let me follow your recipe by taking finite time integration, and taking the limit at the end of the calculation. Assuming the already normalized 3D eigenstate psi_0, I normalize the state psi(x,t)=psi_0(x) over the interval [0,T]. This gives me the normalized state phi=psi/sqrt{T}. Now the probability of finding the particle anywhere in a time interval of length Delta is
\int dx \int_0^\Delta dt |\phi(x,t)|^2 =\Delta/T.
Taking T to infinity tells me that there is a zero probability for finding the particle in any given time interval of length Delta.

What did i do wrong to get this very strange result?

You applied the limit too early, not at what I meant by the "end of calculation". A valid example of an appropriate use of the limit is discussed briefly in the last paragraph of Sec. 2.

Besides, the result is not strange at all. Indeed, it is easy to construct a classical analog of that result, provided that you accept (which you don't) Bayesian view of probability. For example, if the universe will last forever, what is the probability that you live now?

 

[A. Neumaier]

You applied the limit too early, not at what I meant by the "end of calculation".

According to (6), it is a legitimate goal in your calculus to ask for the probability of finding a particle somewhere in spacetime is a legitimate goal. But now you seem to say that the interpretation in (6) is bogus and that your calculus doesn't give _any_ information about the probability of finding a particle somewhere in spacetime.

 

[Demystifier]

According to (6), it is a legitimate goal in your calculus to ask for the probability of finding a particle somewhere in spacetime is a legitimate goal. But now you seem to say that the interpretation in (6) is bogus and that your calculus doesn't give _any_ information about the probability of finding a particle somewhere in spacetime.

It does give some information about the probability of finding a particle somewhere in spacetime, provided that you restrict the probability to a finite region of spacetime.