Photons, particles and wavepackets

[jostpuur]

Can anyone confirm that I got this right?

So when source A and source B both create one photon, the Bose statistics forces these photons to get in superposition, so that each of the photons immediately has amplitude for starting at both sources?

If this is the explanation for the interference of independently emitted photons, that certainly is a convincing proof for the Bose statistics.

It makes sense to me now, and I think I'll believe it unless somebody explains what could be wrong with it. But it would be nicer to be sure. It is difficult to feel confident, when the popular explanations of QM don't bother with this paradox at all

 

[rewebster]

Can anyone confirm that I got this right?



It makes sense to me now, and I think I'll believe it unless somebody explains what could be wrong with it. But it would be nicer to be sure. It is difficult to feel confident, when the popular explanations of QM don't bother with this paradox at all

If I'm reading all this right---what you're REALLY looking for is, "What IS the strongest evidence so far?"

is that about 'right'?

 

[jostpuur]

I somehow missed OOO's #30 post in this thread. Now I think I agree with OOO now with this thing about photons starting in superposition of being at the two sources.

If I'm reading all this right---what you're REALLY looking for is, "What IS the strongest evidence so far?"

is that about 'right'?

hmhm... I'm not sure. It was not my intention to seek evidence for Bose statistics, but I just made the remark, that this interference phenomena turned out to be evidence for it, and it surprised me.

 

[rewebster]

I somehow missed OOO's #30 post in this thread. Now I think I agree with OOO now with this thing about photons starting in superposition of being at the two sources.




hmhm... I'm not sure. It was not my intention to seek evidence for Bose statistics, but I just made the remark, that this interference phenomena turned out to be evidence for it, and it surprised me.

I didn't mean 'Bose statistics'----I was leaning toward the idea of your 'topic'

 

[Anonym]

none of what you quoted has anything to do with a photon size.

f95toli, I would appreciate it greatly obtaining your comment on our debate.

like the ones in the history books.

Do you know the story I mentioned in my post #106 in the “cat in a box paradox” session?

I am not sure about his name (I think P.Wood; my first book on ED at high school was written by him). He was in the middle of the measurements. The performance severely degraded due to dust on the internal surfaces and the tubes were about 11-14 m long. Project! He took the cat, put him inside, thus to show the completely deterministic way out. After 10 min he continues the measurements.

Regards, Dany.

 

[OOO]

I somehow missed OOO's #30 post in this thread. Now I think I agree with OOO now with this thing about photons starting in superposition of being at the two sources.




hmhm... I'm not sure. It was not my intention to seek evidence for Bose statistics, but I just made the remark, that this interference phenomena turned out to be evidence for it, and it surprised me.

Bose statistics is one thing we should always be aware of. But you could also do the experiment with single photons (on average), but of course a single photon wave function is also one aspect of bose statistics.

So let me put it that way: suppose you were sure somehow that a single photon you detect on the screen comes from exactly one of your sources then it is impossible for it to interfere in the sense of the double slit since it must have gone through one slit only. Since in classical terms non-interference means that both sources are not synchronized enough, we could say that knowing for sure were a single photon was emitted from means desynchronization of the sources.

The other way round this means that we have only synchronized the two sources sufficiently if and only if we can't be sure were a photon detected on the screen came from.

You probably might say that a photon always comes from one of the sources, even if they are synchronized but this is truly not the case in a quantum mechanical system. Actually you have to add some measurement device to the sources in order to detect where a photon is emitted from, and this measurement device destroys your coherence.

 

[jostpuur]

My final word on this is, that merely saying "you cannot know where the photon came from" doesn't make the whole point clear, pedagogically. As I said in my original response to the cesiumfrog, there can be several reasons for why we don't know something, and they are not always related to the quantum mechanics itself. I understood originally that we cannot know from which source the photon comes from, but so what, I don't know what you are doing behind your computer either, and that doesn't mean that you are in superposition of doing several things. The symmetry of the wave function, according to the Bose statistics, makes the explanation complete.

 

[rewebster]

Do you know the story I mentioned in my post #106 in the “cat in a box paradox” session?


Regards, Dany.

no, sorry, I hadn't read that thread yet (read the part leading up to your post just now though)

my, my, my--'interpretation' is such an all encompassing word (even when it comes to 'data')

 

[OOO]

My final word on this is, that merely saying "you cannot know where the photon came from" doesn't make the whole point clear, pedagogically. As I said in my original response to the cesiumfrog, there can be several reasons for why we don't know something, and they are not always related to the quantum mechanics itself. I understood originally that we cannot know from which source the photon comes from, but so what, I don't know what you are doing behind your computer either, and that doesn't mean that you are in superposition of doing several things. The symmetry of the wave function, according to the Bose statistics, makes the explanation complete.

Sure you are right, there are several reasons for why we don't know what we'd like to know. For the last 80 or so years quantum mechanics has appeared "as if" its uncertainties were something substantially different from not knowing what somebody does behind his computer. But is it really ?

"Pedagogically" I like to think about these things as some kind of analogy with Maxwell's demon: although in classical physics it might be considered possible to control a system as much to violate the second law of thermodynamics, it is practically impossible so that you come nowhere near violating the second law of thermodynamics. But do not take this analogy too literally. It will probably be easy for one of you to point out the flaw in this argument. As I said "Pedagogically"...

 

[mn4j]

So let me put it that way: suppose you were sure somehow that a single photon you detect on the screen comes from exactly one of your sources then it is impossible for it to interfere in the sense of the double slit since it must have gone through one slit only. Since in classical terms non-interference means that both sources are not synchronized enough, we could say that knowing for sure were a single photon was emitted from means desynchronization of the sources.

How then do you explain "interference" patterns from single slits?

You probably might say that a photon always comes from one of the sources, even if they are synchronized but this is truly not the case in a quantum mechanical system. Actually you have to add some measurement device to the sources in order to detect where a photon is emitted from, and this measurement device destroys your coherence.

I'm sorry but this makes absolutely no sense to me. Just because you don't know the source of a photon does not mean it does not come from a particular source. Probability is a measure of the quality of information our mind has about reality, not a reflection of reality. If you don't know where the photon come from, the probability is 0.5,0.5 It doesn't mean half the photon came from one source andhalf from the other. (see http://bayes.wustl.edu/etj/articles/prob.in.qm.pdf)

 

[OOO]

How then do you explain "interference" patterns from single slits?

I think you got me wrong. It's become a fairly long thread now so it's forgivable that you don't have the thread start in mind anymore. But jostpuur initially talked about putting a wall congruent to the plane between both slits so as to assure that no photon crosses the plane. If you have just one slit then of course you will have interference according to the circumstances of a one-slit experiment. Is that what you meant ?

 

[OOO]

Probability is a measure of the quality of information our mind has about reality, not a reflection of reality.

As far as quantum mechanics is concerned I guess most people would say you're wrong and probability amplitude in quantum mechanics does indeed reflect some physical reality independent of the observer. If this was simply probability in the sense of the maximum entropy compatible with observation then how would you explain the complex probability amplitude ? If you have some solution to this problem, I'd be grateful if you explained it to me.

 

[mn4j]

As far as quantum mechanics is concerned I guess most people would say you're wrong and probability amplitude in quantum mechanics does indeed reflect some physical reality independent of the observer. If this was simply probability in the sense of the maximum entropy compatible with observation then how would you explain the complex probability amplitude ? If you have some solution to this problem, I'd be grateful if you explained it to me.

Could you be kind to define the word "probability". Say the word slowly "probable...ity". There is no physical entity as a probability amplitude. Call it something else, so long as it is a probability amplitude, it is not a real "thing". The fact that the probable outcome of an experiment matches the physical outcome when dealing with an ensemble system does not mean the two are the same.

If you throw a die a single time, only one phase will show, yet you can talk of a probability for each phase showing (6 values). It does not mean the expectation value, which is a probability distribution (what you call a wavefunction or probability amplidue) is a real thing. It is a mathematical description of our uncertainty. The only situation in which it gets close to anything real is when you throw an infinitely large number of dice. You will find that the observed histogram matches the wavefunction of a single die. Even though each die only shows a single face. In other words, the wavefunction of a single particle, is a good mathematical description of how an ensemble of particles will behave, rather than a physical description of an individual particle.

That is why the double slit pattern appears one speck at a time but a large number of particles builds up the pattern. Interpreting the wavefunction as though it were the physical nature of individual particles is not science but metaphysics. That is why nobody has ever explained why the wavefunction collapses at point A and not point B

The first sentence of the wikipedia article on the subject of "probability amplitude" describes it quite well.

Have you ever seen a quantum explanation of a single-slit experiment?

 

[OOO]

Could you be kind to define the word "probability". Say the word slowly "probable...ity". There is no physical entity as a probability amplitude. Call it something else, so long as it is a probability amplitude, it is not a real "thing". The fact that the probable outcome of an experiment matches the physical outcome when dealing with an ensemble system does not mean the two are the same.

If you throw a die a single time, only one phase will show, yet you can talk of a probability for each phase showing (6 values). It does not mean the expectation value, which is a probability distribution (what you call a wavefunction or probability amplidue) is a real thing. It is a mathematical description of our uncertainty. The only situation in which it gets close to anything real is when you throw an infinitely large number of dice. You will find that the observed histogram matches the wavefunction of a single die. Even though each die only shows a single face. In other words, the wavefunction of a single particle, is a good mathematical description of how an ensemble of particles will behave, rather than a physical description of an individual particle.

That is why the double slit pattern appears one speck at a time but a large number of particles builds up the pattern. Interpreting the wavefunction as though it were the physical nature of individual particles is not science but metaphysics. That is why nobody has ever explained why the wavefunction collapses at point A and not point B

The first sentence of the wikipedia article on the subject of "probability amplitude" describes it quite well.

Have you ever seen a quantum explanation of a single-slit experiment?

I did know what probability is before you told me. Only I don't know what you're trying to tell me. Quantum mechanics describes physics by means of probability amplitudes. Period.

If you got some more clever way to describe nature but by using probability amplitudes then feel free to tell us.

By the way, what do you precisely mean by "quantum explanation of a one-slit experiment" ? Is that something different than a "quantum explanation of a one-slit-and-a-bump-on-the-edge-experiment" ?

 

[mn4j]

I did know what probability is before you told me. Only I don't know what you're trying to tell me. Quantum mechanics describes physics by means of probability amplitudes. Period.

Yes, quantum mechanics is a mathematical model of the behavior of nature. Yes you can solve a lot of problems in nature by using probability amplitudes. But it doesn't mean the probability amplitudes are physical entities. That is what I'm trying to tell you. Probability means just that -- probability: a mathematical tool for doing inference from incomplete information. Period.

If you got some more clever way to describe nature but by using probability amplitudes then feel free to tell us.

No I don't. But questioning the establishment is a step in the right direction. All progress in physics starts with a critical look at the establishment. If we are expected to form new theories before questioning existing ones, there will never be progress. Besides I don't question QM. It is a great mathematical tool. I question the prevailing physical interpretation of the mathematics. The fact that the mathematics works does not validate any interpretation of it.

By the way, what do you precisely mean by "quantum explanation of a one-slit experiment" ? Is that something different than a "quantum explanation of a one-slit-and-a-bump-on-the-edge-experiment" ?

However you choose to call it, do you have any pointers to a quantum explanation of single-slit experiments with photons/electrons etc.

 

[jostpuur]

Just because you don't know the source of a photon does not mean it does not come from a particular source.

I agree to some extent, and understand the confusion.

merely saying "you cannot know where the photon came from" doesn't make the whole point clear, pedagogically ...

... The symmetry of the wave function, according to the Bose statistics, makes the explanation complete.

If you are interested to understand why independently emitted photons interfere, which you seemingly do not yet understand because you don't yet seem to believe in the probability concept of QM, you can find the explanation in my post #37

This thread was about one particular kind of experiment, its outcome, and its explanation. There is no point in leading the discussion to the foundations of the QM.

 

[OOO]

Yes, quantum mechanics is a mathematical model of the behavior of nature. Yes you can solve a lot of problems in nature by using probability amplitudes. But it doesn't mean the probability amplitudes are physical entities. That is what I'm trying to tell you. Probability means just that -- probability: a mathematical tool for doing inference from incomplete information. Period.

No I don't. But questioning the establishment is a step in the right direction. All progress in physics starts with a critical look at the establishment. If we are expected to form new theories before questioning existing ones, there will never be progress. Besides I don't question QM. It is a great mathematical tool. I question the prevailing physical interpretation of the mathematics. The fact that the mathematics works does not validate any interpretation of it.

However you choose to call it, do you have any pointers to a quantum explanation of single-slit experiments with photons/electrons etc.

I guess I didn't claim anywhere that a probability amplitude is a sort of physical entity. But on the other hand I believe it is a valid point of view to think of the wave function as a physical entity the square of which is "accidentally" a measure for the probability for what happens in a certain context.

In this sense, one day, we could probably find the reason why there is such parallelism between the actual physical field (the wave function) and the fact that it can trigger a stochastic process with well defined transition probabilities. (we need not extend the discussion to the relation between the terms "physical field" and "wave function", I know about gauge invariance and probably there are some other issues with it, but I think that's not relevant here)

So I think we are not so far apart regarding our skepticism against quantum mechanics. It seems you're tending more towards particles as the fundamental reality whereas I'd prefer the field and collapse viewpoint. But arguing about these views is futile as long as nobody of us has a mathematical formulation that expresses one of these views clearly.

It's fine to question the establishment but one always has to keep in mind that every alternative must be at least as powerful in explaining reality as the established theories.

 

[Demystifier]

I guess I didn't claim anywhere that a probability amplitude is a sort of physical entity. But on the other hand I believe it is a valid point of view to think of the wave function as a physical entity the square of which is "accidentally" a measure for the probability for what happens in a certain context.

In this sense, one day, we could probably find the reason why there is such parallelism between the actual physical field (the wave function) and the fact that it can trigger a stochastic process with well defined transition probabilities.

We already know (a possible) reason for that: Bohmian mechanics.

 

[OOO]

We already know (a possible) reason for that: Bohmian mechanics.

I like Bohmian mechanics but I think it is a bit optimistic to say that it solves the problems discussed here. I guess that was the reason why you put "possible" in parentheses.

For example I know of no consistent relativistic formulation of Bohmian mechanics. So how do you explain particle creation and annihilation with it. And by the way it doesn't explain the collapse of the "guidance field" either. As far as I know it just serves to demonstrate that quantum mechanics does not contradict permanent localization of particles.

 

[Demystifier]

For example I know of no consistent relativistic formulation of Bohmian mechanics. So how do you explain particle creation and annihilation with it. And by the way it doesn't explain the collapse of the "guidance field" either. As far as I know it just serves to demonstrate that quantum mechanics does not contradict permanent localization of particles.

Bohmian mechanics DOES explain the EFFECTIVE collapse of the guidance field, provided that environment induced decoherence is also taken into account.

Concerning relativistic formulation and particle creation/destruction, there are several inequivalent approaches, so the things are not yet settled. In my opinion, the most promising approach is the one pushed forward in
http://xxx.lanl.gov/abs/0705.3542
See also Refs. [16] and [3] for other approaches.

 

[OOO]

Bohmian mechanics DOES explain the EFFECTIVE collapse of the guidance field, provided that environment induced decoherence is also taken into account.

Concerning relativistic formulation and particle creation/destruction, there are several inequivalent approaches, so the things are not yet settled. In my opinion, the most promising approach is the one pushed forward in
http://xxx.lanl.gov/abs/0705.3542
See also Refs. [16] and [3] for other approaches.

I have started to read this paper (I assume it is yours ?) but I get drowned in scepticism when I read such things as "there is no position operator in QFT". How can you have a position operator in a field theory ? If you want to describe a position measurement then you have to define position in terms of the fields, i.e. find some state that represents localized fields. Is it that what you mean with "there is no position operator in QFT" ?

 

[OOO]

In my opinion, the most promising approach is the one pushed forward in
http://xxx.lanl.gov/abs/0705.3542

As far as I can see this paper doesn't do much more than define trajectories from currents. That's of course what the Bohm approach always does and it is ideed interesting. But nothing is said about the coupling between gauge fields and fermions.

So where is there any justification for the claim that this is a promising approach for replacing QFT ?

 

[OOO]

Bohmian mechanics DOES explain the EFFECTIVE collapse of the guidance field, provided that environment induced decoherence is also taken into account.

There seems to be no support for such a claim. Can you cite some references where it is explained in detail how such a collapse should work ?

 

[rewebster]

Do you know the story I mentioned in my post #106 in the “cat in a box paradox” session?

I am not sure about his name (I think P.Wood; my first book on ED at high school was written by him). He was in the middle of the measurements. The performance severely degraded due to dust on the internal surfaces and the tubes were about 11-14 m long. Project! He took the cat, put him inside, thus to show the completely deterministic way out. After 10 min he continues the measurements.

Regards, Dany.

Those DARN dust bunnies!----ahh, maybe there's a dust bunny in the box with the 'cat'

 

[Anonym]

Yes, quantum mechanics is a mathematical model of the behavior of nature. Yes you can solve a lot of problems in nature by using probability amplitudes. But it doesn't mean the probability amplitudes are physical entities. That is what I'm trying to tell you. Probability means just that -- probability: a mathematical tool for doing inference from incomplete information. Period.

I don’t agree with you. QM is not a mathematical model of the behavior of nature; QM is the adequate physical theory in the non-relativistic limit. The probability amplitudes are just unsuccessful terminology pushed through by N.Bohr and you clearly explained why in your posts #129 and #151 “cat in a box paradox” session.

But the terminology is not a matter, call it as you wish. Your problem is that you should obtain the image of the extended object and you have only the points available to do that (it is possible, C.Monet demonstrated that). The nature uses the repeatability and indistinguishability. There is no correlation between the individual samples; they are integral inseparable part of the overall picture.

Using the standard QM terminology I would say that the particle density is the real and therefore observable quantity. And I see no essential difference between the electron wave function and the electromagnetic potentials.

Regards, Dany.

 

[Demystifier]

So where is there any justification for the claim that this is a promising approach for replacing QFT ?

In this paper, I argue that a Bohmian description of particle creation/destruction requires strings. It is well known that strings can replace QFT.

 

[Demystifier]

I have started to read this paper (I assume it is yours ?) but I get drowned in scepticism when I read such things as "there is no position operator in QFT". How can you have a position operator in a field theory ? If you want to describe a position measurement then you have to define position in terms of the fields, i.e. find some state that represents localized fields. Is it that what you mean with "there is no position operator in QFT" ?

No. Nonrelativistic QM should be derivable from relativistic QFT. Therefore, there should exist a position operator that corresponds to localized PARTICLES, not localized fields. See Ref. [1] in the paper for more details.

 

[Demystifier]

There seems to be no support for such a claim. Can you cite some references where it is explained in detail how such a collapse should work ?

See any paper on Bohmian mechanics that includes the THEORY OF MEASUREMENT.
Perhaps this is best explained in the P. Holland's book "The Quantum Theory of Motion". But even the original Bohm's paper (part II) contains it. An even better explanation is provided also in the review
D. Bohm and B. J. Hiley, Phys. Rep. 144, 323 (1987).
Note that these works do not use the word "decoherence", as this word become popular only later. But if you are familiar with the theory of decoherence, you will recognize it in these papers.

 

[OOO]

No. Nonrelativistic QM should be derivable from relativistic QFT. Therefore, there should exist a position operator that corresponds to localized PARTICLES, not localized fields. See Ref. [1] in the paper for more details.

In QFT particles are described by fields. So where is the difference between localized fields and localized particles ? They're just two different names for the same underlying theoretical description. At least if you are referring to QFT and not to some other theory.

 

[OOO]

In this paper, I argue that a Bohmian description of particle creation/destruction requires strings. It is well known that strings can replace QFT.

In this age it seems to be that strings can predict almost anything you want. If this is what you're looking for then you are certainly going to be a happy fellow compared to all those nerds who try to explain the outcomes of experiments. Nevertheless, just mentioning the buzzword strings doesn't help in clarifying anything.

 

[Demystifier]

In QFT particles are described by fields. So where is the difference between localized fields and localized particles ? They're just two different names for the same underlying theoretical description. At least if you are referring to QFT and not to some other theory.

This is not that simple. For example, in nonrelativistic QFT, you CAN introduce a state that corresponds to a localized particle. You can also introduce a state that corresponds to a localized field. However, these two states are VERY different. In particular, the former is a 1-particle state, whereas the latter is a coherent state with an indefinite number of particles. So no, particles and fields are NOT just two different names for the same thing.

 

[Demystifier]

In this age it seems to be that strings can predict almost anything you want. If this is what you're looking for then you are certainly going to be a happy fellow compared to all those nerds who try to explain the outcomes of experiments. Nevertheless, just mentioning the buzzword strings doesn't help in clarifying anything.

Now you shifted your original objection to a (common) objection against string theory. I will not dwell into a discussion for and against strings, because it would belong to the "Beyond the Standard Model" subforum, not this subforum. Let me just note that, in my paper, I indicate how, with a Bohmian-like reformulation of quantum theory, string theory could be tested at low energies, at the level of Standard-Model particles.

 

[OOO]

This is not that simple. For example, in nonrelativistic QFT, you CAN introduce a state that corresponds to a localized particle. You can also introduce a state that corresponds to a localized field. However, these two states are VERY different. In particular, the former is a 1-particle state, whereas the latter is a coherent state with an indefinite number of particles. So no, particles and fields are NOT just two different names for the same thing.

I don't know why you're so keen on nonrelativistic theory, it's just an approximation.

Let's talk more specifically about "relativistic" gauge theory. You have a wave functional that assigns to every gauge field configuration a complex number which you could consider as defining the probability density for having that gauge field configuration:

\Psi: A\to \Psi[A] with \Psi^*[A]\Psi[A]= probability density to find configuration A

So if you have detected a grain of silver on your photographic plate, you know definitely that the gauge field must have been in some configuration A_0 that was localized around the site of your grain of silver somewhere during or immediately after producing the latter.

So your wave functional \Psi[A] which has originally contained the possibility for many different gauge field configurations has now collapsed to a wave functional \Psi_0[A] such that

\Psi[A] = \left\{ \begin{array}{cr} \infty &amp; \qquad A=A_0 \\<br /> 0 &amp; \qquad otherwise \end{array}\right.

where I have introduced a symbolical definition of a "delta-function over function space".

Now where is the need to define something like a position operator here ? If you like to, you may define position as the weighted average of the usual position operator "x" over the electromagnetic energy of your localized gauge field configuration A0:

\bar{x}[A_0]=\frac{\int x T^{00}(F(A_0)) d^4 x}{\int T^{00} d^4 x}

You will not be surprised that this provides you with the position of your grain of silver. So what further insight does this give you ?

 

[Demystifier]

OOO, I think you do not understand what I am talking about. But I will give you a hint. You said that nonrelativistic theory is an approximation. Fine! Now how would you DERIVE nonrelativistic QM as an approximation from your (correct) equations above?

 

[OOO]

OOO, I think you do not understand what I am talking about. But I will give you a hint. You said that nonrelativistic theory is an approximation. Fine! Now how would you DERIVE nonrelativistic QM as an approximation from your (correct) equations above?

There was another thread recently where someone cited a reference on that topic. Unfortunately I can't find it, maybe you can.

Of course my equations above are by no means complete.

But thinking about it in terms of path integrals I'd say you have to take the limit of classical field theory (h to 0) where you obtain the condition that the classical action becomes stationary (the usual thing about phases oscillating rapidly except for the classical path). So you haven't got a wave functional anymore but classical fields.

From the classical field equations it gets quite easy I think, see Bjorken Drell. Sketch: approximate frequency by compton frequency + delta, drop second order terms, redefine the electric potential by absorbing rest mass. Et voilà, you got the Schrödinger equation. So the classical fields described by it define your position in the same sense as the (probabilistically distributed) fields did in the full quantum field theory.

 

[jostpuur]

where I have introduced a symbolical definition of a "delta-function over function space".

btw. wouldn't notation

<br /> \delta^{\mathbb{R}^3}(A-A&#039;) = \prod_{x\in\mathbb{R}^3} \delta(A(x)-A&#039;(x))<br />

be logical? It would work like this

<br /> \int\mathcal{D}A&#039;\;\delta^{\mathbb{R}^3}(A-A&#039;)\Psi[A&#039;] = \Big(\prod_{x\in\mathbb{R}^3} \int dA&#039;(x) \delta(A(x)-A&#039;(x))\Big)\Psi[A&#039;] = \Psi[A]<br />

It could help to use notation A_x = A(x) also.

 

[Demystifier]

There was another thread recently where someone cited a reference on that topic. Unfortunately I can't find it, maybe you can.

Of course my equations above are by no means complete.

But thinking about it in terms of path integrals I'd say you have to take the limit of classical field theory (h to 0) where you obtain the condition that the classical action becomes stationary (the usual thing about phases oscillating rapidly except for the classical path). So you haven't got a wave functional anymore but classical fields.

From the classical field equations it gets quite easy I think, see Bjorken Drell. Sketch: approximate frequency by compton frequency + delta, drop second order terms, redefine the electric potential by absorbing rest mass. Et voilà, you got the Schrödinger equation. So the classical fields described by it define your position in the same sense as the (probabilistically distributed) fields did in the full quantum field theory.

In this way, you will obtain the Schrodinger equation. But how will you recover the probabilistic interpretation of the solutions of the Schrodinger equation? You should not postulate it, but derive from the probabilistic interpretation of QFT. So, how exactly you will do that?

 

[OOO]

btw. wouldn't notation

<br /> \delta^{\mathbb{R}^3}(A-A&#039;) = \prod_{x\in\mathbb{R}^3} \delta(A(x)-A&#039;(x))<br />

be logical? It would work like this

<br /> \int\mathcal{D}A&#039;\;\delta^{\mathbb{R}^3}(A-A&#039;)\Psi[A&#039;] = \Big(\prod_{x\in\mathbb{R}^3} \int dA&#039;(x) \delta(A(x)-A&#039;(x))\Big)\Psi[A&#039;] = \Psi[A]<br />

It could help to use notation A_x = A(x) also.

Yes of course, that's what I meant to say with my short hand notation.

 

[jostpuur]

I would very much like to see how to derive the one particle wave function

<br /> \Psi:\mathbb{R}^3\to\mathbb{C},<br />

its equation of motion, and its probability interpretation, by starting from the field wave functional

<br /> \Psi:X^{\mathbb{R}^3}\to\mathbb{C},<br />

its equation motion, and its probability interpretation. (X defines what is the value of the classical field. It can be X=\mathbb{R},\mathbb{C},\mathbb{R}^4,\ldots or something else.)

 

[jostpuur]

Yes of course, that's what I meant to say with my short hand notation.

Have you seen this notation a lot already? If so, you wouldn't bother mentioning some sources? I mean, I have not seen this anywhere yet.

 

[OOO]

In this way, you will obtain the Schrodinger equation. But how will you recover the probabilistic interpretation of the solutions of the Schrodinger equation? You should not postulate it, but derive from the probabilistic interpretation of QFT. So, how exactly you will do that?

You can't because you've already dropped the information about probabilities in your nonrelativistic approximation. The classical solutions virtually describe wave functionals with infinitely sharp distribution (localized at the classical solution). So I think you'd have to reintroduce the spreading of the wave functional again somehow in order to describe how the time evolution deviates from the classical path.

As to the details I have no clue. But I guess you have.

 

[OOO]

I would very much like to see how to derive the one particle wave function

<br /> \Psi:\mathbb{R}^3\to\mathbb{C},<br />

its equation of motion, and its probability interpretation, by starting from the field wave functional

<br /> \Psi:X^{\mathbb{R}^3}\to\mathbb{C},<br />

its equation motion, and its probability interpretation. (X defines what is the value of the classical field. It can be X=\mathbb{R},\mathbb{C},\mathbb{R}^4,\ldots or something else.)

Well then I'm afraid you'll have to take a look at some QFT textbook.

 

[OOO]

Have you seen this notation a lot already? If so, you wouldn't bother mentioning some sources? I mean, I have not seen this anywhere yet.

Come on, it's a bit OT discussing math 101. I'm sure you know the most stringent definitions of distributions.

 

[jostpuur]

Well then I'm afraid you'll have to take a look at some QFT textbook.

I've already been forced to reinvent the idea of the wave functional on my own, because all that the books tell are the cursed operators and their commutation relations!

 

[jostpuur]

dumb question

Come on, it's a bit OT discussing math 101. I'm sure you know the most stringent definitions of distributions.

What does OT mean?

 

[OOO]

What does OT mean?

off-topic

 

[OOO]

I've already been forced to reinvent the idea of the wave functional on my own, because all that the books tell are the cursed operators and their commutation relations!

Yes, the textbooks have more to say about the Heisenberg picture but it should't be difficult to put it in Schrödinger terms. I wouldn't call this reinvention.

If you don't like operators that much, try out the path integral approach (but of course you'll need some operators to show the equivalence of expectation values in both approaches).

 

[jostpuur]

Yes, the textbooks have more to say about the Heisenberg picture but it should't be difficult to put it in Schrödinger terms. I wouldn't call this reinvention.

It's so relative. There has been incidents where people, who already know QFT in their own opinion, tell me that the wave functional stuff is something that I have come up on my own and that doesn't really belong to the correct QFT. No doubt, because the QFT seems to be operators and Feynman diagrams usually.

 

[Demystifier]

I've already been forced to reinvent the idea of the wave functional on my own, because all that the books tell are the cursed operators and their commutation relations!

Then see
B. Hatfield, Quantum Field Theory of Point Particles and Strings
This pedagogically written textbook on QFT (and string theory, but you don't need to read the part II if you don't like strings) develops the functional Schrodinger representation of QFT in detail.
In addition, it points out some conceptual details on the difference and relation between particles and fields.

 

[Demystifier]

You can't because you've already dropped the information about probabilities in your nonrelativistic approximation. The classical solutions virtually describe wave functionals with infinitely sharp distribution (localized at the classical solution). So I think you'd have to reintroduce the spreading of the wave functional again somehow in order to describe how the time evolution deviates from the classical path.

As to the details I have no clue. But I guess you have.

Yes I do. (Although, it does not really work in the way you sketch above.) But as I already said, first quantization can be deduced from second quantization (QFT) ONLY in the nonrelativistic formulations of both first and second quantizations. This is closely related to the fact that relativistic QM does not have well defined probabilistic interpretation, at least not in the conventional orthodox approach.
It is frequently said that relativistic QFT solves this problem of relativistic QM, but it does not. Instead, it merely sweeps it under the carpet. It is not a problem for most of the practical applications of QFT, but it is a problem as a matter of principle. You cannot just state the usual axioms of RELATIVISTIC QFT and then derive all the rules of nonrelativistic QM as an approximation.