Photons, particles and wavepackets

[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

$$\Psi[A] = \left\{ \begin{array}{cr} \infty & \qquad A=A_0 \\ 0 & \qquad otherwise \end{array}\right.$$

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

$$\delta^{\mathbb{R}^3}(A-A') = \prod_{x\in\mathbb{R}^3} \delta(A(x)-A'(x))$$

be logical? It would work like this

$$\int\mathcal{D}A'\;\delta^{\mathbb{R}^3}(A-A')\Psi[A'] = \Big(\prod_{x\in\mathbb{R}^3} \int dA'(x) \delta(A(x)-A'(x))\Big)\Psi[A'] = \Psi[A]$$

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

$$\delta^{\mathbb{R}^3}(A-A') = \prod_{x\in\mathbb{R}^3} \delta(A(x)-A'(x))$$

be logical? It would work like this

$$\int\mathcal{D}A'\;\delta^{\mathbb{R}^3}(A-A')\Psi[A'] = \Big(\prod_{x\in\mathbb{R}^3} \int dA'(x) \delta(A(x)-A'(x))\Big)\Psi[A'] = \Psi[A]$$

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

$$\Psi:\mathbb{R}^3\to\mathbb{C},$$

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

$$\Psi:X^{\mathbb{R}^3}\to\mathbb{C},$$

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

$$\Psi:\mathbb{R}^3\to\mathbb{C},$$

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

$$\Psi:X^{\mathbb{R}^3}\to\mathbb{C},$$

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.

 

[OOO]

You cannot just state the usual axioms of RELATIVISTIC QFT and then derive all the rules of nonrelativistic QM as an approximation.

You say this. But have you tried hard enough ? I can neither confirm nor refute your claim because there would have to be a no-go theorem or a proof of said NR limit for that.

 

[Demystifier]

You say this. But have you tried hard enough ? I can neither confirm nor refute your claim because there would have to be a no-go theorem or a proof of said NR limit for that.

The argument (not a proof) is actually simple. NR QM contains a NR position operator. It should be a NR limit of the relativistic position operator. However, the latter does not seem to exist. I am not sure if there is a rigorous proof that it does not exist, but I know that the most obvious attempts do not really work, for one reason or another.

On the other hand, in my paper I show that the axioms of nonrelativistic Bohmian mechanics CAN be derived as an approximation of the axioms of relativistic Bohmian mechanics (because the axioms of Bohmian mechanics are not based on operators describing observables). In a sense, this makes Bohmian mechanics more powerfull than the orthodox approach.

 

[OOO]

The argument (not a proof) is actually simple. NR QM contains a NR position operator. It should be a NR limit of the relativistic position operator. However, the latter does not seem to exist. I am not sure if there is a rigorous proof that it does not exist, but I know that the most obvious attempts do not really work, for one reason or another.

You're just shifting the problem from the NR limit to the position operator. The fact that you can't show it doesn't mean a proof doesn't exist.

On the other hand, in my paper I show that the axioms of nonrelativistic Bohmian mechanics CAN be derived as an approximation of the axioms of relativistic Bohmian mechanics (because the axioms of Bohmian mechanics are not based on operators describing observables).

What does it help to prove well known physical theory A from speculation B ? Unless you haven't got a Bohmian equivalent to QFT there is no point in doing that.