Explaining Bohmian Mechanics: Wavefunction, Pilot Wave & Particles

[Demystifier]

I'm not sure how one could even do loops with Fermi's 4-fermion theory. Without the W, there's not much to "loop".

Of course you can do loops with Fermi's theory. Of course, they are not renormalizable, but as I said before, this is not a serious problem. For example, you can take a finite cutoff and just continue to work with it.

 

[humanino]

For humanino:
[10] Phys. Rev. Lett. 93, 090402 (2004)

This and 0904.2287 are a bohmian interpretation of QFT formalism. I already know that, and this is not what I requested. I want to know how you deal with scaling violation in a bohmian interpretation. This involve gauge theories. It is not covered by your article.

The question is pretty simple and it deals mostly with quarks and gluons in a a proton or a neutron. Let us say that at a given scale I count the number of quarks and gluons in a proton as a function of the light cone momentum fraction. When I change the scale, those numbers will change, as you have singlet evolution for the structure functions under the renormalization group. I don't know how to reconcile this with a bohmian interpretation. It would be quite interesting if it can be clarified from a pilot-wave point of view how you turn gluons into quarks. I don't see any reason why it should be impossible or even difficult, but I have never seen it done.

 

[Demystifier]

The question is pretty simple and it deals mostly with quarks and gluons in a a proton or a neutron. Let us say that at a given scale I count the number of quarks and gluons in a proton as a function of the light cone momentum fraction. When I change the scale, those numbers will change, as you have singlet evolution for the structure functions under the renormalization group. I don't know how to reconcile this with a bohmian interpretation. It would be quite interesting if it can be clarified from a pilot-wave point of view how you turn gluons into quarks. I don't see any reason why it should be impossible or even difficult, but I have never seen it done.

Nobody has ever studied it explicitly within the Bohmian interpretation. However, in my understanding of QCD, the change of scale has nothing to do with the change of the number of particles. After all, scales change continuously, while the number of particles can change only in discrete values. Therefore, I do not see why the Bohmian interpretation would need to say anything nontrivial about that.

Anyway, I must admit that I am not an expert for QCD, so I may be wrong. So let us have a deal. You give me a reference where it is explained how the change of scale is related to the change of the number of particles, I will read that reference, and after that I will try to say to you how the Bohmian interpretation interprets that. OK?

 

[humanino]

scales change continuously, while the number of particles can change only in discrete values.

Sure we always have the same number (3) of valence quarks, when integrated over the light-cone momentum fraction, but sea partons (quarks, antiquarks and gluons) vary a lot with scale. Besides, those variations are represented by densities as a function of this light-cone momentum fraction, so-called x-Bjorken. The are governed by integro-differential equation and as you know, densities are continuous.

I am not just nitpicking, I seriously think that you are dealing in this system with not just lite quantum mechanics but brute wild quantum field theory. Those things are well established.

You give me a reference where it is explained how the change of scale is related to the change of the number of particles, I will read that reference, and after that I will try to say to you how the Bohmian interpretation interprets that.

I would most certainly appreciate that.

I suppose QCD evolution equations for parton densities on scholarpedia is an acceptable reference. I think it's quite good for a quick review of the concepts and contains several serious textbook and/or published references. Besides, it has been written by one of the Nobel recipient for this work.

 

[Count Iblis]

Perhaps in an interacting theory, the vacuum, which according to the article consists of "dead particles", do something non trivial.

 

[Demystifier]

Sure we always have the same number (3) of valence quarks, when integrated over the light-cone momentum fraction, but sea partons (quarks, antiquarks and gluons) vary a lot with scale. Besides, those variations are represented by densities as a function of this light-cone momentum fraction, so-called x-Bjorken. The are governed by integro-differential equation and as you know, densities are continuous.

I am not just nitpicking, I seriously think that you are dealing in this system with not just lite quantum mechanics but brute wild quantum field theory. Those things are well established.

I would most certainly appreciate that.

I suppose QCD evolution equations for parton densities on scholarpedia is an acceptable reference. I think it's quite good for a quick review of the concepts and contains several serious textbook and/or published references. Besides, it has been written by one of the Nobel recipient for this work.

Thanks humanino, it is now clearer to me what you mean.
Of course, I have not became a QCD expert during the last night, but here are some comments.

1. In the books that I have seen (e.g. Halzen and Martin, Quarks and Leptons) the Bjorken scaling is introduced phenomenologically, not derived from first principles of QCD. Without a strict derivation from QCD, I am not able to give definite statements on the Bohmian interpretation of it. Nevertheless, I can give some hand-waving arguments.

2. Bjorken scaling is a property of the form-factor. The form factor is essentially the wave function in the momentum space. As you can see, wave functions in the momentum space play an important role in my paper too, even before the Bohmian interpretation. Of course, I have not studied in detail the case of QCD, but I don't think that it is essential here.

3. It is not completely clear to me whether the sea particles in hadrons are real or virtual. But in both cases, the Bohmian interpretation makes a clear interpretation of them. So let us discuss both possibilities.

3a. Let us assume that they are virtual. In terminology of my paper, it means that we deal with a 3-particle (for 3 real quarks) wave function psi_3. The contribution of the virtual particles is included in equations such as (70), because the perturbative expansion of U contains the contributions from virtual particles.

3b. Let us assume that they are real. A natural question is: How many real particles are there? My answer is - probably infinite. Or more precisely, the state of the hadron is probably something like a coherent state (or some generalization of it) which is a superposition of states with different numbers of particles. As I discussed in my paper, in such cases there is an infinite number of trajectories. However, depending on the kind of the measurement once performs (in particular, depending on the scale), most of these particles may have a very small probability of detection. With increasing energy some particles may become more easily detectable, but it does not mean that they did not exist for smaller energies.

Of course, these are only qualitative ideas. A serious answer would require a serious analysis by someone with a deeper understanding of QCD. Nevertheless, I am quite certain that either 3a or 3b is on the right track. I hope that it helps, at least a little.

 

[humanino]

Thanks for your answer, I do appreciate your sharing thoughts.

It is not completely clear to me whether the sea particles in hadrons are real or virtual. But in both cases, the Bohmian interpretation makes a clear interpretation of them. So let us discuss both possibilities.

What I would like from a Bohmian interpretation is to clarify this issue !

The truth is, practical researchers know only for sure of well defined matrix elements. Those are universal so that measuring parton densities (which merely parameterize those matrix elements) in one process allows prediction of amplitudes in another process. Then we can be proud of ourselves and think we have some kind of projection or moment of the true wave-function. But that does not resolve what the wave-function is. Written in the same tower of Fock states, it can be interpreted differently. It is useful to have an interpretation of the parton densities, because it allows us to guess sum rules, or positivity constraints, or dispersion relations, which we can then attempt to prove rigorously, an often quite difficult task (so one need good motivation to attempt the demonstration). All those constraints (like sum-rules, positivity, or dispersion relations for instance) prove to be quite powerful to restrict the full functional space when attempting to fit such a complicated object as a hadronic wave-function (or density matrix).

We switch from thinking of partons in terms of virtual particles to constituent particles permanently, and for all practical purposes for instance, a proton at the LHC is mostly a big bag of glue whose quarks are pretty much irrelevant. In addition, other semi-classical quantities of interest still resist a full clear understanding : for instance we know that the proton spin is shared non-trivially among its constituents in terms of spin and angular momentum, but it is not clear how to perform such a decomposition in a gauge-invariant fashion for the gluons for instance. Worse, how much is carried by what again depends on the scale. All indication points to a scale dependence of what we call virtual and real constituents.

My conclusion is, after thinking about it, I do not really have to complain about the Bohmian interpretation, but why I constantly ask those questions is that I hope they could shed light on issues we have with hadronic structures in general.

 

[Demystifier]

Thanks for your answer, I do appreciate your sharing thoughts.What I would like from a Bohmian interpretation is to clarify this issue !

My conclusion is, after thinking about it, I do not really have to complain about the Bohmian interpretation, but why I constantly ask those questions is that I hope they could shed light on issues we have with hadronic structures in general.

Well, as you can see in my paper, the number of particles associated with a given QFT state is defined BEFORE one imposes the Bohmian interpretation. Hence, it is standard QCD, and not the Bohmian interpretation of it, that should say how many particles are there. Perhaps it is already clear in the advanced QCD literature, but it is just not clear to me.

Anyway, if I remember correctly, you asked me also about the Unruh effect. Yesterday I got an idea how Unruh effect can also be explained with the Bohmian approach (although, I still need to work out the details). I will just give you the hint: The states |xi> in (11) can be eigenstates of the Rindler number operator. Thus, the behavior of the accelerated measuring apparatus described by (12) is as if Rindler particles were real, even though only Minkowski particles are actually real. For example, if the accelerated detector finds the system in the Rindler vacuum, then actually there is an infinite number of Bohmian trajectories of Minkowski particles. The interaction between the (Minkowski) vacuum and the accelerated measuring apparatus (made up of Minkowski particles) creates new Minkowski particles in very special (actually, squeezed) states.

 

[Ilja]

What I would like from a Bohmian interpretation is to clarify this issue !

My conclusion is, after thinking about it, I do not really have to complain about the Bohmian interpretation, but why I constantly ask those questions is that I hope they could shed light on issues we have with hadronic structures in general.

As far as a pilot wave theory can add something for the understanding of concrete phenomena at all (beyond the general insights like that quantum strangeness is largely unnecessary metaphysics) this would depend on the particular choice of the beables. If one uses a field ontology, which seems much more adequate, one probably cannot tell anything interesting about particles.

 

[Dmitry67]

Lets talk about the predicting power.
If you dint know about the Unruh effect, based on the BM you would probably say: the number of REAL particles is an objective reality, it is the same in all inertial and all accelerating frames. Hence, if vacuum does not contain real particles then there is neither Unruh effect nor Hawking radiation. BM can be equivalent to QM in inertial frames but lead to different results (so it is testable) when we include gravity. Also,as I understand, BM should deny the existence of gravitons and their existence is frame-dependent.

 

[Dmitry67]

The interaction between the (Minkowski) vacuum and the accelerated measuring apparatus (made up of Minkowski particles) creates new Minkowski particles in very special (actually, squeezed) states.

is it compatible with

The Unruh effect could only be seen when the Rindler horizon is visible. If a refrigerated accelerating wall is placed between the particle and the horizon, at fixed Rindler coordinate ρ0, the thermal boundary condition for the field theory at ρ0 is the temperature of the wall. By making the positive ρ side of the wall colder, the extension of the wall's state to ρ > ρ0 is also cold. In particular, there is no thermal radiation from the acceleration of the surface of the Earth, nor for a detector accelerating in a circle[citation needed], because under these circumstances there is no Rindler horizon in the field of view

 

[Demystifier]

Lets talk about the predicting power.
If you dint know about the Unruh effect, based on the BM you would probably say: the number of REAL particles is an objective reality, it is the same in all inertial and all accelerating frames. Hence, if vacuum does not contain real particles then there is neither Unruh effect nor Hawking radiation. BM can be equivalent to QM in inertial frames but lead to different results (so it is testable) when we include gravity. Also,as I understand, BM should deny the existence of gravitons and their existence is frame-dependent.

You misunderstood something about BM.
Anyway, irrespective of BM, see this:
http://xxx.lanl.gov/abs/0904.3412

 

[Ilja]

Lets talk about the predicting power.
If you dint know about the Unruh effect, based on the BM you would probably say: the number of REAL particles is an objective reality, it is the same in all inertial and all accelerating frames. Hence, if vacuum does not contain real particles then there is neither Unruh effect nor Hawking radiation. BM can be equivalent to QM in inertial frames but lead to different results (so it is testable) when we include gravity. Also,as I understand, BM should deny the existence of gravitons and their existence is frame-dependent.

Particle approaches to QFT use and have to use variable particle numbers.

Then, pilot wave theory has a preferred frame. Thus, there will be different theories related with different preferred frames. But the empirical predictions will show the same observable symmetries. At least I see absolutely no reason to expect that the equivalence proof does not work in QFT or quantum gravity.

If some version of BM will deny the existence of something observable, like gravitons, it is reasonable to expect that it will deny their existence in a similar way as it "denies" the existence of spin in Bell's version of particle theory with spin. (A theory I don't like, but it is useful to illustrate that one does not need much, one can even deny the existence of spin, but nonetheless prove the empirical equivalence to QM.)

 

[Demystifier]

More seriously, I would have preferred a published paper, but I will certainly take a look at your reference.

If someone is interested, now the final version accepted for publication in Int. J. Mod. Phys. A is available:
http://xxx.lanl.gov/abs/0904.2287

 

[Mentz114]

Thanks. I look forward to reading it.

 

[Dmitry67]

From another thread:

The point is that the number of created particles does NOT depend on the observer. Instead, it depends on whether a particle detector exists and on how does it move. In the case of accelerated particle detector, the created particles exist for any observer, not only for the observer comoving with the detector.
Does it make sense to you? Or will you ask me the same question again?

Wait, I don't understand.

In Unruh effect particles comes not from the detector, but from apparent cosmological horizons.

http://en.wikipedia.org/wiki/Unruh_effect

The Unruh effect could only be seen when the Rindler horizon is visible. If a refrigerated accelerating wall is placed between the particle and the horizon, at fixed Rindler coordinate ρ0, the thermal boundary condition for the field theory at ρ0 is the temperature of the wall. By making the positive ρ side of the wall colder, the extension of the wall's state to ρ > ρ0 is also cold. In particular, there is no thermal radiation from the acceleration of the surface of the Earth, nor for a detector accelerating in a circle[citation needed], because under these circumstances there is no Rindler horizon in the field of view.

How your theory can explain that?

 

[Demystifier]

As I said, there are two inequivalent versions of the Unruh effect. The one you mention is not measurable and the Bohmian theory cannot explain it. The Bohmian theory can only explain the other, measurable, version of the Unruh effect.

Let me briefly explain the two versions of the Unruh effect. The first version is defined with respect to various choice of spacetime coordinates and does not rest on properties of the measuring apparatus. It is based on Rindler quantization. In the second version there is only one definition of particles, which are standard Minkowski particles, while the Unruh effect manifests as clicks in an accelerated particle detectors. In the case of uniform acceleration, both approaches predict the thermal distribution of particles with the same temperature. Yet, the two approaches are not completely equivalent.

And of course, don't trust everything you find on wikipedia.

By the way, if you accept that the existence of many worlds in MWI only makes sense in the context of decoherence, then MWI also cannot explain the first version of the Unruh effect. The first version of the Unruh effect only makes sense in some variants of the Copenhagen interpretation (and in the Tegmark mathematical-universe interpretation, but this is much more than MWI because it asserts that any mathematical structure exists). As you probably know, decoherence makes sense only if you have some objective environment (measuring apparatus), so a vague notion of an "observer" not described by wave functions cannot explain decoherence, which is why decoherence-based MWI cannot explain Unruh effect based on such a vague notion of "observers".

 

[Dmitry67]

1 Why it is not measurable? Say, if I accelerate a ball, its front (and rear?) side will melt from the Unruh effect (theoretically). If you accelerate a ball in the box, then box melts but the ball is unaffected, correct?

2 What is a problem with MWI and the first type of Unruh effect? Why do we need a measurement (CI)?

 

[Demystifier]

1 Why it is not measurable? Say, if I accelerate a ball, its front (and rear?) side will melt from the Unruh effect (theoretically). If you accelerate a ball in the box, then box melts but the ball is unaffected, correct?

2 What is a problem with MWI and the first type of Unruh effect? Why do we need a measurement (CI)?

1. Well, nothing is measurable without a measuring apparatus. If you include a QUANTUM description of the measuring apparatus in your theory, then fine. But nobody has done it yet for the first version.

2. I assume that MWI needs decoherence. (Do you agree?) On the other hand, the first version does not include it.

Even worst, the first version does NOT even assert that the total state is a superposition of a state without particles and a state with particles. Therefore, MWI is not applicable. The first version asserts that THE SAME STATE has two different interpretations (vacuum interpretation and thermal bath interpretation), depending on motion of the classical observer NOT DESCRIBED BY QUANTUM MECHANICS. It is unacceptable in MWI because MWI asserts that everything is described by quantum mechanics. There are no classical observers in MWI, only quantum states in the Hilbert space. Therefore, I think you should be among the first who reject the first version of the Unruh effect.

All this reinforces what we already know, that MWI and BI are almost the same. :-)

 

[Dmitry67]

'Ball melted' is macroscopic event. Why do we talk about the measurement here while the effects of the Unruh radiation are macroscopic?
Lets talk only about the macroscopically observable version.

So, what happens to to ball and to the ball in the box? I provided you my version of what is going to happen. What do you think?

 

[Count Iblis]

You can treat your melting ball experiment in an inertial frame. In that frame the ball accelerates and there are no Unruh particles. The effect is then related to the Casimir effect. It is not the same as the "dynamic Casimir effect", but essentially it is just a matter of the acceleating object imposing certain boundary conditions on the fields.

 

[Dmitry67]

What? If we agree on the macroscopic realism, then if ball melted in one frame, it must melt in all others. If ball is accelerating then it can not be in inertial frame

Well, technically, in GR you can use any frames, but in any frame co-moving with a ball there will be rindler horizons.

 

[Dmitry67]

BTW, how particle can be accelerated? What we see in "accelerators" is a sum-over-histories of charged particles, exchanging virtual photons with magnets, increasing the momentum after each interactions. But between interactions the momentum is conserved.

 

[Count Iblis]

What I mean is that you an describe the accelerating ball in an intertial frame and do the calculations in tat frame. Then there is no Unruh radiation, but the ball will still melt.

See here for an example worked out from the two points of views:

http://arxiv.org/abs/gr-qc/0104030

 

[Demystifier]

'Ball melted' is macroscopic event. Why do we talk about the measurement here while the effects of the Unruh radiation are macroscopic?
Lets talk only about the macroscopically observable version.

So, what happens to to ball and to the ball in the box? I provided you my version of what is going to happen. What do you think?

First, I don't understand the word "macroscopic" except in terms of decoherence.
Second, I think that accelerated ball will melt and that nothing will happen with the box. This is because the forces act on the ball and not on the box.

 

[Demystifier]

See here for an example worked out from the two points of views:

http://arxiv.org/abs/gr-qc/0104030

I partially disagree with this paper, i.e., I do not think that the Unruh effect is really necessary to describe this process in the accelerated frame. I have actually submitted a comment on this paper, but PRL has rejected it.