How are electrons considered waves?

[zonde]

The idea that "particles" somehow cause "events" is not necessarily the best way to understand Physics. An alternative is to think of there being a field that causes the events in the measurement apparatuses.

Seems reasonable. But what is the role of quanta in this? Is it for all distinguishable things the same as "particle" or is it something different?

 

[Peter Morgan]

Seems reasonable. But what is the role of quanta in this? Is it for all distinguishable things the same as "particle" or is it something different?

Good question. Quanta come to two things: (1) some operators have discrete spectra. We observe discrete spectra that can be modeled rather well as the eigenvalue spectrum of an operator, such as the energy spectrum of the Hamiltonian of an electron in a bound state.

I personally think that bound states are problematic in the Standard Model of quantum field theory, insofar as perturbation theory largely deals with scattering, whereas the mathematics of bound states is much less developed. Nonetheless, we can introduce various central potentials and produce quite nice results, including fine details like the Lamb shift. The introduction of a particular central potential, however, is problematic from a QFT perspective, in which such forces ought to be derived as a result of gauge fields. As far as I know, it is mostly a matter of faith that we will eventually be able to derive from the QCD Lagrangian what the effective central forces are for specific nuclei. [Given the nontrivial magnetic moments of nuclei and electrons, however, it's clear that the effective forces are not central forces. There are a lot of details to keep track of!]​

(2) Just as delicately, superselection sectors give a discrete structure in quantum field theory. However creation and annihilation operators of modes of the electromagnetic field do not map between superselection sectors, so that there is no natural discrete structure associated with the electromagnetic field. The same is true for simple Boson fields such as the Klein-Gordon field. This has been a debated issue in the quantum field theory literature for at least 20 years, with a large section of the literature coming down against there being discrete structure (and certainly against there being particle structure) for the electromagnetic field --- Feynman diagrams and the ways that Physicists talk about photons notwithstanding. [Maybe it's helpful to point out that just because a continuous curve can be decomposed as the sum of Fourier components doesn't necessarily mean that there is a natural discrete structure to the continuous curve.]

Fermion fields and non-abelian gauge fields are associated with specific discrete charges, electric charge in the case of electrons, so there is this discrete structure to be found for almost all fields except the electromagnetic field. The closest analogy to the superselection sectors of QFT that I know of are the topological sectors of continuous fields. Determining which superselection sector a given state is in requires global measurement, just as when we try to determine which topological sector a given state is in (think of a mobius strip -- locally, we can't tell whether there's a twist or not; only by checking the whole loop can we determine whether there's a twist or not). Unfortunately, continuous functions are not adequate to model Physics, because it is not possible to introduce thermal and quantum fluctuations in a mathematically acceptable way (probability distributions over continuous fields result in problems). There is a classical object called a random field that allows us to model thermal and quantum fluctuations as well as a quantum field can, but, again, this is an idea that is so far only to be found in a few published papers, all by me (which means that there are a few referees who thought the papers were not so ridiculous that they wouldn't let them be published, not that the referees thought the papers were brilliant -- they definitely had reservations). The papers are available through my web-page (links to both arXiv and published versions). I explain random fields multiple times in those papers; they could be thought of as what you need to do to deal with proper mathematical care with probability densities over a generalized function space (certainly a more generalized function space than just the space of continuous functions, so that everything is measure theoretical instead of differentiation). After years of working with random fields, they seem very simple and pretty mathematics to me, but there is a sophistication to them that is pretty much equal to that of quantum fields (as you should expect would be needed), and indeed my latest papers show that there are random fields that are empirically equivalent to the complex Klein-Gordon quantum field (published) and to the quantized electromagnetic field (so far unpublished).

The final discrete structure (3!) is the engineered thermodynamics of the detection apparatuses, which I would say has nothing to do with quanta. When we construct an apparatus that has two thermodynamic states, a ready state and an excited state, which are macroscopically discernible in the sense that we can see that it is on one or the other thermodynamic state, then, inevitably, we observe discrete events. When we tune the two states so that the apparatus may be as delicately balanced as a pencil on its tip (with a feedback mechanism for getting it back to vertical as soon as something knocks it out of its unstable thermodynamic ready state), we observe different statistics for transitions depending on what other apparatus we put near it. Modern Physics is about determining the response of different kinds of such delicately balanced thermodynamically nontrivial apparatus to different kinds of preparation apparatus (which is generally about as nontrivial in its own way). [I've been looking for an alternative to the words "measurement apparatus" that doesn't imply so strongly that there is something that is measured, perhaps "response apparatus" is OK.] It is fundamental that all this happens in the presence of quantum fluctuations, which are distinct from thermal fluctuations.

It's amazing to me that this kind of detailed analysis of what happens in a measurement is not common parlance amongst Physicists, but to my knowledge it's not. A study of the changes of the response of an apparatus to different inputs is a commonplace for control systems.

 

[ZapperZ]

Indeed it is difficult to explain how can entity dissolve into the field and form back without admitting that it can actually vanish into and emerge from the field.

But you're just illustrating the fact that there isn't a "paradox", but rather a conceptual difficulties by some people (you). You should not confuse those two - they are not identical.

Zz.

 

[zonde]

Peter,
Your answer was very interesting to read but I didn't found direct answer. I took out some sentences that as I understand hint about possible answers:

some operators have discrete spectra
...
superselection sectors give a discrete structure in quantum field theory. However creation and annihilation operators of modes of the electromagnetic field do not map between superselection sectors, so that there is no natural discrete structure associated with the electromagnetic field.
...
Fermion fields and non-abelian gauge fields are associated with specific discrete charges, electric charge in the case of electrons, so there is this discrete structure to be found for almost all fields except the electromagnetic field.

From these sentences I guess that the aim is to make field so discrete that it will allow to speak about something like particles.
But I will try to answer myself:
Particles has certain energy (mass) and spatial borders so that we can unambiguously identify what energy belongs to particle and what does not belong.
For quanta we have certain energy so there is match for particle. Unclear thing for me is whether quanta can spatially overlap with other quantas.
You mentioned superselection sectors but this is mathematical concept and does not give me any hint about this overlapping question. My guess is that quanta can overlap with other quanta.

The final discrete structure (3!) is the engineered thermodynamics of the detection apparatuses, which I would say has nothing to do with quanta.

I understand that empirical approach is safer meaning that there we are free from uncertain assumptions but in case of quanta that would mean that we should talk about black body radiation. However there are a lot of indirect empirical data for meaningful speculations about quanta as it seems to me.

There is a classical object called a random field that allows us to model thermal and quantum fluctuations as well as a quantum field can

I understand that this is a topic that you would really want to discuss. But I just don't really understand at what your approach is aimed. I guess it's related to some QFT questions.

 

[Peter Morgan]

Peter,
Your answer was very interesting to read but I didn't found direct answer. I took out some sentences that as I understand hint about possible answers:

To be honest, my post was long enough that I expected that I was writing for myself, trying to clarify issues in my head that I expect to write about in papers that I will hope to be published. I took your post as a starting point, but at least in part I got carried away. I'm glad that you read all that!

From these sentences I guess that the aim is to make field so discrete that it will allow to speak about something like particles.
But I will try to answer myself:
Particles has certain energy (mass) and spatial borders so that we can unambiguously identify what energy belongs to particle and what does not belong.

I would say that it's difficult to say that energy (or mass, stuff, or whatever) identifiably belongs to one particle or another. To revert to (the problematic analogy of) the topology of a strip paper that has two twists in it, there is no natural way to say which part of the paper is one of the twists and which part of the paper is in the other twist. If we introduce a coordinate system in the 3-space that the paper loop is embedded in, we may be able to identify different parts of the paper to assign to each of the twists relative to that coordinate system, but I would say that introducing a strong structure such as a coordinate system (or some other structure that allows an assignment of parts of the whole to individuals) is not to be done lightly.

Separability is well-known to be extremely delicate in quantum mechanics, so that it seems better to think in terms of particle number as a global topological property (insofar as one thinks in terms of analogies instead of taking the safer road of just using whatever mathematics turns out to be empirically effective). I take it to be one of the most important properties of the empirical data that the discrete particle number is conserved over time. Properties of a whole system are not necessarily expressible in terms of properties of parts of the whole system taken separately.

All that said, I take it that the de Borglie-Bohm interpretation would do much of what you are asking for, but there are too many reasons not to like it (the way that it uses configuration space instead of real space, the way in which nonlocality is introduced, and whether a relativistic version exists at all). I'm more impressed by Nelson stochastic mechanics and by Stochastic electrodynamics, but I prefer not to introduce point-like particles embedded in a field, which effectively rules out these classes of models.

For quanta we have certain energy so there is match for particle. Unclear thing for me is whether quanta can spatially overlap with other quantas.
You mentioned superselection sectors but this is mathematical concept and does not give me any hint about this overlapping question. My guess is that quanta can overlap with other quanta.

Because I don't take particular parts of energy or space-time to be allocated to quanta, this is a non question for me. If you can find a specific natural way to allocate parts of energy and space-time to particular quanta, I'm sure that would answer your question. I suspect that you won't be able to, but of course I've been prejudiced in favor of random field models for too long to be any help to you.

I understand that empirical approach is safer meaning that there we are free from uncertain assumptions but in case of quanta that would mean that we should talk about black body radiation. However there are a lot of indirect empirical data for meaningful speculations about quanta as it seems to me.

What we take to be the empirical data is a very important question! What answer we give goes a long way to determine what our theory will look like. I am not an empiricist in the 1920s positivist style. The post-positivist critique that was mostly constructed through the 1950s to the 1970s includes a strong claim that our descriptions of experiments are theory-laden, which I personally think is undeniable. [There are other aspects to the post-positivist critique that I also think have weight, incommensurability, underdetermination, and the pessimistic meta-induction, but those are not directly relevant to this particular point.]

If we say that one empirical idea is an empirical principle, as Einstein did with the speed of light in vacuum, we set it above other empirical ideas. If other people espouse different empirical principles, the mathematics they do will look very different. If one goes to Foundations of Physics conferences, or looks at the table of contents of the journal Foundations of Physics, it is very striking that very different parts of the empirical data are taken to be most important, less important, rather indirect, etc.

I effectively take random fields on Minkowski space as a starting point, because I take it that there must be some mathematical structure that can be taken to be the substrate for the thermodynamics of response apparatus events, but I'm not willing to use particles because I see the mathematics of classical particles leading to trouble. Part of the reason for taking random fields as a starting point is that they are the closest classical mathematics to quantum fields, which I take to be empirically strongly enough supported that they have to be taken seriously -- at least.

I understand that this is a topic that you would really want to discuss. But I just don't really understand at what your approach is aimed. I guess it's related to some QFT questions.

Oh yes, QFT is a major target, because there are some senses in which I think it looks easier to understand/interpret QFT than it is to understand/interpret non-relativistic quantum mechanics.

I see that you were the person who took ZapperZ to task for suggesting that Art Hobson's "Teaching Quantum Mechanics without Paradoxes" is a good way to go. I agree with you that Art doesn't get there, certainly not enough for Zz to cite him as a definitive source, but I personally find Art's attempt to use quantum field theory as the way to go very refreshing. I read some of Art's ideas as somewhat counter-culture. I think I read them generously, however, in the sense of bringing in other field-related ideas, and particularly random fields, which I find enough to make sense of him.

 

[zonde]

I take it to be one of the most important properties of the empirical data that the discrete particle number is conserved over time.

Can you expand this?
Do you mean particle number is conserved statistically? And what empirical data do you have on mind?

Because I don't take particular parts of energy or space-time to be allocated to quanta, this is a non question for me. If you can find a specific natural way to allocate parts of energy and space-time to particular quanta, I'm sure that would answer your question. I suspect that you won't be able to, but of course I've been prejudiced in favor of random field models for too long to be any help to you.

I am not sure I understand your position. Where do you see physical significance of quanta then?
Surely you must take that there is some physical significance of quanta at least in interactions (photon absorption by electron for example).

I have to say that I have some rude ideas how energy can be physically allocated to quanta but these are for my own comfort so to say.

And do you have some link with introduction in random fields? I looked into wikipedia but it is very short about this topic. Maybe there is any of your own papers that are not very specific?

 

[zonde]

But you're just illustrating the fact that there isn't a "paradox", but rather a conceptual difficulties by some people (you). You should not confuse those two - they are not identical.

From wikipedia:
A paradox is a statement or group of statements that leads to a contradiction or a situation which defies intuition.

I would say it's subjective what is considered "paradox".

 

[Peter Morgan]

Can you expand this?
Do you mean particle number is conserved statistically? And what empirical data do you have in mind?

By particle number, I mean the number of particles minus the number of anti-particles in a state. Strictly, I would say that charges are conserved, which imply the conservation of particle number. I would take the conservation of electrical charge —absolutely, not statistically— to be an empirical principle that is essentially unquestioned. Electric charge is of course intimately related to the U(1) gauge group in the standard formalisms. Physical states apparently can be mixtures of states that have different charges, but they cannot be superpositions of states that have different charges, which is just to say that there is a superselection principle for electric charge.

As far as the conservation of other charges is concerned, of electroweak and color charges, or perhaps of hadron number conservation, I would regard these as more open to question, but, in the absence of any definite reason, it suits me for now to give the Standard Model of Particle Physics the benefit of the doubt. My understanding of the detailed phenomenology of Particle Physics needs a lot of brushing up.

I take it to be a significant key to how we should construct theories that electric charge is an integer multiple of the charge on the electron in almost all circumstances.

I am not sure I understand your position. Where do you see physical significance of quanta then?
Surely you must take that there is some physical significance of quanta at least in interactions (photon absorption by electron for example).

I have to say that I have some rude ideas how energy can be physically allocated to quanta but these are for my own comfort so to say.

And do you have some link with introduction in random fields? I looked into wikipedia but it is very short about this topic. Maybe there is any of your own papers that are not very specific?

Where did you last see a photon absorbed by an electron? It's a standard way of talking, of course, justified by a naive interpretation of a tree level Feynman diagram, but if we introduce loop level Feynman diagrams, and get past renormalization, a similarly naive interpretation would have to say that there are infinite numbers of electrons and photons of infinitely varied energies, both on and off shell, interacting together, which is just messy.

At the empirical level, I would be (somewhat) more happy to say that a photon was absorbed by a macroscopic object, causing a thermodynamic transition from a ready state to an excited state that I can see with a microscope, or that is amplified electrically to the point that a computer memory bit is modified. There's always a question whether any given event was caused by a cosmic ray, was a result of an internal fluctuation of the macroscopic object, was caused by a stray electron, neutron, neutrino, or whatever, from an unshielded piece of apparatus in the room, etc.. If we see the rate of events change just after we turn on a light, we can only say that a particular discrete thermodynamic transition was quite likely caused by the light being turned on, not that it was definitely caused by a photon. I'm happy saying that the change of the statistics was caused by turning on the light, but we should be careful what we say about individual events.

As I said a few messages back, I take there to be three fairly incontrovertible discrete structures, charges, discrete spectra (something like the spectrum of Hydrogen, say, which are modeled as the eigenvalues of observables), and thermodynamic transitions of macroscopic objects. The idea that there are "quanta" is too vague to be thrown around without any indication of what mathematics we're using. Insofar as Planck's constant is what we're talking about when we talk about "quanta", I take it to be a measure of irreducible quantum fluctuations, which have effects on most small-scale Physics, which can be observed in large-scale Physics when we take appropriate steps to engineer amplification. My papers talk about this in an evolving way that is not entirely coherent over time.

As far as references on random fields are concerned, please get back to me immediately if you find something accessible on the web or in the literature! My own attempts at explaining random fields only scratch at the surface of the mathematics. There are a number of books in the Yale libraries,
Preston, Christopher J., Random fields, Springer-Verlag, 1976.
Vanmarcke, Erik., Random fields, analysis and synthesis, MIT Press, c1983.
Rozanov, Yu. A, Random fields and stochastic partial differential equations, Kluwer Academic, c1998.
Spitzer, Frank Ludwig, Random fields and interacting particle systems, Mathematical Association of America, 1971.

Another book is Stanley P. Gudder, Stochastic Methods in Quantum Mechanics, North-Holland, 1979, which I have taken to citing in papers, referring particularly to chapter 6, because the discussion of random fields is quite nicely done and geared to quantum field theory ways of thinking.

As you see, most of these references are from 20-30 years ago.

The algebraic methods I'm using to construct random fields and states over them, of creation and annihilation operators, are not in the literature in any direct way to my knowledge, although they are not particularly difficult mathematics. People haven't much thought it would be interesting to work with random fields, so finding effective ways of using them has not been investigated. The mathematics I'm using is partly just that of commutative *-algebras, for the field observables, whereas the algebra of creation and annihilation operators are very similar to what is used in quantum field theory. The literature of C*-algebras to a limited extent applies also to *-algebras, although the lack of a norm for the local observables has plenty of consequences.

Andrei Khrennikov has been using a mathematics that can be thought of as random fields, although I have not liked his formalism much. His papers can be found on arXiv.

Random fields are used in Physics at the Cosmological scale (indeed it was someone who works on Cosmology who was the second person to point out this mathematics to me, about six years ago), but I don't know whether there is a standard reference for random fields in that literature. So I've written to that person to find out whether there's a standard reference to cite or for sending graduate students to it.

 

[ZapperZ]

From wikipedia:
A paradox is a statement or group of statements that leads to a contradiction or a situation which defies intuition.

I would say it's subjective what is considered "paradox".

Intuition? I can show you plenty of situations that defy your "intuition", but will later makes sense after you UNDERSTAND the physics. So relying on your intuition to challenge a theory, much less, a VALID theory, is not a valid challenge, nor is it a valid argument against something.

Even using that definition, where is the "contradiction" here? The existence of a paradox is claimed, but that was simply a statement without justification. It takes zero effort and knowledge to make such statement. I can say you're wrong and left it at that without bothering to explain why. Do we expect physics to be as flimsy as political discussion you get on TV?

Zz.

 

[encrypted]

Howdy everyone, too much to read for a business major with ADD. I don't know what most of the physics majors in this thread have posted and I couldn't manage to read past page 1 cause it got too detailed for me (thanks ADD) and I didn't read past it anyways. But I have such an undying love for the world differentiating the quantum world vs. Einstein's general relativity word, which is how I ended up on this forum.

In layman's terms as I understand it, electrons (whether you consider them particles of matter or a quantum wave at the sub-atomic level are pretty much the same thing and) are possibilities of an outcome with endless possibilities resulting from a certain action, in the absence of an observer i.e. when someone is not watching (or not). A particle (whether a piece of matter or otherwise) can be in several places at the same time (i.e. a million outcomes of a single action and be it at several positions/locations at the same time). As a result, we find the "quantum wave" tends to answer so many more mathematical un-naturalities and potentials in the sub-atomic world.

But what makes it so intriguing is that when you as an individual or a measuring device is placed at a critical phase during the experiment, the particle "realizes" that it's being observed and changes itself to behave like an ordinary particle atom i.e. going into a "yes or no" outcome of an experiment.

The simple act of "observing" a particle basically made if behave differently, as though the particle was aware if was being observed, thanks to us being the "observer".

Which begs the ultimate question, does the particle behave differently in the eyes of the ultimate observer (i.e. God) vs. us (as humans) and yet, furtherly different in the absence of no state of observence.

 

[zenith8]

Which begs the ultimate question, does the particle behave differently in the eyes of the ultimate observer (i.e. God) vs. us (as humans) and yet, furtherly different in the absence of no state of observence.

Hi encrypted. The most obvious interpretation of what the equations mean - it was so obvious that it was the first one postulated in 1927 by de Broglie - is that the electrons are just perfectly ordinary particles 'guided' or pushed around by a wave (the usual quantum mechanical 'wave function'). Quantum mechanics is then just the same as classical (statistical) mechanics but with an extra force. All the whizzo 'isn't this weird' God observer cat stuff was tacked on by Bohr and his mates in the late 1920s - essentially because they were influenced by a temporarily fashionable philosophical prejudice called positivism (which was later violently disproved by the philosophers themselves in a process almost entirely unnoticed by the physics community.).

You probably think I sound like a crackpot. But that's how Bohr and the others won - by kindly accusing the behaviour of people like me as being due to our having had a full-frontal lobotomy or something like that.

Seriously - you think electrons don't have trajectories and only manifest themselves when an observer looks at them? Try following the streamlines of the probability current. Might be a revelation. Look up de Broglie-Bohm 'pilot wave' theory.

 

[Phrak]

You probably think I sound like a crackpot. But that's how Bohr and the others won - by kindly accusing the behaviour of people like me as being due to our having had a full-frontal lobotomy or something like that.

Do you think there are only two alternative camps?

Seriously - you think electrons don't have trajectories and only manifest themselves when an observer looks at them? Try following the streamlines of the probability current. Might be a revelation. Look up de Broglie-Bohm 'pilot wave' theory.

And then comes relativity. Where are the Lorentz invariant pilot waves?

 

[zenith8]

Lazy, knee-jerk criticism of something you plainly know nothing about.

And then comes relativity. Where are the Lorentz invariant pilot waves?

In the Lorentz invariant pilot-wave theory. Read the literature.. (or just allow it to be Lorentz invariant on average).

 

[zonde]

So relying on your intuition to challenge a theory, much less, a VALID theory, is not a valid challenge, nor is it a valid argument against something.

Prove that.

For that purpose I propose to consider photon EPR experiment with two sites with PBSes and 4 detectors at each output of two PBSes. In one of four channels between detector and PBS we insert wave plate that rotate polarization angle by 90 deg. Does this additional wave plate change outcome of experiment (compared to outcome without this plate)?
What says your understanding of physics and what says your intuition?
My intuition says that wave plate for this modified channel will invert correlations with other site's two channels.

 

[Peter Morgan]

Lazy, knee-jerk criticism of something you plainly know nothing about.

In the Lorentz invariant pilot-wave theory. Read the literature.. (or just allow it to be Lorentz invariant on average).

I'm curious what you take to be the Lorentz invariant pilot-wave theory literature. Do you mean Shelly Goldstein et al? In any case, I'd be interested in, say, 3 citations, that you think best make the case. arXiv or published would be OK.

I'm also curious whether anyone has been careful enough with the mathematics to get a paper published on allowing "it to be Lorentz invariant on average". Again, I'd be interested in arXiv or published.

It is of course possible not to know all the literature despite an honest attempt to do so. In response to someone who apparently doesn't know about a given literature, it's better to give citations than to tell them to "read the literature" if you want to be taken seriously. It also could be said to be less lazy, perhaps, and surely to be more polite? And, I suppose, there is more chance of persuading anyone of your point of view.

Physicists' perception of the inadequacies of de Broglie-Bohm models is, as I presume you know, not just a question of whether there is a Lorentz-invariant version of the theory, which as you suggest is routinely trotted out as a get-rid-of-this-fool reason for rejecting de Broglie-Bohm models. It's irritating, and don't diss the minority laws ought to be applied against such ex Cathedra statements, but don't let it get to you. Personally, I simply feel more comfortable with Hilbert space mathematics than working with the trajectory equations of de Broglie-Bohm, and I feel somewhat uncomfortable with the use that is made of configuration space, both because it's 3N dimensional and because it's not phase space. I follow the literature more-or-less, though not actively, but I always come away dissatisfied.

It may be that I'm not seeing what you see in the de Broglie-Bohm approach, but, given that I've done my best to see whether there's something interesting there, I think that means that you have more to do to make it clearer what you see. That means that you have to make a serious attempt to clarify, in papers on arXiv at least, but preferably with the care required to get something published (yes, with enough care papers on de Broglie-Bohm approaches have been published), why the various reasons people have for not thinking de Broglie-Bohm approach as seriously as, say, Nelson trajectories or the trajectories of Stochastic Electrodynamics, or as seriously as other interpretations of quantum theory, for all their failings. Two line rejoinders don't make much impression.

 

[Naty1]

A comment on
"WHAT does it mean to consider an electron as a wave?"

also posts 64 (Zapper) and perhaps reply in part to post #80 at the same time.

The following quote is from Roger Penrose celebrating Stephen Hawking 60th birthday in 1993 at Cambridge England...this description offered me a new insight into quantum/classical relationships:

Either we do physics on a large scale, in which case we use classical level physics;the equations of Newton, Maxwell or Einstein and these equations are deterministic, time symmetric and local. Or we may do quantum theory, if we are looking at small things; then we tend to use a different framework where time evolution is described... by what is called unitary evolution...which in one of the most familiar descriptions is the evolution according to the Schrodinger equation: deterministic, time symmetric and and local. These are exactly the same words I used to describe classical physics.

However this is not the entire story... In addition we require what is called the "reduction of the state vector" or "collapse" of the wave function to describe the procedure that is adopted when an effect is magnified from the quantum to the classical level...quantum state reduction isnon deterministic,time-asymmetric and non local...The way we do quantum mechanics is to adopt a strange procedure which always seems to work...the superposition of alternative probabilities involving w,z, complex numbers...an essential ingredient of the Schrodinger euqation. When you magnify to the classical level you take the squared modulii (of w, z) and these do give you the alternative probabilities of the two alternatives to happen...it is a completely different process from the quantum (realm) where the complex numbers w and z remain as constants "just sitting there"...in fact the key to..keeping them sitting there is quantum linearity...
QUOTE]
and he goes on to relate this linearity and superposition to the double slit experiment.

So I finally "get" what Zapper was stating in another thread about quantum consistency...
the "ambiguity" is in the classical to quantum interface and conversion...YET

Penrose goes on to say

My own view is that quantum theory is an approximate theory and we have to seek some new theory which supplants all three procedues.. classical, reduction and quantum...

(He subsequently notes lots of people would not agree)

(The above comes from The Penrose lecture, The problem of spacetime singularities:implications for quantum gravity, pages 63-67, THE FUTURE OF THEORETICAL PHYSICS AND COSMOLOGY, 1993..)

 

[zenith8]

That means that you have to make a serious attempt to clarify, in papers on arXiv at least, but preferably with the care required to get something published (yes, with enough care papers on de Broglie-Bohm approaches have been published)..

Stop treating me like a child, Morgan. Look - no offence, but I can't be bothered to take you on. These things have all been discussed ad nauseam in recent threads, and one just loses the will to live (particularly since this thread is specifically not about this issue).

Detailed references for many of the er.. several thousand papers that the naughty Bohmians have been lucky enough to have published are included on the website for the http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" . Click the 'Further Reading' tab. For relativistic stuff you should look under the section "Relativistic stuff and non-locality" - a good summary/review is included in the opening part of the recent paper "A Dirac sea pilot-wave model for quantum field theory" S. Colin, W. Struyve (2007). Some of the modern general review articles also have good discussions.

I'm also curious whether anyone has been careful enough with the mathematics to get a paper published on allowing "it to be Lorentz invariant on average". Again, I'd be interested in arXiv or published.

How about er.. D. Bohm, Prog. Theo. Phys. 9, 273 (1953). Quite a recent development then.