Are virtual particles really there?

[tom.stoer]

Let me summarize why the question is problematic:
1) in QFT perturbation theory results in the invention of virtual particles; why do we not invent virtual apples in Newtonian theory of gravity?
2) w/o perturbation theory nobody would care about virtual particles
3) the discussion shows that especially the Coulomb potential and the virtual particles related to it are gauge dependend, so cannot be "real" in the sense that everybody has the same understanding; a "virtual photon" in Coulomb gauge and in axial gauge are two different "things"
4) in physics it's always difficult to explain what "is real" and "why it is the way it is"; what we can do is to calculate experimentally testable phenomena, but not an ontology

 

[kexue]

Since this virtual sticky thread became real sticky now, I like to contribute with two more answers I just received to my question which was the following:

I'm a physics student with a quick question.

Are virtual/ off mass particles really out there, do they really exist or are they just mathematical artifacts of perturbation theory and thus fictious?

I would be very grateful for any answer.

the first very short answer from Steven Weinberg

They are, in your words, mathematical artifacts of perturbation theory. SW

the other from David Politzer

This is precisely the kind of question you should be asking as
you're learning about relativistic quantum mechanics. And in
trying to find answers, you'll surely learn quite a bit of
physics.

However, as you work on it, I suspect that the more of the
physics you understand, the less relevant the initial question
becomes.

I do not mean to be cryptic just for its own sake, but, in
finding answers, you'll first find that there are problems with
the question.

In particular, the most problematic words and concepts in your
note to me are: exist, fictitious, and real.

You probably already know that there is some quantum funny
business that concerns the relation of energy uncertainties and
time intervals (or at least those are the terms that are usually
used). So is your question just a matter of degree? Are rho
mesons real particles? Or are they just mathematical artifacts
whose role is to simplify our account of the behavior of "real"
particles? How about the rho''? Pions? W's? Protons are known
to live much longer than the current age of the universe. But
what if they're unstable, too? Are quarks mathematical
artifacts? Some people would insist yes. But where does that
get them? Some people will tell you that you can't construct a
gauge invariant, correct state of a single electron. In some
sense they're right, and one can imagine structuring all physical
calculations in terms of space-time correlations between gauge
invariant sources. But insisting on that is just plain foolish.

In practice, you will be in a better position to confront and
answer questions of the sort you asked me all by yourself once
you have mastered the rudiments of the theoretical calculations
under discussion and (very importantly) how those calculations
are used to confront the physical (i.e., real) world.

If you really and absolutely believe that quantum mechanics has
anything to do with uncertainties and probabilities and that wave
functions collapse upon the making of observations (as opposed to
all of those being conveniences for us who are not all that
smart after all), then you believe that quantum mechanics doesn't
apply to you. However, the only argument in favor of that
proposition is that it makes you comfortable. There is not a
shred of experimental evidence. Rather, there is an enormous
body of evidence to the contrary.

We are all Schroedinger's cats!

D.P.

 

[nismaratwork]

Since this virtual sticky thread became real sticky now, I like to contribute with two more answers I just received to my question which was the following:



the first very short answer from Steven Weinberg


the other from David Politzer

"mathematical artifacts of perturbation theory"... is the mainstream view in line with current theory, and with all due respect, all D.P. commits to saying is that TCI shouldn't be regarded as a valid description of reality... which is fine, but unhelpful in this context.

 

[kexue]

Here goes another Nobel prize winner with his take on virtual particles!

Gerad t'Hooft answer

Virtual particles have little to do with perturbation expansion. They "are really out there" in the sense that their contribution certainly affects the amplitudes of particle transitions. But all of quantum mechanics is based on "states" that are not usually there in the classical sense. It's just like the two slit experiment. The particle goes through one slit or through the other, while nevertheless the behavior afterwards is determined by the fact that there were two slits. Similarly, virtual particles may have been present or absent.
Some scattering events may be entirely due to the exchange of a virtual particlke; in that case, it is hard to denay that the particle was there. Sometimes, you don't know whether it was a particle going from A to B, or an antiparticle going from B to A, this happens for instance when charged particles attract or repel one another by the exchange of a photon.

I doubt whether this helps,
G. /'t Hooft

 

[nismaratwork]

It gets better, here goes another Nobel prize winner with quite another take on this! (even though that leaves me even more confused than I was before I startet my little "survey" titled, I think, what do the most eminent quantum field theorists of the world think of virtual particles".)

Gerad t'Hooft answer

Are we playing, "Quote The Physicist"? Maybe you should try "Understand the physicists" first. Beyond that, what are you doing, making a poll? This isn't something that's up for grabs no matter how many people you email, with or without doctorates. Being brilliant doesn't mean you automatically get to be right (see: Spukhafte Fernwirking), and failing to engage in the substance while endlessly polling physicists is deeply unhelpful.

 

[tom.stoer]

@kexue: have you ever derived Feynman rules based on canonical quantization or based on the path integral? have you ever calculated and amplitude based on Feynman rules? have you ever seen non-perturbative effects like instantons that cannot be described via perturbation theory?

If you go through all that stuff by yourself you will have a totally different understanding of virtual particles than you get based on reading textbooks, popular books, e-mails, ... PF threads, ...

 

[kexue]

I don't have to share these emails here with you, if it is not appreciated. I find them interesting and enlightening. If you don't or even find them unhelpful, I'll stop from now on.

To Tom, yes I worked through the solved problems and excercises in Maggiore's QFT book, now I fight my way through Srednicki.

 

[tiny-tim]

Are we playing, "Quote The Physicist"? … what are you doing, making a poll? This isn't something that's up for grabs no matter how many people you email, with or without doctorates. Being brilliant doesn't mean you automatically get to be right …

Yes, we are playing "Quote The Physicist" … and why not?

Physics does change according to the majority accepted opinion, either because physicists change their minds, or because the stubborn old physicists eventually die off!

It's not conclusive, obviously, but it certainly helps to see how eminent reliable physicists approach the same question.

And this isn't simply a poll, with a yes/no answer … each physicist is giving reasons, and each reason in itself is a good starting-point for further study and discussion.

Perhaps in the end, all the quotes should be separated and copied onto a new sticky, as a permanent record of eminent opinions, and a useful research and historical document? ​

(kexue, could you possibly use the INDENT rather than the QUOTE tag in future (and perhaps even on those you can still edit), so that we don't have to read them in italics? And perhaps put the scientist's name in the post title? )

 

[unusualname]

I don't have to share these emails here with you, if it is not appreciated. I find them interesting and enlightening. If you don't or even find them unhelpful, I'll stop from now on.
.

Actually it is very interesting and enlightening. You should perhaps mention that you may post the replys for public viewing and discussion.

I agree with tom.stoer's point that doing the mathematical calculations gives you the best understanding and "feel" for what might be going on down there in microworld or what might be mathematical artifact.

No harm in asking a few people who have spent years truly mastering the subject what their understanding and feelings are though

 

[tom.stoer]

The meaning of brilliant physicists is always relevant in such a discussion.

@kexue: is you question more formal (what are virtual particles in a certain approach? how are these concepts to be translated? ...) or ontological? (what "are" particles? what "is" a quantum state? what "is" xyz? ...)

 

[nismaratwork]

Yes, we are playing "Quote The Physicist" … and why not?

Physics does change according to the majority accepted opinion, either because physicists change their minds, or because the stubborn old physicists eventually die off!

It's not conclusive, obviously, but it certainly helps to see how eminent reliable physicists approach the same question.

And this isn't simply a poll, with a yes/no answer … each physicist is giving reasons, and each reason in itself is a good starting-point for further study and discussion.

Perhaps in the end, all the quotes should be separated and copied onto a new sticky, as a permanent record of eminent opinions, and a useful research and historical document? ​

(kexue, could you possibly use the INDENT rather than the QUOTE tag in future (and perhaps even on those you can still edit), so that we don't have to read them in italics? And perhaps put the scientist's name in the post title? )

Eminent physicist have already make their approaches known in their published work, which I prefer to informal emails in this case. I don't mind these emails as much as I mind their use as support for some kind of general confusion or a particular definition. Even if Kexue's question to each is the same: "Are virtual particles real?", some might take that in a literal QFT manner, and others assume it's an ontological question... who knows. Some of these emails seem to trend more towards the latter, and while interesting, that should probably be in the philosophy section.

 

[Delta2]

I haven't read a lot about QFT or QED (feynman diagrams and path integrals still sound chinese to me ) but i myself was puzzled how quantum physics explain the electrostatic interaction since there are no time varying field in this case so there can be no photons emitted or absorbed.

So the virtual photons which explain it, if they don't exist for real, doesn't that mean that this is a big problem for QED?. Seems to me like we trying to force a theory to be in agreement with experiment by introducing the virtual photons.

In general why we can't detect virtual particles?. Is it due to limitations of technology or because virtual particles doesn't really exist, and we invent them to cover up the inefficiency of QFT?

 

[nismaratwork]

I haven't read a lot about QFT or QED (feynman diagrams and path integrals still sound chinese to me ) but i myself was puzzled how quantum physics explain the electrostatic interaction since there are no time varying field in this case so there can be no photons emitted or absorbed.

So the virtual photons which explain it, if they don't exist for real, doesn't that mean that this is a big problem for QED?. Seems to me like we trying to force a theory to be in agreement with experiment by introducing the virtual photons.

In general why we can't detect virtual particles?. Is it due to limitations of technology or because virtual particles doesn't really exist, and we invent them to cover up the inefficiency of QFT?

First... by definition a virtual particle is unobservable. Second, you're right, there are huge issues with QFT and QED, in part because of a reliance on math to lead the way... but it works. There is only a "huge" problem when the tricks in the math stop working, which they are yet to do... QM as a whole is like this. I think most people expect a 'next theory' to either eliminate those timescales (something like String Theory), or describe in a full manner what is happening.

The internal lines of a Feynman Diagram only exist in that diagram, even though they describe to a high degree of fidelity, a real interaction. The interaction is real, the description of that interaction as REALLY being comprised of virtual particles is not and was never meant to be.

 

[Delta2]

First... by definition a virtual particle is unobservable. Second, you're right, there are huge issues with QFT and QED, in part because of a reliance on math to lead the way... but it works.

What do you mean by definition unobservable? This is the nicest trick of all, to introduce particles which we say are unobservable but we base our theory on em. Then no one can prove the theory wrong at least not by trying to observe these particles because the theory will tell us "Stop! What you trying to do is meaningless , I told you the particles are unobservable! "

 

[tom.stoer]

@Delta²: Try to understand the paper I proposed in the thread https://www.physicsforums.com/showthread.php?t=445730 post #5,7 and my short summary in post #48 in this thread. You will see that the static potential is naturally explained via solving an equation of constraint and is rather similar to what one would expect in Maxell's theory (except for the fact that in QED the fields are always field operators!) w/o the need to introduce virtual particles.

This is not a problem, neither for QED nor for QCD. The problem is rather that one is taking these virtual particles and Feynman diagrams too literally and that one does not keep in mind that they are not to be understood as the definition of the theory but only as a calculational tool that is gauge dependend and that is valid in a certain limit only but will eventually break down!

Neither in QED nor in QCD is the perturbation expansion well-defined: even if the single terms are finite (or can be made finite via regularization / renormalization) the whole series (suming over infinitly many terms) will diverge! In QCD there are well know non-perturbative phenomena (e.g. instantons, vacuum tunneling) which do not scale with the coupling constant g but with 1/g which means that the limit g=0 (which is used in perturbation theory) is certainly not well defined. There are so-called Gribov ambiguities which spoil the standard path integral quantization but can be neglected in the perturbative regime [as an example: the pole in (1-x)^-1 is irrelevant for x~0]. There are low-energy phenomena like color confinement below the QCD scale which means that they are in principle beyond perturbation theory as this concept will break down near the QCD scale.

Once one forgets about perturbation theory, virtual particles become rather uninteresting. Looking at a non-perturbative definition of QCD (not in the strict mathematical sense) which is used in lattice gauge theories there are no virtual particles any more (OK, I have to admit that due to some computational issues on the lattice there may be fermions at tree level = quenched approx. or one-loop level).

 

[TrickyDicky]

Kexue, would you care to tell us what is your interpretation of virtual particles? Do you think they are literally particles popping in and out of existence as they are depicted in some popular science media or do you rather consider them a representation of some physical phenomenons not well known yet (which would explain why the perturbative way doesn't work so well in QCD or gravitation), if so do you have an alternative view that you can share with us?

 

[nismaratwork]

What do you mean by definition unobservable? This is the nicest trick of all, to introduce particles which we say are unobservable but we base our theory on em. Then no one can prove the theory wrong at least not by trying to observe these particles because the theory will tell us "Stop! What you trying to do is meaningless , I told you the particles are unobservable! "

Yeah, this isn't just some giant scam... I think tim.stoer outlined the rest quite well. To be literal, it's not a trick to say that a virtual particle would be unobservable... what's the big deal? It seems that you're trying to say that QFTs are somehow designed to be impossible to falsify, which is a pretty crazy claim to make without serious proof.

 

[haushofer]

Neither in QED nor in QCD is the perturbation expansion well-defined: even if the single terms are finite (or can be made finite via regularization / renormalization) the whole series (suming over infinitly many terms) will diverge!

I've never really understood what this means for QFT. Naïvely you would say that renormalization still doesn't make sense, because the series as a whole still becomes infinite. Could you elaborate a bit more on that? :)

 

[tom.stoer]

It has nothing to do with renormalization, but with the definition of the perturbation series itself. Let's write an amplite as

$$A(s,t) = \sum_n g^n A_n(s,t)$$

with

$$A_n(s,t) = \sum_i A_{ni}(s,t)$$

The sum over i is the sum over all Feynman diagrams for fixed order n. What renormalization does is to provide a means to regularize all these amplitudes such that

$$A^{\ast}(s,t) = \sum_n g^{\ast}(s)^n A^{\ast}_n(s,t)$$

where now all quantities are energy dependent and finite; infinities have been removed consistently. But that does not mean that the series

$$A(g) = \sum_n g^n A_n$$

as a whole is finite! Afaik in QFT one believes that perturbation series are divergent asymptotic expansions in the coupling constant

They are nevertheless interesting as the partial sums = truncation to finite N

$$A_{(N)}(g) = \sum_n^N g^n A_n$$

are an arbitrarily good approximation for small enough g.

Refer e.g. to http://mathworld.wolfram.com/AsymptoticSeries.html

 

[kexue]

Kexue, would you care to tell us what is your interpretation of virtual particles? Do you think they are literally particles popping in and out of existence as they are depicted in some popular science media or do you rather consider them a representation of some physical phenomenons not well known yet (which would explain why the perturbative way doesn't work so well in QCD or gravitation), if so do you have an alternative view that you can share with us?

Again, it is pretty much what selfadjoint described in his post.

Here's the idea. In quantum mechanics nothing is really real unless you can observe it or measure it. In order to be observable, a particle has to have some minimum amount of energy for some minimum amount of time; this comes out of the uncertainty principle that says the product of those two things has to be bigger than a certain number.

So it's possible to conceive of a particle whose energy is not big enough or whose lifetime is not long enough to permit a true quantum measurement, but still both of them could be greater than zero. The world could be full of such particles, and the measurements would never show it.

Well, quantum field theory takes those particles seriously. It says they interact with observable particles, for example they make the electron which emits and absorbs them a bit heavier, and a bit more sluggish in motion, than it would be if they didn't exist.

Furthermore, QFT says that the virtual particles are the ones that carry the forces. For example with photons, the "real" photons make light, and other forms of electromagnetic radiation, but the virtual photons carry the electric force; a charged particle is charged because it emits virtual photons. And the other bosons, that carry the weak and strong forces, behave the same way. Real particles interact with each other by exchanging virtual bosons.

This is the story quantum field theory tells, and the justification, the reason you should at least consider beliving in it, is that it makes fantiastically correct predictions. That bit above where I said that interacting with virtual particles made the electron sluggish? It's called the anomalous moment of the electron, and the prediction, based on virtual particles, matches experiment to six decimal places.

Do you disagree with what he says?

Furthermore, what selfadjoint calls real particles should be more correctly called slightly off-shell particles, since there are no asymptotic states in the real world, every particle is "virtual", an off-shell particle. On-shell particles are an idealization that never occurs in practice.

 

[kexue]

Once one forgets about perturbation theory, virtual particles become rather uninteresting.

Not at all. They transcend perturbation theory. They allows us to have one coherent quantum field theoretical picture of nature. Virtual photons, are not less real, not less an mathematical idea than the electromagnetic field. Can we really see an electromagnetic field? No. We can just feel its effects on how it changes charges.

QFT says that a sea of virtual photons, which are the excitations of an quantized electromagnetic field transmits momentum between two charges. But compared to the picture of an electomagnetic field moving the charges, this picture comes with the huge benefit in that it gives us one picture, a picture that describes field and particle behaviour. That is because when we more and more shake one of the two charges, we get more and more 'less off-shell photons', we turn increasingly "virtual" into "real" photons, we can detect more and more clicks in our measurement apparatus. Very few for radio waves, many more for waves with higher frequencies. These less off-shell photons can travel much farther until they get absorbed, they don't fall off with 1/r^2 as the "virtual" photons, the much more off-shell photons in the Coulomb field.

Only if a photon lives forever, moves forever, it would be on-shell. Every photon that gets created and absorbed is not.

This one beautiful picture of how nature works, and that is the picture of quantum field theory. Amen

 

[tom.stoer]

I think you didn't understand what I am talking about; I am questioning that perturbation theory and the definition of virtual particles that comes along with it is a safe means to define a QFT.

Can you give us a non-perturbative definition of a virtual photon?

 

[kexue]

I think you didn't understand what I am talking about; I am questioning that perturbation theory and the definition of virtual particles that comes along with it is a safe means to define a QFT.

Can you give us a non-perturbative definition of a virtual photon?

Didn't I just do that? I explained the 'off shell particle' view, the quantum field theory view on nature. No perturbation theory needed for that. As much there is no perturbation or non-perturbation calculation needed, to see that there is an energy-time uncertainty relation and the rule in quantum physics "everything that can happen, happens", which together implies to me the existence of so-called virtual particles.

 

[tom.stoer]

Hm, I don't see it; can you write down a formula that contains "non-perturbative virtual particles"?

 

[lugita15]

I don't understand the point of this argument. Surely there will be no contradiction with experiment if we assume that virtual particles are really there. Also, the terms in the perturbative expansion are easier to understand if we think about them in terms of virtual particles. That's a good enough reason to assume they exist.

There are few things in physics that we can guarantee really exist. Can you prove that quarks exist, or for that matter the wave function?

Without using virtual particles, how would you answer the question of why the bare mass of an electron differs from the actual mass?

 

[tom.stoer]

Also, the terms in the perturbative expansion are easier to understand if we think about them in terms of virtual particles. That's a good enough reason to assume they exist.

As I tried to explain above the whole perturbation expansion is ill-defined in many cases. But as soon as one goes to non-perturbative techniques the whole concept of virtual particles ceases to exist. That's why I was asking for a formula that shows what a "non-perturbative virtual particle" is. I am really interested to see that.

Let me quote chapter 9.3 from

http://lanl.arxiv.org/abs/quant-ph/0609163v2
Quantum mechanics: Myths and facts
H. Nikolic
(Submitted on 21 Sep 2006 (v1), last revised 16 Apr 2007 (this version, v2))

9.3 Virtual particles?
The calculational tool represented by Feynman diagrams suggests an often abused picture according to which “real particles interact by exchanging virtual particles”. Many physicists, especially nonexperts, take this picture literally, as something that really and objectively happens in nature. In fact, I have never seen a popular text on particle physics in which this picture was not presented as something that really happens. Therefore, this picture of quantum interactions as processes in which virtual particles exchange is one of the most abused myths, not only in quantum physics, but in physics in general. Indeed, there is a consensus among experts for foundations of QFT that such a picture should not be taken literally. The fundamental principles of quantum theory do not even contain a notion of a “virtual” state. The notion of a “virtual particle” originates only from a specific mathematical method of calculation, called perturbative expansion. In fact, perturbative expansion represented by Feynman diagrams can be introduced even in classical physics [52, 53], but nobody attempts to verbalize these classical Feynman diagrams in 33 terms of classical “virtual” processes. So why such a verbalization is tolerated in quantum physics? The main reason is the fact that the standard interpretation of quantum theory does not offer a clear “canonical” ontological picture of the actual processes in nature, but only provides the probabilities for the final results of measurement outcomes. In the absence of such a “canonical” picture, physicists take the liberty to introduce various auxiliary intuitive pictures that sometimes help them think about otherwise abstract quantum formalism. Such auxiliary pictures, by themselves, are not a sin. However, a potential problem occurs when one forgets why such a picture has been introduced in the first place and starts to think on it too literally.

 

[kexue]

Hm, I don't see it; can you write down a formula that contains "non-perturbative virtual particles"?

The canonical quantization process, where we have quantized classical fields is intrinsically perturbative, non-perturbartive terms can not be computed.

The path integral quantization is in principle non-perturbative, but we integrate over paths in the space of classical field configurations.

So we have two different ways of doing the calculations.

I *speculate* that it is like asking the question in quantum mechanics, what is true, the path integral approach or the canonical quantization? Both are true!

If you do not like the picture of quantized fields acting on the vacuum and popping out particles, then take paths in classical function space, but make sure to integrate over all paths even the virtual ones, paths that are not allowed by classical mechanics.

 

[weejee]

The canonical quantization process, where we have quantized classical fields is intrinsically perturbative, non-perturbartive terms can not be computed.

The path integral quantization is in principle non-perturbative, but we integrate over paths in the space of classical field configurations.

So we have two different ways of doing the calculations.

I *speculate* that it is like asking the question in quantum mechanics, what is true, the path integral approach or the canonical quantization? Both are true!

If you do not like the picture of quantized fields acting on the vacuum and popping out particles, than take paths in classical function space, but make sure to integrate over all paths even the virtual ones, paths that are not allowed by classical mechanics.

Being non-classical (in terms of path integral) and being virtual (in terms of perturbation theory) are two different things.

 

[nismaratwork]

The canonical quantization process, where we have quantized classical fields is intrinsically perturbative, non-perturbartive terms can not be computed.

The path integral quantization is in principle non-perturbative, but we integrate over paths in the space of classical field configurations.

So we have two different ways of doing the calculations.

I *speculate* that it is like asking the question in quantum mechanics, what is true, the path integral approach or the canonical quantization? Both are true!

If you do not like the picture of quantized fields acting on the vacuum and popping out particles, than take paths in classical function space, but make sure to integrate over all paths even the virtual ones, paths that are not allowed by classical mechanics.

Can you write the formula or not? Let the formula do the talking, because this is getting really old, REALLY fast.

 

[kexue]

Being non-classical (in terms of path integral) and being virtual (in terms of perturbation theory) are two different things.

Elaborate, please.

 

[weejee]

Elaborate, please.

Even for a free field theory, which doesn't involve any virtual particles whatsoever, we need to integrate over all possible paths.

 

[tom.stoer]

The canonical quantization process, where we have quantized classical fields is intrinsically perturbative, non-perturbartive terms can not be computed.

The path integral quantization is in principle non-perturbative, but we integrate over paths in the space of classical field configurations.

I am sorry to say that but this is simply wrong!

Canonical quantization is used in QCD to calculate non-perturbative effects like chiral symmetry breaking, confinement etc. There are explicit expressions for the Hamiltonian in several gauges. There are explicit effects scaling with 1/g. There is no reason why this should not work in this formalism.

The path integral as we know it from standard QCD textbooks is typically perturbative only as it suffers from Gribov ambiguities which are not well under control. Exponentiating the Fadeev-Popov determinant somehow hides these shortcomings. I agree that these issues can be resolved along the same lines as in the canonical approach, but unfortunately this is not always not taken into account properly.

 

[kexue]

Can you write the formula or not? Let the formula do the talking, because this is getting really old, REALLY fast.

Doesn't this forum has any mentors that could point out to this poster that a civilized and respectful tone is helpful in discussions?

 

[kexue]

I am sorry to say that but this is simply wrong!

Canonical quantization is used in QCD to calculate non-perturbative effects like chiral symmetry breaking, confinement etc. There are explicit expressions for the Hamiltonian in several gauges. There are explicit effects scaling with 1/g. There is no reason why this should not work in this formalism.

The path integral as we know it from standard QCD textbooks is typically perturbative only as it suffers from Gribov ambiguities which are not well under control. Exponentiating the Fadeev-Popov determinant somehow hides these shortcomings. I agree that these issues can be resolved along the same lines as in the canonical approach, but unfortunately this is not always not taken into account properly.

I can not judge this. But it is written down in Michele Maggiore A Modern Introduction to Quantum Field Theory, page 219.

 

[nismaratwork]

Doesn't this forum has any mentors that could point out to this poster that a civilized and respectful tone is helpful in discussions?

As tom and others keep pointing out, for page after page... you continually make statements as though they're fact, when they are blatantly wrong. When asked simply to support your view with a formula, you evade. So, will you write it out, or not? You're clearly not some hapless newcomer to QM, so it seems odd that you make these sweeping generalizations, share a number of emails, but you won't write out an equation to support your point when politely asked by tom.stoer? I don't think you want mentors going over your posts kexue, you're no exactly being the most helpful conversationalist.