Are virtual particles created spontaneously out of the vacuum?

[Rhizomorph]

When QM physicists refer to virtual particles being created spontaneously out of the vaccum, which particles are the referring to? I know of examples with photons, electrons and positrons, but does this occur with other particles as well - gravitons, quarks, WZ Bosons? Have these been observed?

 

[marlon]

You got to be careful here on what them virtual particles really are. Basically they are NOT physical thus you will never observe them. Then why are they used ?

Well, they are nothing more then a mathematical trick used to describe the interactions between all possible elementary particles. So virtual particles occur in ALL interactions between ALL elementary particles.

Now, there is a possibility to turn the virtual particles into real physical particles for a very short time. What you need is enough "external" energy that is used to make them real. The energy gives them a reason to exist, so to speak...This is possible because in QM and in QFT, energy-conservation can be violated for a very short while due to the Heisenberg-uncertainty-principle...

Now in QFT the vacuum is not empty, it is constituted out of an infinite amount of virtual particles that can become real when there is enough energy available. These virtual particles can be all the possible force carriers like photons, gluons, vector-bosons because they mediate the interactions between the fermionic matter-fields like the electrons, positrons, and quarks.

Ofcourse fermions can also be virtual and then become real when enough energy is available...You only got to make sure that the conservation laws for spin, letpon-number, and so on are always respected. Only energy-conservation can be violated for a short while...
regards
marlon

 

[ZapperZ]

I would like to point out that there is a risk in indicating that something isn't "real" or "practical", and that it is simply a "mathematical trick" or convenience. If that is true, then the concept of virtual particles would not predict anything new or deviate from conventional picture of "fields". Now we know that this is certainly isn't the case. Using such virtual particle interactions, QED can make some amazing predictions BEYOND classical field theory, so much so that we are gasping at the accuracy of the value of the electronic gyromagnetic ratio between theory and experiment. The concept of virtual particles also play a role in predicting the Casimir effect, which simply would not come about if we just stick to classical fields.

So are virtual particles "real" or not? Are they merely a mathematical invention? Can a mathematical artifact predict new physical phenomena? They may not be "observed" directly (but then again, what does?), but if their presence can affect and correct existing measurements, and also predict new phenomena, I certainly would not want to say they are not "physical".

Zz.

 

[marlon]

I would like to point out that there is a risk in indicating that something isn't "real" or "practical", and that it is simply a "mathematical trick" or convenience. If that is true, then the concept of virtual particles would not predict anything new or deviate from conventional picture of "fields". Now we know that this is certainly isn't the case. Using such virtual particle interactions, QED can make some amazing predictions BEYOND classical field theory, so much so that we are gasping at the accuracy of the value of the electronic gyromagnetic ratio between theory and experiment. The concept of virtual particles also play a role in predicting the Casimir effect, which simply would not come about if we just stick to classical fields.

So are virtual particles "real" or not? Are they merely a mathematical invention? Can a mathematical artifact predict new physical phenomena? They may not be "observed" directly (but then again, what does?), but if their presence can affect and correct existing measurements, and also predict new phenomena, I certainly would not want to say they are not "physical".

Zz.

Hmm,
I think this post will make this thread evolve into a discussion on personal opinions...let's focus on the facts...

Stating that a mathematical trick never leads to something new is just wrong. The virtual particles were introduced in QED in order to calculate the interactions between fermionic matterfields using perturbation theory and the corresponding Feynmann-diagrams. They are certainly not to be looked at as real physical particles. This is a misconception that occurs very often so it is necessary to stress this difference, especially to student who are just getting to know these concepts.

Besides nobody is questioning the accuracy of QED here, i think everybody studying this subject will be familiar with Feynmann's analogy of the distance between the top/bottom of our head to the moon.

Virtual particles are NOT REAL yet they do mediate interactions and physical phenomena that ARE REAL and observable. The Casimir-effect is explained using the virtual particles but it does not show that virtual particles are real...
A particle being not real does not mean it does not exist...
In the description of the interaction between elementary particles in quantum field theory, a virtual particle is a temporary elementary particle, used to describe an intermediate stage in the interaction. A virtual particle can never be the end-result of a process, that is what i mean that they are not real.

I also pointed out that virtual particles can only become real for a very short while.

If they were real then they would be they result of some interaction and they would not be an intermediate stage during the interaction...

regards
marlon

 

[selfAdjoint]

Marlon, as you say, this view of virtual particles is a personal one, if increasingly popular with the younger generation of physicists. But it is not universally held. I would like to repeat my question, what is your opinion of the dressed electron, and its different mass from the bare electron? Are these real differences? Since there is experimental measuremant how can they not be? And then what is your account of the Casimir effect?

 

[marlon]

Rhizo,

This site gives a good example of the socalled vacuum fluctuations involving virtual particle - antiparticle pairs that are created out of "nothing" and die a short period later. During their existence they can influence real physical phenomena...Check out the famous Casimir-effect.

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

regards
marlon

 

[ZapperZ]

Stating that a mathematical trick never leads to something new is just wrong. The virtual particles were introduced in QED in order to calculate the interactions between fermionic matterfields using perturbation theory and the corresponding Feynmann-diagrams. They are certainly not to be looked at as real physical particles. This is a misconception that occurs very often so it is necessary to stress this difference, especially to student who are just getting to know these concepts.

Besides nobody is questioning the accuracy of QED here, i think everybody studying this subject will be familiar with Feynmann's analogy of the distance between the top/bottom of our head to the moon.

Virtual particles are NOT REAL yet they do mediate interactions and physical phenomena that ARE REAL and observable. The Casimir-effect is explained using the virtual particles but it does not show that virtual particles are real...
A particle being not real does not mean it does not exist...
In the description of the interaction between elementary particles in quantum field theory, a virtual particle is a temporary elementary particle, used to describe an intermediate stage in the interaction. A virtual particle can never be the end-result of a process, that is what i mean that they are not real.

I also pointed out that virtual particles can only become real for a very short while.

If they were real then they would be they result of some interaction and they would not be an intermediate stage during the interaction...

regards
marlon

I'm not saying they are real. I just don't think they should be looked at simply as a mathematical artifact or "unphysical". By saying that you are implying that we made it up, and that there shouldn't be any problem if we simply eliminate them, since they weren't "real" in the first place. The fact that we do not "observe" them directly should not be used as a criteria for something to be "real" or not.

Let's discuss this another way to make sure we are not simply talking about preferences. Let's say I have a charge q at a distance d from an infinite conducting plane. The typical issue here would be to find the E field in the space between the charge and the conducting plane, and the charge distribution on the conducting plane itself. Now if we try to solve the Poisson's equation for this, it will be a pain. However, we know via the Uniqueness theorem that if I can find a similar situation that satisfies the same Dirichlet or Neumann boundary conditions, then I will have found a unique solution up to an additive constant. That is why we can replace the conducting plane with an image charge. This is a much simpler situation to solve. The image charge isn't real, it is unphysical, and purely a mathematical convenience/artifact. If the conducting plane is at z=0 and z>0 represents the space that the charge q is in, then z<0 is an unphysical region. We can solve for z<0 region, but this solution is purely a mathematical artifact. There's nothing meaningful about it.

Now, if for some odd reason, by replacing the conducting plane with an image charge, I find that not only did it satisfy all the Dirichlet and/or Neumann boundary conditions as the original problem, I also find that by doing so, it gave EXTRA values that was not in the orginal situation for z>0 region, then something is different here. There is now a distinct deviation between the two situations. If I make a measurement and verify that this deviation is actually present, then what I orginally thought to simply be a mathematical entity (the image charge), is no longer that. There is a component of some reality to this situation that is closer to what Nature is then the original situation of solving a charge q in front of a conducting plane. Replacing the plane with an image charge is CLOSER to reality than the original description. So in this case, while the image charge is STILL not something you "observe" directly, its description is more accurate!

Of course, this doesn't occur in E&M with image charge, but this is what is occurring when we replace the classical fields with virtual interactions. By including them in, we get a zoo of ADDITIONAL interactions that simply were not predicted by classical fields. Real particles can scatter off these virtual particles, producing corrections that are simply not there in the classical picture. These particles may be virtual, but they have a real set of properties that we can detect and measure via their interactions with others.

While calling them real, unphysical, mathematical, etc. may be a question of semantics, I don't think their existence is a matter of tastes or preferences.

Zz.

 

[jcsd]

Usually whether or not these 'mathematical conviences' are 'real' or not is just a matter of ontology, but aren't virtual particles artifacts specifically of applying peturbation theory to QED and therefore not necessarily a prediction of non-peturbative QED?

 

[ZapperZ]

Usually whether or not these 'mathematical conviences' are 'real' or not is just a matter of ontology, but aren't virtual particles artifacts specifically of applying peturbation theory to QED and therefore not necessarily a prediction of non-peturbative QED?

The application of virtual particles is more general than that - it is more of a QFT, rather than for a specfic QED technique. So anything that uses QFT - even in condensed matter - would make use of virtual interactions.

Secondly, I don't think it is restricted to simply perturbative scenario. Even without including any higher order corrections, the "coupling" constant at each vertex of interaction involves the emission or absorption of a virtual particle.

Zz.

 

[marlon]

I would like to repeat my question, what is your opinion of the dressed electron, and its different mass from the bare electron? Are these real differences?

Well, what is renormalization ?
If someone would ask me this question the answer would be : Renormalization is a "mathematical" system that turns the infinite unphysical bare parameters of the Lagrangian of the field theory into finite dressed parameters that are physical. Why are the dressed parameters the fysical ones ? Well because they are the "quantities" that we measure in experiments. The naked parameters are unfysical because they do not describe properties of the particles that are DIRECTLY measurable...

The view of QFT is that the naked property let's take the mass is surrounded by a cloud of virtual particles that undergo all kinds of creation, annihilation and scattering-processes and as a consequence of this they dress the naked particle into the real physical particle that we observe. Here it is again : the virtual particles on themselves are NOT real, yet their contributions due to their interactions is REAL. So virtual particles DO deliver a contribution that we can observe when it is summed up with the naked parameters. We do not observe the virtual particles themselves, we only see (indirectly) a manifestation of their INTERACTIONS (just like in the Casimir-effect). All these interactions are lumbed up into the socalled self-interaction, which itself can be infinite in perturbationtheory. Well, as a matter of fact it is infinite. We solve this by stating that the naked mass for example is infinite itself (with opposite sign as the infinity of the self-energy) so these two concepts (naked mass + self-energy) sum up to an finite one : the dressed real mass that we observe.

These are examples of coupling-constant-renormalization (ie introduction of the surrounding virtual cloud) and mass-renormalization.

Basically that is my view (and all other younger physicist born after 1975 )

regards
marlon

 

[marlon]

Real particles can scatter off these virtual particles, producing corrections that are simply not there in the classical picture. These particles may be virtual, but they have a real set of properties that we can detect and measure via their interactions with others.

While calling them real, unphysical, mathematical, etc. may be a question of semantics, I don't think their existence is a matter of tastes or preferences.

Zz.

i agree with this statement

 

[marlon]

Usually whether or not these 'mathematical conviences' are 'real' or not is just a matter of ontology, but aren't virtual particles artifacts specifically of applying peturbation theory to QED and therefore not necessarily a prediction of non-peturbative QED?

This is correct, this is how virtual particles were "born" yet they are certainly used in non-perturbative QED and QFT generally

regards
marlon

 

[humanino]

what is your opinion of the dressed electron, and its different mass from the bare electron? Are these real differences? Since there is experimental measuremant how can they not be?

With all the respect due to a scientist such as you sA, I was wondering how the question can be addressed at all ? We have to renormalize because we must admit our ignorance of what could happen at very high energy or very small distance. Then, how could we say the naked particle has a physical meaning at all ? We cannot measure it. The naked parameters are unphysical to me

 

[marlon]

I share the same opinion as Humanino sA, i wrote this down in the above post on renormalization, although we can manage the properties of phenomena at high energies (certainly in theory)

marlon

 

[ZapperZ]

With all the respect due to a scientist such as you sA, I was wondering how the question can be addressed at all ? We have to renormalize because we must admit our ignorance of what could happen at very high energy or very small distance. Then, how could we say the naked particle has a physical meaning at all ? We cannot measure it. The naked parameters are unphysical to me

At the risk of putting words into SelfAdjoint's mouth, I think you are misinterpreting what he is asking.

If a "naked particle" is unphysical, and yet, a "dressed particle" is real since we can measure it, but yet again, it is dressed due to all the virtual particles surrounding it, then how can we say virtual particles aren't real? I think that was selfAdjoint's original question to Marlon. If the dressed particle is real, how can we then turn around and say the dressing isn't?

SelfAdjoint can correct me if I misread his intentions...

Zz.

 

[humanino]

You had 666 posts zZ, it scared me

Thank you for the precision. To me (this is a personal opinion of a young person) the virtual particles are not real, neither the naked ones. Only the dressed ones makes sens. The dressing process by itself would not correspond to anything else than a mathematical clever trick, to make our model consistent with reality.

This is all a matter of taste and color however

 

[humanino]

I could not be so short it is confusing.

I do attribute reality to the effectiveness of saying "one virtual particule is exchanged in a deep inelastic process". I do not attribute reality to imagining that this correspond to a single exchange of a massive photon for instance. As Zee put it "the vacuum is boiling sea of emptiness, full of sound and fury, and significating a great deal" or something close. When we try to measure the charge of an electron by "getting a probe too close to it" we trigger many flucuations around which cannot be accurately computed, only effectively described by a dressing process.

 

[marlon]

it is dressed due to all the virtual particles surrounding it, Zz.

Well, this is my whole point. A real particle is NOT dressed by the virtual particles surrounding it, it is dressed due to the INTERACTIONS of these virtual particles...

The interactions are real and observable, the virtual particles themselves are not...

regards
marlon

 

[jcsd]

The application of virtual particles is more general than that - it is more of a QFT, rather than for a specfic QED technique. So anything that uses QFT - even in condensed matter - would make use of virtual interactions.

Secondly, I don't think it is restricted to simply perturbative scenario. Even without including any higher order corrections, the "coupling" constant at each vertex of interaction involves the emission or absorption of a virtual particle.

Zz.

So then it is just a matter of ontology then!

 

[ZapperZ]

Well, this is my whole point. A real particle is NOT dressed by the virtual particles surrounding it, it is dressed due to the INTERACTIONS of these virtual particles...

The interactions are real and observable, the virtual particles themselves are not...

regards
marlon

Not if you accept the Higgs mechanism. Then a naked particle has NO MASS until it is "dressed" around a cloud of higgs bosons. In a condensed matter system, you NEVER measure the "naked" particles. All that you measure are the dressed particles. Based on this, an undressed particle is NOT real, since you never measure them, at least not directly.

Zz.

 

[marlon]

Not if you accept the Higgs mechanism. Then a naked particle has NO MASS until it is "dressed" around a cloud of higgs bosons. In a condensed matter system, you NEVER measure the "naked" particles. All that you measure are the dressed particles. Based on this, an undressed particle is NOT real, since you never measure them, at least not directly.

Zz.

I certainly accept the Higgs-mechanism as the mass-giving mechanism in QFT(who am i to deny it, i ain't no Stephen Hawking and even he is not always right, certainly not about the information-paradox). I never said that a naked particle is physical and indeed you never measure it, so i don't see why You say that. In my previous post (the long one as a reply to the question of sA) i clearly stated this fact.

Your statement on a naked particle being surrounded by Higgs-bosons is wrong in my opinion. All elementary particles are massless because mass mixes up the two different chiralities and these two exhibit clear and different physical properties. Once again, the mass is NOT acquired because of this surrounding, it is acquired because of interactions between the elementary particles and the quanta of the Higgs-field. The Higgsbosons are absorbed by the massless elementary particles and this PROCESS yields the acquired and NAKED mass of a particle.In QFT mass is the coupling constant of the Higgs-mechanism expressing the strength of the interaction involved here...

It seems to me you are mixing things up here : the bare and dressed particle is not the same as the massless elementary particle that acquires mass through the Higgs mechanism. I think you are forgetting effects like dynamical mass generation described by the virtual cloud and the corresponding self-energy ONCE the spontaneous symmetry breaking has occurred and the Higgs-mechanism has done its work, yielding the naked mass-parameter of the field theory.

Look, basically the Higgs field is just another trick . It is there in the QCD-vacuum to make sure that the vacuum state is degenerate so that sponateous symmetry-breaking can definitely occur...

The dressing itself has a real physical meaning (once it is renormalized) ("the fluctuations") yet it is not directly observable...
regards
marlon

 

[humanino]

Ignorance is no shame

That is what Zee writes on the blackboard to introduce renormalization. I do have better references than Zee's nutshell, but I like this book. I hope you won't get too annoyed by this. It illustrates to me the necessity to cut off our ignorance of high energy spurious contributions from the model, whose validity will always be bounded by experimental accessible energies. We have to put on blurry glasses to read QFT. He also quotes Bjorken and Drell's on the renormalization group (1965) :

Therefore, conclusions based on the renormalization group arguments ... are dangerous and must be viewed with due caution. So is it with all conclusions from local relativistic field theories.

Of course, this "immensely rich and multifaceted"subject has made huge progress since then, but it illustrates how misleading conclusions can be easily done. I have an experimentalist point of view, from which only renormalized quantities are physical. The naked quantities should not, neither the "blurry glasses" one has to put. The dressing itself correspond to a physical phenomena of increasing fluctuations at short distances, yet we cannot access to a detailed description of this dressing, it is by definition not accessible.

 

[ZapperZ]

The dressing itself has a real physical meaning (once it is renormalized) ("the fluctuations") yet it is not directly observable...
regards
marlon

Can you give me an example of something that you consider to be "directly observable"?

Zz.

 

[marlon]

Can you give me an example of something that you consider to be "directly observable"?

Zz.

What i was trying to say here is that in QFT the "dressing" and more over the corresponding interactions are described in the same way as the vacuum fluctuations (well, in an analogue manner). The vacuum fluctuations are directly observable in the Casimir-effect yet the dressing of particles is not. We only "see" the contributions that give rise to the total dressed particle that we observe...

 

[ZapperZ]

What i was trying to say here is that in QFT the "dressing" and more over the corresponding interactions are described in the same way as the vacuum fluctuations (well, in an analogue manner). The vacuum fluctuations are directly observable in the Casimir-effect.

But really, what do you consider to be something that is directly observable, and thus, "real"?

Zz.

 

[marlon]

Can you give me an example of something that you consider to be "directly observable"?

Zz.

This is not what I "consider" to be directly observable, it is a well established fact...this is not some personal opinion...let's stay to the facts

 

[marlon]

But really, what do you consider to be something that is directly observable, and thus, "real"?

Zz.

Err, i could have sworn i answered that question a few seconds ago?

Again : THE CASIMIR EFFECT

 

[ZapperZ]

Err, i could have sworn i answered that question a few seconds ago?

Again : THE CASIMIR EFFECT

So what makes the casimir effect "directly observable" and thus "real"? Because you can measure a set of properties and observations that fit into a description of what it is?

Zz.

 

[humanino]

But really, what do you consider to be something that is directly observable, and thus, "real"?

:uhh:
Light approximate straight path, so lightlike intervals should be a good idea.
I could not figure anything more directly "observable"

 

[marlon]

So what makes the casimir effect "directly observable" and thus "real"? Because you can measure a set of properties and observations that fit into a description of what it is?

Zz.

?

Is this a rethorical question ?

A phenomenon predicted and described by theory and backed up by experimental results or the other way around.

Finally, i ask the same question to YOU

 

[humanino]

Marlon, I think zZ might refer to the fact that evidences for the Casimir effect are much involved, they require a complicated set-up with several levels of devices. So, "directly observable" might be misadaptated to qualify the Casimir effect evidence.

 

[marlon]

Marlon, I think zZ might refer to the fact that evidences for the Casimir effect are much involved, they require a complicated set-up with several levels of devices. So, "directly observable" might be misadaptated to qualify the Casimir effect evidence.

Ok, i see your point and i agree. Nevertheless this effect IS observable and that is what I meant...

Thanks for your remark, maybe I was seeing things wrong here...

regards
marlon, ton collègue dans le show-bizz

 

[ZapperZ]

?

Is this a rethorical question ?

A phenomenon predicted and described by theory and backed up by experimental results or the other way around.

Finally, i ask the same question to YOU

I have given you the answer of what I consider to "exist". EVERYTHING that we know of are describe by a set of properties. How do I know that I'm observing an electron? Because an electron has so-and-so properties. I also know when the electron has been "renormalized", because I know all the "rules" that it has to follow, while keeping other set of properties constant.

My point here being that THAT is how we define everything, via their set of properties that they must exhibit. Now how do I define a quark? I define a quark by so-and-so properties. The same can be said about the casimir effect, superconductivity, etc.. etc. EAch one of these have a set of properties and characteristics that make them what they are. I can easily apply that to any virtual particles. Each one of them has a set of properties that define what they are, what they can do, and how they interact.

Now in NONE of these properties above is there any such thing as a "direct observation". That phrase is meaningless and vague. Everything is defined by its set of properties and how these properties manifest themselves VIA interactions. So to say that virtual particles are "unphysical" because you can't observe them directon is misleading, because it implies that there ARE things you can "observe directly". That is why I asked you for an example of what you could categorize as a "direct obseration".

The issue here isn't the confusion between you and me and the rest of the "community" here who have studied these things and understood what they are beyond just words and phrases. The issue here is that we have people reading this who may not be aware of such distinctions. I hate to think that our explanations give misleading impression of what we mean, which is why a careful choice of words and phrases we use should always be kept in mind.

Zz.

 

[humanino]

The issue here isn't the confusion between you and me and the rest of the "community" here who have studied these things and understood what they are beyond just words and phrases. The issue here is that we have people reading this who may not be aware of such distinctions. I hate to think that our explanations give misleading impression of what we mean, which is why a careful choice of words and phrases we use should always be kept in mind.

Yes that is very true and very important. Thank you for refreshing my memory at least. We cannot casually discuss here like in MSN chat. :uhh:

 

[marlon]

I have given you the answer of what I consider to "exist". EVERYTHING that we know of are describe by a set of properties. How do I know that I'm observing an electron? Because an electron has so-and-so properties. I also know when the electron has been "renormalized", because I know all the "rules" that it has to follow, while keeping other set of properties constant.

My point here being that THAT is how we define everything, via their set of properties that they must exhibit. Now how do I define a quark? I define a quark by so-and-so properties. The same can be said about the casimir effect, superconductivity, etc.. etc. EAch one of these have a set of properties and characteristics that make them what they are. I can easily apply that to any virtual particles. Each one of them has a set of properties that define what they are, what they can do, and how they interact.

Now in NONE of these properties above is there any such thing as a "direct observation". That phrase is meaningless and vague. Everything is defined by its set of properties and how these properties manifest themselves VIA interactions. So to say that virtual particles are "unphysical" because you can't observe them directon is misleading, because it implies that there ARE things you can "observe directly". That is why I asked you for an example of what you could categorize as a "direct obseration".

The issue here isn't the confusion between you and me and the rest of the "community" here who have studied these things and understood what they are beyond just words and phrases. The issue here is that we have people reading this who may not be aware of such distinctions. I hate to think that our explanations give misleading impression of what we mean, which is why a careful choice of words and phrases we use should always be kept in mind.

Zz.

I agree with the last paragraph but i already stated this in a previous post so that is not really something new. This is the very reason I think there should be a clear distinction between a virtual particle and a "non-virtual" particle, if you will...

You are saying things i never said though. I never said that just because you cannot see something it is not real. That is a very simplistic view and it is just wrong in many cases. But i think that most of the people here are aware of that.

You most certainly CAN NOT apply "a set of properties and characteristics that make them what they are" as the criterium as to whether a particle is real or not, it is rather vague and meaningless(i mean by this :"based upon personal opinion" : this is saying everything and nothing at the same time). The big difference is this one (i stated this before) : virtual particles are just an intermediate stage in interactions between real physical particles (both bosonic force carriers as well as fermionic matter-particles). They will never be the "end-result" of such interactions because of the "restoration" of energy-conservation...

Indeed you know an electron because it has so and so properties, i think that is general knowledge. Yet you do NOT know some virtual particle because it has so and so properties that are of the same "structure" as these of the electron. I mean you know a virtual particle because there is a so and so interaction going on and a so and so conservation law needs to be respected. Ofcourse a virtual particle has certain properties (but here we are again with the "saying of everything and nothing at the same time"), that is something everybody knows. My point is that these properties are of totally different nature as those used to describe or "recognize" the electron.
You will never "acquire" a real particle because of a short-term violation of the energy-conservation law...that is a FUNDAMENTAL difference, not just some other property...

The difference is so big that a fundamental distinction needs to be made when it comes to the classification of these two types of particles. Just look at the effort it took the QFT-guys to find the difference between a socalled elementary particle and some particle that is not elementary. I think this fantastic road that theoretical physics has traveled is the best proof for a clear and strong distinction between virtual and real...

marlon