Misconceptions about Virtual Particles - Comments

[A. Neumaier]

It appears to me that here the classical stance about particles instead of the quantum field view is being used to create an artificial debate about the "existence" of particles whether "real" or "virtual".

Nothing in the article assumes classical systems; everything is about the standard formalism for describing quantum particles. These appear even in QFT to describe asymptotic scattering states. The asymptotic description is valid once the distances are mesoscopic, i.e., before and after collisions.

During interactions, the particle picture breaks down completely. It cannot be used to justify virtual particles as agents in what happens.

The number of particles is not conserved

Nothing in the article assumes a conservation law for particle number.

there exists no non-perturbative relativistic QFT in physical 4-space

The frequently used resummation of infinitely many diagrams and renormlization group techniques are non-perturbative feature; so are lattice calculations. That no mathematically rigorous existence proof exists has no impact at all on the very broad research on nonperturbative aspects of QFT.

Thus all your comments are inappropriate.

 

[Tendex]

Nothing in the article assumes classical systems; everything is about the standard formalism for describing quantum particles. These appear even in QFT to describe asymptotic scattering states. The asymptotic description is valid once the distances are mesoscopic, i.e., before and after collisions.

Asymptotic states are mathematical objects as "real" or not as propagators(VEV of time-ordered products which is the math object usually associated to the "virtual particle" concept by field theorists that use that unfortunate terminology)

During interactions, the particle picture breaks down completely. It cannot be used to justify virtual particles as agents in what happens.

I don't know who tries to justify such nonsense as stated. As I said my considerations apply already for free complex scalar quantum fields. I just don't know why certain mathematical abstractions used to obtain certain results seem more "existent" or "real" than others to you. Math objects are math objects, makes no sense to say that an state at future infinity is more real than a vacuum expectation value of time-ordered product. Certainly all the mathematical machinery including particularly the Feynman propagators is needed to justify observations within QFT. The fact that you seem to like better some mathematical objects than others according to some ill defined philosophy of yours is irrelevant here.

Nothing in the article assumes a conservation law for particle number.

Your use of quantum states to justify your philosophy about notions usually considered in this site as outside physics, like reality and existence suggested to me that you are considering only states with fixed numbers of particles when already for quantized free Klein-Gordon complex scalar fields this is untenable.

The frequently used resummation of infinitely many diagrams and renormlization group techniques are non-perturbative feature; so are lattice calculations. That no mathematically rigorous existence proof exists has no impact at all on the very broad research on nonperturbative aspects of QFT.

That a broad research for a mathematically valid non-perturbative theory is ongoing just goes to restate what I wrote about its inexistence, so your reply is not adding anything to my correct statement.

Thus all your comments are inappropriate.

I don't see why. What I think is inappropriate in a physics discussion is introducing cheap philosophical opinions about what "exists" means or about what's "real" disguised as science while apparently nobody else is allowed to do it. More so when it is done ignoring what the causality of measuremnts/detections means in RQFT

 

[A. Neumaier]

I just don't know why certain mathematical abstractions used to obtain certain results seem more "existent" or "real" than others to you.

Well, I gave a clear rationale: To be able to talk about anything in a quantitative way one must at least be able to calculate probabilities for it. Thus the ability to calculate probabilities (in principle) is a suitable criterion for distinguishing the real from the purely conceptual. That's why the state is the key to assign reality, whether to particles or to fields.

What is regarded as real by everyone (e.g., macroscopic objects, unstable atoms, particles colliding, the electromagnetic field) clearly has states determining what is observable about it (e.g., expectation values, decay probabilities, reaction cross sections, linear response).

On the other hand, virtual particles lack all this. For example, to assign to them existence for a short time (how short?) is purely fictitional, and cannot be backed up by any calculation.

That a broad research for a mathematically valid non-perturbative theory is ongoing just goes to restate what I wrote about its inexistence

It shows its existence. Approximate nonperturbative results are still nonperturbative.

If you only count as physics what is rigorously and exactly established, very little of the large edifice of physics remains.

 

[Tendex]

Well, I gave a clear rationale: To be able to talk about anything in a quantitative way one must at least be able to calculate probabilities for it. Thus the ability to calculate probabilities (in principle) is a suitable criterion for distinguishing the real from the purely conceptual. That's why the state is the key to assign reality, whether to particles or to fields.

What is regarded as real by everyone (e.g., macroscopic objects, unstable atoms, particles colliding, the electromagnetic field) clearly has states determining what is observable about it (e.g., expectation values, decay probabilities, reaction cross sections, linear response).

On the other hand, virtual particles lack all this. For example, to assign to them existence for a short time (how short?) is purely fictitional, and cannot be backed up by any calculation.

You are again using vague philosophical distinctions of what is real or not real that are not accepted in mathematical physics discussions , what is your mathematical definition of "virtual particle" for a quantized KG free field? Is it not related to the mathematical conditions(causal propagators) that allow to use them to obtain vacuum expectation values?

It shows its existence. Approximate nonperturbative results are still nonperturbative.

The existence of "approximate nonperturbative results" whatever that means or a research program to obtain them is not what I talked about, I mentioned a mathematically valid nonperturbative theory, which is inexistent at the moment. Why play word games to avoid such true and simple statement?

 

[A. Neumaier]

what is your mathematical definition of "virtual particle" for a quantized KG free field?

An internal line in a graph depicting a Feynman diagram. It has no properties except for the properties a labelled edge in any graph has: two end vertices and a label. Everything else about it is fiction.

I mentioned a mathematically valid nonperturbative theory, which is inexistent at the moment.

Since nothing in elementary particle physics is covered by a mathematically valid nonperturbative theory, the latter (or its nonexistence) cannot be used to argue about conceptual issues in elementary particle physics.

 

[Tendex]

An internal line in a graph depicting a Feynman diagram. It has no properties except for the properties a labelled edge in any graph has: two end vertices and a label. Everything else about it is fiction.

A Feynman diagram is just a pedagogical graphic aid to teach perturbation theory in QFT. A calculational tool. I see no point splitting hairs about what parts of a didactic graph are more or less fictitious, unless it can be traced to some actual conceptual parts of the actual theory that are well defined like free field QFT which you refuse to acknowledge.

Since nothing in elementary particle physics is covered by a mathematically valid nonperturbative theory, the latter (or its nonexistence) cannot be used to argue about conceptual issues in elementary particle physics.

Right. See the first instance I mentioned it, it was to wonder why another poster used it as argument.

 

[A. Neumaier]

A calculational tool.

Yes, this is what the article says it is. Not something real. It is of the same nature as the coefficient of $x^2$ in a power series expansion used to calculate real exponential growth, where nobody claims that it mediates this growth, just because it can be used to calculate it.

I see no point splitting hairs about what parts of a didactic graph are more or less fictitious.

Then why do you split hairs in your comments on my article?

I wrote this Insight article since many stories for lay people give far more reality to virtual particles than they can ever have. They talk about them as if they were entities with well-defined though unobservable dynamical properties, just because they are useful to represent some integrals used to calculate observable things. This is misleading nonsense.

 

[Tendex]

Then why do you split hairs in your comments on my article?

You forgot to quote the complete sentence, it continued with something like "unless it can be traced to the free field theory", but you also forgot my second question: isn't the internal line of a Feynman diagram representing a Feynman propagator?, because if it is so it can be traced to the causal propagator that can be already seen in complex scalar free qft, are you also claiming either yourself or in your article that such propagator is "non-existent" misleading nonsense? Or is it only when represented as an internal line of Feynman diagram?

 

[A. Neumaier]

"unless it can be traced to the free field theory"

Internal lines cannot be traced to the free field theory. They represent a factor in a formal (mathematically ill-defined) integral expression relevant for the interacting theory, not to the free theory. That the factor is a free Feynman propagator doesn't give a deeper physical meaning to the internal line. It rather emphasizes that it is an artifact of expanding around a nonphysical free field.

isn't the internal line of a Feynman diagram representing a Feynman propagator?

It represents what I just explained. But this representation does not imply anything significant. Just as representing a city on a globe by a dot labeled with a name does not make the dot have more properties than the name of the city and the coordinates on the globe.

On the other hand, the Feynman propagator of an interacting quantum field - the object that can be equipped with a physical meaning - is a nonperturbative object, not existing in your rigor mortis philosophy. The free Feynman propagator exists but is physically meaningless.

In its perturbative approximation, the physical Feynman propagator is a weighted asymptotic sum of infinitely many terms, each one given by a complete Feynman diagram, and each one evaluating to infinity if it contains loops. Only the complete sum of all terms with a given number of loops can be given a numerical meaning after renormalization. But the sum over all loops diverges. Thus any physical meaning is very far removed from the internal line picture.

To extract high quality approximations one needs nontrivial partial resummation tricks - all based on assuming the nonperturbative theory to exist. This is needed already to see that the Feynman propagator has poles defining the physical particle mass, and hence needed to make sense out of the standard renormalization conditions.

Without assuming that the nonperturbative theory exists (even though one cannot prove it rigorously at the moment. neither can it be disproved), nothing of interest can be done in quantum field theory. One cannot even get started: Already the derivation of the Feynman rules assume the existence of the nonperturbative theory!

 

[Tendex]

The free Feynman propagator exists but is physically meaningless.

You cannot construct the interacting Feynman propagator without the free one in the same way you cannot construct interacting QFTs without free QFTs. I do believe this is physically meaninful, but I admit you already have confused me enough with your strange use of "existence" and "physical" as conveniently flexible terms, I thought I had gathered you linked "existence" to some kind of "physical" meaning but nope.

 

[A. Neumaier]

You cannot construct the interacting Feynman propagator without the free one in the same way you cannot construct interacting QFTs without free QFTs.

You with your rigor mortis standards can construct neither the interacting Feynman propagator nor interacting QFTs. Thus you cannot even begin writing down the perturbation series for it - which motivated Feynman diagrams and internal lines!

But all books on relativistic QFT assume both and construct practically very useful approximations for both - it is the only reason books on relativistic QFT exist.

I thought I had gathered you linked "existence" to some kind of "physical" meaning but nope.

I linked nonexistence to lack of physical meaning and existence to states that allow one to make quantitiative statements.

 

[Tendex]

You with your rigor mortis standards can construct neither the interacting Feynman propagator nor interacting QFTs. Thus you cannot even begin writing down the perturbation series for it - which motivated Feynman diagrams and internal lines!

But all books on relativistic QFT assume both and construct practically very useful approximations for both - it is the only reason books on relativistic QFT exist.

Please avoid ad hominem attacks, they only reflect nervousness/insecurity. I am obviously not saying that interacting QFTs can't be assumed or constructed, that's just a straw man cop out.
I simply used a trivial enough statementent, that mathematically the interacting QFTs are built from free QFTs, i.e. in this case that an internal line is mathematically a free Feynman propagator so in this, trivial if you want, sense it can be traced to it, and since you can't contradict it directly without some degree of ridicule you resort to a straw man.

I linked nonexistence to lack of physical meaning and existence to states that allow one to make quantitiative statements.

This is in contradiction with the sentence I quoted in my previous post where you link existence to lack of physical meaning instead.

 

[A. Neumaier]

Please avoid ad hominem attacks, they only reflect nervousness/insecurity.

I am neither nervous nor insecure, and didn't attack you personally but only your insistence on a for the topic under discussion inappropriate amount of rigor.

I simply used a trivial enough statementent, that mathematically the interacting QFTs are built from free QFTs,

Now you use 'mahematical' without the rigor mortis setting, in its formal meaning appropriate for quantum field theory.

Mathematically in this formal sense, interacting QFTs are not built from free QFTs, but can be built from free QFTs. They can also be built from lattice approximations leading to Euclidean formulations without any Feynman diagrams, Schwinger-Dyson equations with completely different diagrams, the Schwinger-Keldysh closed time path approach which gives the physical (i.e., interacting) propagators without any detour over free fields, etc.. Their conceptual definition, which gives rise to all these approaches, is independent of any of these constructive ways to get approximations.

i.e. in this case that an internal line is mathematically a free Feynman propagator

It represents a free Feynman propagator. This is quite different from being a free Feynman propagator. Moreover, as concepts from a noninteracting theory, the latter have no physical meaning, so one cannot derive alleged physical properties of virtual particles from them.

 

[Tendex]

Mathematically, nteracting QFTs are not built from free QFTs, but can be built from free QFTs.They also can be built from lattice approximations leading to Euclidean formulations without any Feynman diagrams,

The context is relativistic QFT so I'm not referring to lattices here.

Schwinger-Dyson equations with completely different diagrams,

equivalent to Feynman's as Dyson showed.

the Schwinger-Keldysh closed time path approach which gives the propagators without any detour over free fields, etc..

Not familiar with this approach but I guess it will have some kind of time-ordering which is the key feature of the free Feynman propagator.

the latter have no physical meaning, so one cannot derive physical properties of virtual particles from them.

Mathematical objects like contour integrals have no physical properties that I know of.Anyway, since you mentioned QFT textbooks I'll quote a popular one, Peskin&Schroeder page 31:"the formalism we have developed is extremely important, since the free theory forms the basis for doing perturbative calculations in the interacting theory."

 

[A. Neumaier]

The context is relativistic QFT so I'm not referring to lattices here.

But I am. The Osterwalder-Schrader theorem on which Euclidean field theory is based, is relativistic. Extrapolated lattice QCD uses this, hence is relativistic too. It predicts meson and baryon masses to 5%, wheras nonrelativistic approaches only work for hadrons build from heavy quarks.

equivalent to Feynman's as Dyson showed.

Only in one direction - if you treat Schwinger-Dyson equations in perturbation theory one recovers standard perturbation theory. One cannot go from traditional perturbation theory to Schwinger-Dyson equations.

Not familiar with this approach but I guess it will have some kind of time-ordering which is the key feature of the free Feynman propagator.

Sharing a simple property (also shared by the interacting Feynman propagator) does not mean that there are relevant relations. Free Feynman propagators never appear in the CTP approach., only interacting ones.

Mathematical objects like contour integrals have no physical properties that I know of.

What has this to do with our arguments?

Peskin&Schroeder page 31:"the formalism we have developed is extremely important, since the free theory forms the basis for doing perturbative calculations in the interacting theory."

Yes, for the perturbative way of building QFTs. But not for the other ways mentioned.

 

[Tendex]

But I am. The Osterwalder-Schrader theorem on which Euclidean field theory is based, is relativistic. Extrapolated lattice QCD uses this, hence is relativistic too. It predicts meson and baryon masses to 5%, wheras nonrelativistic approaches only work for hadrons build from heavy quarks.

Only in one direction - if you treat Schwinger-Dyson equations in perturbation theory one recovers standard perturbation theory. One cannot go from traditional perturbation theory to Schwinger-Dyson equations.

Free Feynman propagators never appear in the CTP approach., only interacting ones.

Yes, for the perturbative way of building QFTs. But not for the other ways mentioned.

I always had in mind the perturbative way in this discussion, being the most accurate. If one relaxes accuracy enough one can include all kinds of theories.

Sharing a simple property (also shared by the interacting Feynman propagator) does not mean that there are relevant relations.

I think this is a relevant enough property for both the free and interacting case to make the case for a relevant relation, but this is a matter of opinion about what is relevant for someone or not.

What has this to do with our arguments?

Honestly, we seem not to be arguing about the mathematics of it, only about philosophy and you don't like mine and I don't like yours(and like even less that you seem to disguise it as physics in your writings which was what first prompted me to comment). So I guess we can agree to disagree on this.

 

[Tendex]

Above I was responding to: "But I am. Extrapolated lattice QCD is relativistic, predicting meson and baryon masses to 5%. Nonrelativistic approaches only work for hadrons build from heavy quarks" and missed your edit to "But I am. The Osterwalder-Schrader theorem on which Euclidean field theory is based, is relativistic. Extrapolated lattice QCD uses this, hence is relativistic too." Euclidean field theory is not relativistic in the sense that it needs an analytic continuation to Minkowski space and the Schwinger functions meeting the Osterwalder-Schrader theorem conditions haven't been found yet in 4 dimensions. The extrapolated lattice QCD is also quite a stretch mathematically from what it is to be relativistic but that will also seem to you to be rigor mortis to mention it.

 

[A. Neumaier]

I always had in mind the perturbative way in this discussion, being the most accurate.

For QCD, the accuracy of perturbative methods is good only at very high energies. Bound state properties cannot even be contemplated to be attacked perturbatively since poles appear only at infinite order.

When writing the article, and in the present discussion, I always had in mind the whole spectrum of particle physics, including the nonrelativistic sector. I propose that you reread my articles in this light.

the Schwinger functions meeting the Osterwalder-Schrader theorem conditions haven't been found yet in 4 dimensions.

Now you are indeed back to rigor mortis. I think everything has been said on both sides.

 

[Tendex]

When writing the article, and in the present discussion, I always had in mind the whole spectrum of particle physics, including the nonrelativistic sector. I propose that you reread my articles in this light.

This sound like quite a retreat. If you had all those theories in mind we could have talked about the free Feynman propagator all along(unless you also had in mind absence of charge), since it is what ultimately guarantees the unitarity of the Hamiltonian making sure of the operator products correct time-ordering and you admitted that internal lines represent free Feynman propagators, but you insisted on the perturbative Feynman diagrams context so this almost "anything goes" strikes me as odd.

Now you are indeed back to rigor mortis.

Knew it!

I think everything has been said on both sides.

I think so too.

 

[Tendex]

I just realized that you never answered my objections about why the asymptotic states cannot be used as a valid rationale for any "reality" or "physicality" within interacting RQFT, certainly not in a Feynman diagram. Basically once the distributional Feynman propagator, that requires manifestly Lorentz invariant spacetime smearing as it includes the locality (microcausality) criterion, is used in the Feynman diagram the "physical" and mathematical distinctions between external and internal lines is empty since states can no longer single out time specifically like asymptotic states do.
Either the whole Feynman diagram is "real" in some sense or not. Since all its components are needed mathematically there is no sense picking one as "real" to the detriment of another.

 

[A. Neumaier]

I just realized that you never answered my objections about why the asymptotic states cannot be used as a valid rationale for any "reality" or "physicality" within interacting RQFT

Asymptotic states (represented by external lines in Feynman diagrams) are on-shell and are real by my criteria, unlike virtual particles (represented by internal lines). The lines themselves are not real.

Either the whole Feynman diagram is "real" in some sense or not. Since all its components are needed mathematically there is no sense picking one as "real" to the detriment of another.

A Feynman diagram is not real in any sense (except for the reality of the ink used to draw it). It just represents a recipe to compute a (most often divergent) term the standard recipe for computing the Dyson expansion.

 

[Tendex]

Asymptotic states (represented by external lines in Feynman diagrams) are on-shell and are real by my criteria, unlike virtual particles (represented by internal lines). The lines themselves are not real.

Please address my argument. In interacting RQFT with spacetime smearing(distributional Feynman propagator and manifest Lorentz invariance) what the external lines represent(states at time infinity) cannot be bona fide states in the perturbative approach since they single out time. Constant states in the Heisenberg representation are used in this scenario that can't single out space or time and that by Haag are unitarily inequivalent to other representations. It is for some reason it is said that state vectors and the Schrodinger rep are pretty useless in this interacting perturbative field theory.

 

[vanhees71]

That's the point: A particle interpretation is only viable for asymptotic free states but not for any kind of interpretation of the "transient states". You always need the "Gell-Mann-Low switching" to define the perturbative S-Matrix elements properly.

If massless particles are involved as, e.g., in QED you also have to deal with the IR divergences, which are partly due to using naively plane-wave rather than "infra-particle" aymptotic states, but that's another story.

 

[Tendex]

That's the point: A particle interpretation is only viable for asymptotic free states but not for any kind of interpretation of the "transient states". You always need the "Gell-Mann-Low switching" to define the perturbative S-Matrix elements properly.

If massless particles are involved as, e.g., in QED you also have to deal with the IR divergences, which are partly due to using naively plane-wave rather than "infra-particle" aymptotic states, but that's another story.

Exactly, that's why it is absurd to unlink external lines from internal lines, since at finite times (actual detections/measurements) we only have Lorentz invariant "transient states", that need in an essential way the residues of the poles in the Feynman propagators contour integrals. I think it is not useful to get attached to the "particle interpretation" if one is going to use perturbative interacting RQFT. The asymptotic state view of particles separated from the rest of elements of the Feynman diagram is as incorrect as giving some sort of individual existence to internal lines as particles.

 

[A. Neumaier]

Please address my argument. In interacting RQFT with spacetime smearing (distributional Feynman propagator and manifest Lorentz invariance) what the external lines represent (states at time infinity) cannot be bona fide states in the perturbative approach since they single out time.

That the perturbative treatment has limitations is well-known.

But by the scattering theory of Haag and Ruelle, asymptotic state in relativistic QFT are bona fide states from a free relativistic QFT involving one field for each bound state, at least when the theory has a mass gap. Nothing singles out time.

by Haag are unitarily inequivalent to other representations.

Haag's theorem only invalidates the unrenormalized interaction picture. The renormalization limit destroys the Fock space and replaces it by another (in 4D as yet poorly defined) Hilbert space with an inequivalent representation of the field algebra. In effect, Haag's arguments therefore say nothing less or more than that renormalization is necessary. However, renormalization does not change the fact that the asymptotic Hilbert spaces are still Fock spaces. The states entering the S-matrix are, by definition, always free, asymptotic states.

The description in terms of asymptotic states is experimentally valid, and the particle picture is appropriate, if and only if the particles are separated well enough so that their interactions can be neglected. The true interacting states are then irrelevant. In scattering experiments, this description is valid except for short times. Indeed, all applications of QFT to the prediction of reaction cross sections are based on assuming this validity, and the many successful predictions confirm the validity.

 

[Tendex]

That the perturbative treatment has limitations is well-known.
But by the scattering theory of Haag and Ruelle, asymptotic state in relativistic QFT are bona fide states from a free relativistic QFT involving one field for each bound state, at least when the theory has a mass gap. Nothing singles out time.

In a previous post you were saying that the free field is unphysical but now you use it to justify asymptotic states reality. I'm perfectly aware they are valid states of the free field theory but again the use of their states "appearance" of time asymptoticity is just that, a fictional way to hide that there is no state for finite time t in the S-matrix(this is part of the limitationsof the perturbative approach you mention above) and to justify mathematically this we still need the time-ordered products given by the Feynman propagator.

We are discussing the "physicalness" you grant to external lines in contrast to internal lines just to save some "particle appearance" that you are attached to. But it is all the math behind the Feynman diagram representation (as a whole) of terms in the S-matrix what gives it physical meaning perturbatively. There is no need for unobservable particles at infinite times interpretations(nor for "virtual particles" for similar reasons) in a field theory. And for physical detections, clicks or observations the whole mathematical apparatus is necessary to explain them, not isolated portions.

 

[A. Neumaier]

In a previous post you were saying that the free field is unphysical but now you use it to justify asymptotic states reality.

The free field is unphysical when considering interactions. it is of course physical when the interactions can be neglected.

I'm perfectly aware they are valid states of the free field theory but again the use of their states "appearance" of time asymptoticity is just that, a fictional way to hide that there is no state for finite time t in the S-matrix

But this is because of the way an S-matrix is defined. It has nothing to do with quantum fields, as it happens already for a single particle in an external potential. For the same reason it has nothing to do with perturbation theory, since the S-matrix is a nonperturbative object.

We are discussing the "physicalness" you grant to external lines in contrast to internal lines

I never granted "physicalness" to lines in a diagram. I granted "physicalness" to properties such that statistical information about them can be computed in principle, since they have a state. For particles, asymptotically defined objects, cross sections, life times, etc. can be computed from their state, so these are physical, and hence real. Whereas for virtual particles, one cannot do it, so they are unphysical. For lines drawn on paper as part of a Feynman diagram, the only physical properties are those made up by their ink. For example, one can talk about their length or color but not about their lifetime.

There is no need for unobservable particles at infinite times interpretations (nor for "virtual particles" for similar reasons) in a field theory.

I never claimed that. But there is a need for asymptotic states to interpret the S-matrix in terms of experimental results.

 

[Tendex]

For the same reason it has nothing to do with perturbation theory, since the S-matrix is a nonperturbative object.

As far as perturbative (calculations of the) S-matrix assume a non-perturbative theory that is supposed to be approximating, the S-matrix can be a non-perturbative object, but most of what we've discussed, the way the actual Dyson series and Feynman diagrams, and perturbative calculations are constructed has everything to do with perturbation theory. Either this or all the QFT texts where the authors often refer to "the perturbative (calculations of the) S-matrix" or "perturbative S-matrix" for short must have some serious misunderstanding.

I never claimed that. But there is a need for asymptotic states to interpret the S-matrix in terms of experimental results.

Oh sure, just as there is a need for Feynman propagators and other objects to mathematically and physically interpret correctly those very states of the perturbative S-matrix(the one that makes plausible that there is a non-perturbative S-matrix) and its results including such properties as causality and unitarity for those same experimental results.

 

[A. Neumaier]

As far as perturbative (calculations of the) S-matrix assume a non-perturbative theory that is supposed to be approximating, the S-matrix can be a non-perturbative object, but most of what we've discussed, the way the actual Dyson series and Feynman diagrams, and perturbative calculations are constructed has everything to do with perturbation theory. Either this or all the QFT texts where the authors often refer to "the perturbative (calculations of the) S-matrix" or "perturbative S-matrix" for short must have some serious misunderstanding.

Oh sure, just as there is a need for Feynman propagators and other objects to mathematically and physically interpret correctly those very states of the perturbative S-matrix (the one that makes plausible that there is a non-perturbative S-matrix) and its results including such properties as causality and unitarity for those same experimental results.

You are setting up everything upside down.

The basic object in scattering theory is the S-matrix, defined from the start nonperturbatively, first in elementary 1 DOF quantum mechanics and then in more and more complex contexts.

The Dyson series is only the most elementary approximation method beyond the Born approximation, and is known to work well only for simple problems, even for 1 DOF. Using it in relativistic QFT in 1928 immediately ran into divergence problems, making its plausibility very questionable. It took nearly 20 years to resolve the associated difficulties by renormalization heuristics - a clear sign that the Dyson series approach has nothing of the fundamental nature you want to assign to it.

A valid nonperturbative definition of the S-matrix in relativistic QFT was given in the 1950s by Bogoliubov, and used later by Epstein and Glaser to give a mathematically rigorous perturbative approach now called [URL='https://www.physicsforums.com/insights/causal-perturbation-theory/']causal perturbation theory[/URL]. In this approach, the perturbative approximation to the S-matrix is rigorously constructed order by order, preserving causality and unitarity at every stage of the construction. But the traditional Feynman diagrams do not appear at all, since the ill-defined Dyson series is completely avoided. The alternative approach deriving scattering information in terms of the 2PI formalism, or their subsequent computation, also do not involve Feynman diagrams.

That textbooks prefer the old, infinity-ridden approach is just because long traditions are slow to change, and because the old approach requires only much less sophisticated mathematical machinery.

 

[Tendex]

I don't really know what you're ultimate goal is then. You seem to despise the whole perturbative approach as not rigurous enough(while calling this rigor mortis when I do it with the broader non-perturbative theory according to the actual valid standard of mathematics), but at the same time you don't have any quibbles with asymptotic states that are precisely an awkward example of a perturbative hack in the diagrams of perturbative QFT because perturbatively the interaction must be switched on-off so a narrative about particles at unobservable infinite times must be invented while the actual observations are in finite time. Then you lament that textbooks are not as modern and rigurous as you'd like them...while any suggestion that all this is kind of moot with the actual measure of rigor which is plain mathematical rigor, instead of the ladder of more or less sophisticated mathematical narratives that you may find to your liking, is rejected violently by you.
I'd say that as long as the calculations perturbative way gives us good predictive approximations and as long as it is not backed by valid mathematics it is perfectly ok regardless of the way it is explained as long as it gets the results right. This will change the moment there is a solid math theory behind it(one that at least mathematicians can understand).

 

[A. Neumaier]

you don't have any quibbles with asymptotic states that are precisely an awkward example of a perturbative hack in the diagrams of perturbative QFT because perturbatively the interaction must be switched on-off

This only holds in the textbook approach. In Haag-Ruelle theory, asymptotic states are mathematically and physically impeccable. And in causal perturbation theory they produce an impeccable perturbation series without any switching on or off. Thus there is nothing to complain about, except for the lack of convergence of the series. But the latter doesn't matter for low energy QED or high energy QCD, where the partially resummed asymptotic series converges so fast that a few terms suffice.

any suggestion that all this is kind of moot with the actual measure of rigor which is plain mathematical rigor, instead of the ladder of more or less sophisticated mathematical narratives that you may find to your liking, is rejected violently by you.

Both rigor and formal arguments have their place.

I do not reject rigor where rigorous results are not yet known, since nonrigorous results are much better than nothing where they lead to good predictions. I prefer rigor where useful things can be rigorously stated and/or proved. Rigor mortis is where you kill a useful and hence legitimate approach or argument by insisting on rigor where it is clearly not appropriate.

But this has nothing to do with my Insight article series, under discussion in this thread. There I define reality (or physicalness if you want to avoid the philosophically loaded term) by having a state that can be used to make successful predictions. Particles have states, and quantum fields have states, hence both have reality in this sense, leading to meaningful physical properties. But virtual particles and propagators don't have states, hence they only have their defining properties - endpoints for virtual particles, mass and spin for propagators, but nothing else.

 

[Tendex]

Particles have states, and quantum fields have states, hence both have reality in this sense, leading to meaningful physical properties. But virtual particles and propagators don't have states, hence they only have their defining properties - endpoints for virtual particles, mass and spin for propagators, but nothing else.

And forgetting for a moment the loaded "particle" ontology, and since I've always agreed that the "virtual particles" picture of the math behind internal lines is quite idiotic, don't you think that properties that Feynman propagators help to ensure mathematically such as microcausality and boundedness of the Hamiltonian, to the extent they allow time-ordering of fields keeping Lorentz invariant locality and positive energies, lead also to meaningful physics?

 

[A. Neumaier]

that Feynman propagators help to ensure mathematically such as microcausality and boundedness of the Hamiltonian

Microcausality and Hamiltonians are associated with field operators, not with propagators. The mathematical properties of the latter are derived from the former, not vice versa.

I've always agreed that the "virtual particles" picture of the math behind internal lines is quite idiotic

Then why this whole discussion here? The point of these Insight articles is precisely to get this message across, by defining (in this Insight article) all terms used in the discussion, including 'real', in a consistent way and pointing out their consequences.

Your first contribution was:

It appears to me that here the classical stance about particles instead of the quantum field view is being used to create an artificial debate about the "existence" of particles whether "real" or "virtual". [...]
Summarizing, this debate is addressing a non-issue and doing it introducing wrong information as it insists on the "existence" of "real particles" suspiciously similar to classical "billiard" particles

I consider particles as real because, having a state, they have stochastically predictable dynamical properties. Particles were detected and considered real long before there was a working relativistic quantum field theory. They didn't stop existing simply because a new theory was established; so the terminology is appropriate. I never claimed any resemblance to classical "billiard" particles, which seems to be inherent only in your personal view of what a real particle should be. You should not confound your deviating views with how I defined my terminology.

 

[vanhees71]

Instead of saying "something is real" one should say "something is observable", because this is what physics is about. When discussing "reality" you enter the shallow waters of philosophy, where not two philosophers seem to agree upon the meaning of a word like "reality".

 

[A. Neumaier]

Instead of saying "something is real" one should say "something is observable", because this is what physics is about.

Then galaxies too distant, but also the deep interior of the Earth or the Sun, would be nonreal. What is real would be a matter of available technology...

When discussing "reality" you enter the shallow waters of philosophy, where not two philosophers seem to agree upon the meaning of a word like "reality".

This is why I gave a precise definition of what to understand by reality.