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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 10th 2013

added to S-matrix a useful historical comment by Ron Maimon (see there for citation)

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeSep 10th 2013

Sounds interesting, but what’s the key point? String theory ought to be considered as a S-matrix theory to make best sense of it?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeSep 10th 2013
• (edited Sep 10th 2013)

To me a key point is that the ways of the scientific community can be convoluted (reminds me of Nolte’s account of the history of the notion of “phase space”) and that one should trust in genuine understanding and not just the in the current folk lore, as that may be confused in subtle ways over decades. In particular it’s important to know what is actually known, what is just expected.

Yes, perturbative string theory is an S-matrix theory. This is so by definition and by design. The point here is that it is a curious irony of history: first S-matrix theory is overthrown and abandoned in favor of quantum field theory, then later some people start to say that quantum field theory needs to be refined by string theory – which in turn is an S-matrix theory that contradicts the claim that every sensible S-matrix is that of a field theory.

One statement of Maimon that particularly resonated with me is the “…the main S-matrix theory, string theory, is not properly explained and remains completely enigmatic even to most physicists.” This is a feeling that have all the time: so much discussion of string theory in the public domain, and so much confusion about what it really is. It may be all wrong, but not for the alleged reasons most commonly voiced about it.

So I think it’s good to be aware of the history here and beware of common lore without thorough scrutinization.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeNov 12th 2013

added a quote from a talk by Weinberg in 2009 to the History-section at S-matrix (search for “Weinberg” to find it)

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeAug 7th 2017
• (edited Aug 7th 2017)

I have tweaked the Idea-section at S-matrix a little and collected the historical comments in a single History-section and then moved that further towards the end of the entry. Then I copied over the technical discussion of the S-matrix in quantum mechanics and perturbative quantum field theory from the entry interaction picture.

(This means that now these two entries have a considerable overlap. But that seems better than one of the two lacking this basic information. Eventually the entries will grow in different directions.)

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeAug 8th 2017
• (edited Aug 8th 2017)

If

$\vert \psi(t)\rangle_I \coloneqq \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \vert \psi(t)\rangle_S$

shouldn’t

$\vert \psi(t) \rangle_S = \exp\left({\tfrac{t}{i \hbar} H}\right) \vert \psi(t) \rangle_I$

have $H_{free}$?

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeAug 8th 2017

Thanks for catching this. Absolutely, as the surrounding text explained, the point is that we have $H_{free}$ here. I have fixed it now.

I have also added another paragraph of text below the definition of the state in the interaction picture, here.

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeAug 8th 2017

Since that section had been copied (#5) from Dirac interaction picture, those same changes are needed there.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeAug 8th 2017

Yes, done now. I have also done a bunch of other edits to this section.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeAug 28th 2017

Just in case you are watching the logs and wondering:

I am working on bringing in content on the S-matrix in field theory at S-matrix.

But I am in the middle of it. Don’t have a look unless you are serious about joining in non-trivial editing. I hope to have something coherent, consistent and readable later this week.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeSep 3rd 2017
• (edited Sep 3rd 2017)

I have been further working on S-matrix.

There is now a complete proof spelled out that causal perturbation theory indeed does yield a local net of quantum observables. It culminates in the sub-section Causal locality and Quantum observables The full proof of this crucial statement of pAQFT is somewhat hidden in the literature, it needs the computation of the support of the “generating” retarded products in section 8.1 of the old Epstein-Glaser article, now reproduced as this prop in the entry.

I have also spelled out now in detail how the S-matrix is expressed by the Feynman pertubation series over Feynman diagrams “away from the diagonal” (this section). What remains is discussion of how these products of Feynman propagator distributions are extended to the diagonal, but that discussion should probably be kept only at renormalization, not to overburden the entry on the S-matrix. I’ll get to that.

Also, the first sections on spacetime causality and on free field algebras are not yet expository at the moment, but just terse collection of what is needed later on. I’ll eventually expand on that to make it more readable.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeSep 18th 2017
• (edited Sep 18th 2017)

I have added further details to the subsection Perturbative S-matrix and Time-ordered products, such as to complete the full details of the proof of the statement (this prop) that the causal factorization axiom on the time-ordered products implies the causal additivity axiom on the S-matrix.

• CommentRowNumber13.
• CommentAuthorDavid_Corfield
• CommentTimeNov 20th 2017

I have a student interested in the story of the paths of the S-Matrix and QFT approaches. Is there anything to say in retrospect about the structures involved as to why we should have this strange history, leaving aside all the sad history of the psychology drives of the participants. Do we now see why it isn’t so surprising that Veneziano wrote out that amplitude, when apparently thinking about the strong force?

Or to put it another way, is there a “rational reconstruction” which has these two approaches find what each needs in the other, allowing a friendly merger?

There’s been a similar reworking of the phlogiston/oxygen theories of combustion where the best of each survives and hastens the development of chemistry.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeNov 20th 2017
• (edited Nov 20th 2017)

On the one hand, Ron Maimon’s comment is spot on in pointing out that even though people eventually realized that most S-matrices come from local Lagrangian field theory, examples from string theory show that not all do. On the other hand, it seems to have remained underappreciated that the only rigorous construction of these QFTs is (or was, until Collini 16) by axiomatization of their S-matrices, namely via causal perturbation theory.

This causal perturbation theory is really what unifies the two perspectives: On the one hand it is just an axiomatization of the S-matrix, on the other hand the key axiom (“causal additivity”) relates the S-matrix to spacetime causality.

But apart from this I feel like the situation remains very interesting in the sense of still requiring deeper understanding. For instance the surprising story of Veneziano’s amplitude that you mention by and large remains just that: endlessly surprising. It feels like despite all these developments, we are still just at the beginning of understanding the full picture.

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeNov 20th 2017
• (edited Nov 20th 2017)

By the way, you may enjoy reading Geoffrey Chew’s physico-philosophical essays, such as “Quark or Bootstrap

• CommentRowNumber16.
• CommentAuthorDavid_Corfield
• CommentTimeNov 20th 2017

Thanks! Your second paragraph should be of great interest.

• CommentRowNumber17.
• CommentAuthorDavid_Corfield
• CommentTimeDec 19th 2017

Is the Wikipedia article accurate on why the assumption of analyticity is made?

1. Crossing: The amplitudes for antiparticle scattering are the analytic continuation of particle scattering amplitudes.
2. Dispersion relations: the values of the S-matrix can be calculated by integrals over internal energy variables of the imaginary part of the same values.
3. Causality conditions: the singularities of the S-matrix can only occur in ways that don’t allow the future to influence the past (motivated by Kramers–Kronig relations)
4. Landau principle: Any singularity of the S-matrix corresponds to production thresholds of physical particles.

If as you say,

This causal perturbation theory is really what unifies the two perspectives: On the one hand it is just an axiomatization of the S-matrix, on the other hand the key axiom (“causal additivity”) relates the S-matrix to spacetime causality,

this means (3) above was not enough to deliver what is needed causally? Does analyticity persist somehow in causal perturbation theory?

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeDec 19th 2017
• (edited Dec 20th 2017)

I am not yet expert enough on the analytic properties of the S-matrix, I’ll try to come back to this later.

But generally the analyticity condition and more generally the crossing symmetry condition on the S-matrix is a reflection of what in the field theoretic construction is the causal factorization property.

The standard argument for this is somewhat simple, recalled as equations (1.1.1) to (1.1.5) and the paragraph below that in Eden-Ladshoff-Olive-Polkinhorne 66.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeDec 19th 2017
• (edited Dec 20th 2017)

In Weinberg’s QFT textbook, section 10.8, there is a more detailed argument offered on how it is the microcausality of the S-matrix when based on fields on spacetime which gives rise to its analyticity when regarded more abstractly as a function of asympotic scattering states.

I have added a paragraph with these pointers to the entry here.

Maybe there is more on this question in Wightman-Streater. Will check now, if the gods of WiFi admit…

• CommentRowNumber20.
• CommentAuthorDavid_Corfield
• CommentTimeDec 19th 2017

Thanks! Maybe then a further issue is

“One of the most remarkable discoveries in elementary particle physics has been that of the existence of the complex plane.” J. Schwinger (1970)

That’s how Zee opens a section on ’Dispersion relations and high frequency behavior’ in QFT in a Nutshell

Considering that amplitudes are calculated in quantum field theory as integrals over products of propagators, it is more or less clear that amplitudes are analytic functions of the external kinematic variables.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeDec 19th 2017

What do you mean by “further issue”?

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeDec 19th 2017
• (edited Dec 20th 2017)

The statement that analyticity of the S-matrix reflects (micro-)causality is explicit in section 1.1.2 of Gribov 69, where it says:

It is assumed that the scattering amplitude A is an analytic function of its arguments (for instance it cannot contain terms like $\Theta(p_{i 0})$). This assumption is a manifestation of the causality principle.

But no further justification is give there. So the only substantial argument that I have seen so far remains that in Weinberg QFT I, section 10.8.

• CommentRowNumber23.
• CommentAuthorDavid_Corfield
• CommentTimeDec 19th 2017

Oh, was that comment (which it seems is only reported by Schwinger) already including the discovery that not only is the complex plane involved but also that certain functions are analytic?

The causal input allows the argument that the given function is in two parts, each analytic on half of the plane, because an integrand is zero outside, and then the exponential caps the integrand inside.

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeDec 19th 2017
• (edited Dec 20th 2017)

Sorry, I am still not sure if I understand what you are asking. The quote by Schwinger in your #20 is a humorous way of saying that analytic continuation plays a role in field theory. But I thought the question we are after is how in the case of S-matrix elements this analyticity is related to causality?

By the way, it feels like the analytic dependency appearing here is not unrelated to that in the Paley-Wiener-Schwartz theorem, but I don’t quite know yet how to put my finger on it.

• CommentRowNumber25.
• CommentAuthorDavid_Corfield
• CommentTimeDec 19th 2017

I thought there might be a more general reason for the appearance of analytic continuation in field theory, and then a particular association relating causality and analyticity of a specific function. Maybe not.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeDec 20th 2017
• (edited Dec 20th 2017)

The textbook R. G. Newton 82 “Scattering Theory of Waves and Particles” has a lot on analyticity of S-matrices in classical field theory and in quantum mechanics. Unfortunately, as far as I see, it does not consider the S-matrix in (perturbative) quantum field theory.

• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeJan 7th 2018
• (edited Jan 8th 2018)

The title of this book here sounds as if it might have the answer regarding analyticity of S-matrices in QFT:

• Daniel Iagolnitzer, The Analyticity Program in Axiomatic Quantum Field Theory, in A.B. de Monvel, Detlev Buchholz, Daniel Iagolnitzer, Moschella U. (eds.) Rigorous Quantum Field Theory, Progress in Mathematics, vol 251. Birkhäuser, Basel, 2007 (web)

I haven’t had a chance to look at it yet.

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeJan 8th 2018
• (edited Jan 8th 2018)

This is interesting: Arkani-Hamed et al. 06 argue that analyticity of the S-matrix of a Lagrangian field theory is what rules out global causality violation of non-renormalizable local Lorentz invariant Lagrangians

• CommentRowNumber29.
• CommentAuthorDavid_Corfield
• CommentTimeJan 9th 2018

That’s the paper Jacques Distler discussed here.

• CommentRowNumber30.
• CommentAuthorUrs
• CommentTimeJan 12th 2018
• (edited Jan 12th 2018)

Thanks, David. I’ll have to chat about these matters with Igor, when I find the leisure…

• CommentRowNumber31.
• CommentAuthorUrs
• CommentTimeJan 12th 2018
• (edited Jan 12th 2018)

I am now polishing the details at S-matrix in the section In causal perturbation theory.

I removed a chunk of background material that has meanwhile gotten its own stand-alone entries, and instead added a brief recollection with pointers to these entries. Then I began to update notation accordingly. (Not quite done yet, from prop. 2.14 on there remain “$L$“s that should be changed to “$S$“s. But have to interrupt for the moment.)

After def. 2.2 I have added a bunch more remarks on what it all means.

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeJan 14th 2018

I have further polished and expanded that list of remarks (trying to put the concept of the S-matrix in causal perturbation theory in perspective), so that I now gave it its own subsection “Remarks”, for ease of navigation.

• CommentRowNumber33.
• CommentAuthorUrs
• CommentTimeJan 16th 2018

I have been further polishing, harmonizing and expanding at S-matrix. The following sections should be coherent now:

In the section

I have expanded and completed the proof of the key proposition (here) but this section deserves being expanded further.

• CommentRowNumber34.
• CommentAuthorUrs
• CommentTimeJan 16th 2018
• (edited Jan 16th 2018)

okay, now I have worked a bit more on the section Feynman diagrams.

What is there now is pretty polished, I think, but some topics are missing. I should still say something about effective actions, connected diagrams and 1PI diagrams. Maybe tomorrow.

Also eventually I might spell out some Lamb shift-computation as an example. Maybe later…

• CommentRowNumber35.
• CommentAuthorUrs
• CommentTimeJan 17th 2018

have added another section: Effective action

• CommentRowNumber36.
• CommentAuthorUrs
• CommentTimeJan 18th 2018

I think I am getting close to finalizing the section Feynman perturbation series. In particular I tried to spell out the underlying labeled-graph business in detail.

• CommentRowNumber37.
• CommentAuthorUrs
• CommentTimeJan 18th 2018
• (edited Jan 18th 2018)

I have now worked on finalizing the section that is (now) called Interacting field observables, which shows that

$\array{ \array{ \text{gauge-fixed} \\ \text{Lagrangian} \\ \text{field theory} } & \overset{ \array{ \text{causal} \\ \text{perturbation theory} \\ } }{\longrightarrow}& \array{ \text{perturbative} \\ \text{algebraic} \\ \text{quantum} \\ \text{field theory} } \\ \underset{ \array{ \text{(Becchi-Rouet-Stora 76,} \\ \text{Batalin-Vilkovisky 80s)} } }{\,} & \underset{\text{(Epstein-Glaser 73)}}{\,} & \underset{ \array{ \text{ (Il'in-Slavnov 78, } \\ \text{Brunetti-Fredenhagen 99,} \\ \text{Dütsch-Fredenhagen 01)} } }{\,} }$

Hence I have harmonized notation, expanded the proofs (the key lemma is this prop.) and polished (in fact completely rewrote) the lead-in paragraphs.

• CommentRowNumber38.
• CommentAuthorUrs
• CommentTimeJan 20th 2018
• (edited Jan 20th 2018)

I have expanded the section Feynman perturbation series, bringing out in more detail the step from plain multigraphs to labeled multigraphs (hence Feynman diagrams); added a corresponding lead-in comment here

• CommentRowNumber39.
• CommentAuthorUrs
• CommentTimeJan 20th 2018
• (edited Jan 20th 2018)

Somebody signing “Anonymous” had left a request in the entry (rev 122) for more details on the derivation of the Schrödinger equation in the interaction picture (this equation).

So I have added the detailed manipulation now, a few lines below that equation.

• CommentRowNumber40.
• CommentAuthorUrs
• CommentTimeJan 20th 2018

I have added a section Vacuum diagrams, culminating in the proof that in a stable vacuum state the contribution of the vacuum diagrams to the scattering amplitudes cancels out.

(This is often implemented by hand, here I am deriving it from the interacting field observables, that’s where the necessity of a stable vacuum stated comes in.)

This concludes (I’d hope) the discussion of how the Feynman perturbation series serves to organize the abstractly axiomatized S-matrix by increasing orders of $\hbar$ and $g$.

• CommentRowNumber41.
• CommentAuthorDavid_Corfield
• CommentTimeJan 21st 2018

That’s interesting that you see this as a synthetic approach:

But we may think of the axioms for the S-matrix in causal perturbation theory (def. 2.4) as rigorously defining the path integral, not analytically as an actual integration, but synthetically by axiomatizing the properties of the desired outcome of the would-be integration.

Are there other cases of “synthetic physics”? You later point out that FQFT is defined in the same vein.

• CommentRowNumber42.
• CommentAuthorDavid_Corfield
• CommentTimeJan 21st 2018
• (edited Jan 21st 2018)

Of course there’s your own Synthetic Quantum Field Theory. I guess I’d be interested to know about historical cases that could be construed as adopting a synthetic approach, even if it wasn’t explicitly mentioned at the time.

• CommentRowNumber43.
• CommentAuthorDavid_Corfield
• CommentTimeJan 21st 2018

That would make for a project in the future: What kinds of philosophical motivations drive synthetic approaches to theory-building?

If, as above, we should see S-matrix theory as another instance, one might think, along with Weinberg, that positivism is at play:

The hope of S-matrix theory was that, by using the principles of unitarity, analyticity, Lorentz invariance and other symmetries, it would be possible to calculate the S-matrix, and you would never have to think about a quantum field. In a way, this hope reflected a kind of positivistic puritanism:we can’t measure the field of a pion or a nucleon, so we shouldn’t talk about it, while we do measure S-matrix elements, so this is what we should stick to as ingredients of our theories.

(Given the positivist-verificationist connection, that reminds me to look up the origin of that verificationist/pragmatist distinction for natural deduction, introduced by Dummett it seems, and its connection to polarity, see e.g. Zeilberger’s thesis.)

Relating to the issue of analyticity, we were discussing earlier, Weinberg says

I think that the emphasis in S-matrix theory on analyticity as a fundamental principle was misguided, not only because no one could ever state the detailed analyticity properties of general S-matrix elements, but also because Lorentz invariance requires causality (because as I argued earlier otherwise you’re not going to get a Lorentz invariant S-matrix), and in quantum field theory causality allows you to derive analyticity properties. So I would include Lorentz invariance, quantum mechanics and cluster decomposition as fundamental principles, but not analyticity.

• CommentRowNumber44.
• CommentAuthorUrs
• CommentTimeJan 21st 2018
• (edited Jan 21st 2018)

The comment that you quote in #41 is meant to explain to the reader brought up with the traditional informal picture how it can be that Epstein-Glaser constructed pQFT just as commonly understood, but without invoking an ill-defined path integration. I thought the distinction between analytic and synthetic perspectives is useful to appreciate what is going on here.

Just to caution that even so, causal perturbation theory is far from being a fully synthetic formulation of pQFT, say such as it would make sense to ask to implement in some cohesive topos or the like. Those axioms on the S-matrix are nice and simple in themselves, but they invoke the free field Wick algebra and its subspace of local polynomial observbables for which it is unclear how synthetically these may be phrased.

Also it’s not so clear how fundamental the concept of the S-matrix really is. While it does lend itself to elegant axiomatization and to direct comparison with scattering experiments, not every experiment is a scattering experiment, and the S-matrix axioms are not manifestly related to any systematic definition of quantization.

It is a non-trivial theorem that secretly the S-matrix axioms do connect to a general systematic concept of quantization (Hawkins-Rejzner, prop. 5.4, Collini 16).

From this perspective the fundamental concept would be formal deformation quantization, possibly obtained as a special case of geometric quantization of symplectic groupoids, and it is the latter that connects to the idea of fully synthetic QFT that I have been talking about back when I had the leisure to entertain such thoughts. Along this route the S-matrix would be more a clever mathematical convenience than a fundamental concept, and its conceptual superiority over the informal concept of the path integral would mainly be a reflection of the deficiency of the latter.

This state of affairs is amplified by the incarnation of the S-matrix in string theory. Here we have a theory all based on taking the concept of the S-matrix as fundamental (perturbative string theory) and interesting as that happens too be, it turns out that the key open problem this theory is facing is to go beyond the S-matrix (say in string field theory or M-theory).

• CommentRowNumber45.
• CommentAuthorUrs
• CommentTimeJan 21st 2018
• (edited Jan 21st 2018)

I have now expanded the section (“Re”-)Normalization just enough so that all the ingredients that go into the formulation of the “main theorem of perturbative renormalization” are introduced. (The discussion of the proof of this theorem and of methods for making (“re”-)normalization choices should go instead to the entry renormalization, naturally.)

In particular I added statement of the concept of “renormalization condition” (here), which neeeds to be specified for the “main theorem” to hold and to determine how large the resulting Stückelberg-Petermann renormalization group comes out. In the process I gave renormalization conditions its own little entry, for ease of referencing.

So far I only state three renormalization conditions. The two elementary ones “field independence” and “translation invariance” are needed for the “main theorem” to apply at all. The third condition “quantum master equation”/”master Ward identity” is needed to make renormalization of gauge theories be consistent.

Accordingly, I also added a section BV-differential and Ward identities which derives the form of this renormalization condition. (The material for this I essentially grabbed from the entry quantum master equation, just harmonizing cross-links such as to fit it neatly into the S-matrix-entry.)

• CommentRowNumber46.
• CommentAuthorUrs
• CommentTimeJan 23rd 2018
• (edited Jan 23rd 2018)

I have made explicit a formal definition of “renormalization conditions” (here) and hence of “quantum anomaly” (the failure of these conditions).

(Currently the entry normalization condition itself has just an Idea-section and a pointer to S-matrix for the details, while “quantum anomaly” itself is an old version that deserves to be improved)

• CommentRowNumber47.
• CommentAuthorUrs
• CommentTimeFeb 9th 2018

I have added this prop. spelling out the generating functional for interacting field observables in a stable vacuum state, and then this example explaining how that relates, under favorable assumptions, to the “quantum effective action” whose critical points are given by the vevs of the interacting fields.

• CommentRowNumber48.
• CommentAuthorUrs
• CommentTimeFeb 9th 2018

And I have given the warning on terminology, regarding the two different meanings of “effective”, its stand-alone numbered remark: here

• CommentRowNumber49.
• CommentAuthorUrs
• CommentTimeJul 9th 2018
• (edited Jul 9th 2018)

added pointer to today’s Bottino 18 (history of T. Regge’s contributions to analytic S-matrix theory)

• CommentRowNumber50.
• CommentAuthorUrs
• CommentTimeDec 17th 2019
• (edited Dec 17th 2019)

added a new subsection under References: “Classification of long-range forces”, with this content:

Classification of possible long-range forces, hence of scattering processes of massless fields, by classification of suitably factorizing and decaying Poincaré-invariant S-matrices depending on particle spin, leading to uniqueness statements about Maxwell/photon-, Yang-Mills/gluon-, gravity/graviton- and supergravity/gravitino-interactions:

Quick review:

• CommentRowNumber51.
• CommentAuthorUrs
• CommentTimeDec 18th 2019

added also pointer to this review:

will make an !include-entry with these references at

classification of long-range forces – references

for ease of including them in other relevant entries

• CommentRowNumber52.
• CommentAuthorUrs
• CommentTimeNov 12th 2020

• Alexander S. Blum, The state is not abolished, it withers away: how quantum field theory became a theory of scattering (arXiv:2011.05908)

and completed publication data for

• CommentRowNumber53.
• CommentAuthorUrs
• CommentTimeDec 8th 2020

added these recent pointers on the S-matrix for mesons in chiral perturbation theory:

• Andrea Guerrieri, Joao Penedones, Pedro Vieira, S-matrix Bootstrap for Effective Field Theories: Massless Pions (arXiv:2011.02802)

• Eef van Beveren, George Rupp, Modern meson spectroscopy: the fundamental role of unitarity (arXiv:2012.03693)

• CommentRowNumber54.
• CommentAuthorUrs
• CommentTimeMay 10th 2021

• Kenzo Ishikawa, Kenji Nishiwaki, Kin-ya Oda, New effect in wave-packet scatterings of quantum fields: Saddle points, Lefschetz thimbles, and Stokes phenomenon (arXiv:2102.12032)

following Zoran’s announcement here

• CommentRowNumber55.
• CommentAuthorUrs
• CommentTimeMay 10th 2021

also to:

• Kenzo Ishikawa, Yutaka Tobita, Matter-enhanced transition probabilities in quantum field theory, Annals Phys. 344 (2014) 118-178 (arXiv:1206.2593)