added also pointer to this review:

- Claus Kiefer, section 2.1.3 of:
*Quantum Gravity*, Oxford University Press 2007 (doi:10.1093/acprof:oso/9780199585205.001.0001, cds:1509512)

will make an `!include`

-entry with these references at

*classification of long-range forces – references*

for ease of including them in other relevant entries

]]>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:

Steven Weinberg,

*Feynman Rules for Any Spin. 2. Massless Particles*, Phys. Rev. 134 (1964) B882 (doi:10.1103/PhysRev.134.B882)Steven Weinberg,

*Photons and Gravitons in $S$-Matrix Theory: Derivation of Charge Conservationand Equality of Gravitational and Inertial Mass*, Phys. Rev. 135 (1964) B1049 (doi:10.1103/PhysRev.135.B1049)Steven Weinberg, _Photons and Gravitons in Perturbation Theory: Derivation of Maxwell’s and Einstein’s Equations,” Phys. Rev. 138 (1965) B988 (doi:10.1103/PhysRev.138.B988)

Paolo Benincasa, Freddy Cachazo,

*Consistency Conditions on the S-Matrix of Massless Particles*(arXiv:0705.4305)David A. McGady, Laurentiu Rodina,

*Higher-spin massless S-matrices in four-dimensions*, Phys. Rev. D 90, 084048 (2014) (arXiv:1311.2938, doi:10.1103/PhysRevD.90.084048)

Quick review:

- Daniel Baumann,
*What long-range forces are allowed?*, 2019 (pdf)

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

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

]]>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.

]]>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)

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.)

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).

]]>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.

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.

]]>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.

]]>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 $g$.

]]>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.

]]>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

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

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

]]>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.

have added another section: *Effective action*

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…

]]>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.

]]>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.

]]>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.

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

]]>That’s the paper Jacques Distler discussed here.

]]>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

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