# Introduction to Perturbative Quantum Field Theory

This is the beginning of a **series** that gives an introduction to perturbative quantum field theory (pQFT) on Lorentzian spacetime backgrounds in its rigorous formulation as **locally covariant perturbative algebraic quantum field theory**.

This includes the theories of quantum electrodynamics (QED) and electroweak dynamics, quantum chromodynamics (QCD), and perturbative quantum gravity (pQG) — hence the standard model of particle physics — on Minkowski spacetime (for particle accelerator experiments) and on cosmological spacetimes (for the cosmic microwave background) and on black-hole spacetimes (for black hole radiation).

This first part introduces the broad idea and provides a commented list of references. The next part will start with general discussion of a pivotal part of the theory: the “S-matrix” in causal perturbation theory, see below for the quick idea.

**Perturbation and Non-perturbation.**

Often “perturbative quantum field theory” (pQFT) is referred to simply as “quantum field theory” (QFT). However it is worthwhile to make the distinction explicit.

The word “perturbative” means that both the interactions between the fields/particles as well as the quantum effects they exhibit are assumed to be tiny — in fact infinitesimal — perturbations of the free (non-interacting) classical fields, hence of the undisturbed (matter-)waves freely propagating through the universe, with well-defined amplitudes at each spacetime point. More precisely this means that the observables of the theory (i.e. the numerical predictions that it makes about phenomena seen in experiment) are not true functions of the coupling constant ##g## (indicating the strength of the interaction) and of Planck’s constant ##\hbar## (indicating the strength of quantum effects), but just non-converging *formal* power series, at best “asymptotic series“.

This sounds like a drastically coarse approximation to the actual interacting and quantum world that we inhabit — and indeed it is. However, a remarkable mathematical fact is that this drastically coarse approximation is already extremely rich in phenomena and demanding in mathematical techniques; and a remarkable experimental fact about the observable universe is that this extremely coarse approximation suffices to explain/predict essentially all phenomena that are seen in high energy particle scattering experiments, and to high numerical precision. Hence while on the one hand pQFT is a dramatic triumph of pure thought over reality, on the other hand it amplifies the vastness of the presently unknown reality that must still lie beyond our present understanding: In a mathematically precise sense, pQFT desribes only the infinitesimal neighbourhood of the space of classical and free field theories inside the full space of quantum field theories.

Indeed some extremely basic aspects of observed physical reality are invisible to pQFT: Notably the curious phenomenon of QCD called asymptotic freedom means that it completely fails to describe the bound nature of the hadronic matter that all the world around us it made of (the confinement of quarks); it only applies well for high energy scattering processes seen in particle accelerators. This is believed to be related to a special non-perturbative nature of the QCD vacuum known as the instanton sea, to which we briefly turn below at the very end.

Hence we will eventually need to understand non-perturbative quantum field theory. This is by and large a wide open problem, both conceptually and physically. Presently not a single example of an interacting non-perturbative Lagrangian quantum field theory has been constructed in spacetime dimension ##\geq 4## (besides numerical simulation, such as lattice gauge theory). For the case of 4d Yang-Mills theory (such as QED and QCD) one single aspect of its non-perturbative quantization (the expected “mass gap”) is among the list of “Millenium Problems” listed by the Clay Mathematics Institute. Full non-perturbative Yang-Mills theory might well be a ##10^4## year problem, and full non-perturbative quantum gravity might be a ##10^5## year problem. But every journey needs to start with a first step in the right direction, and therefore a conceptually clean understanding of pQFT theory should be a helpful stepping stone towards these big open problems.

Unfortunately, even pQFT has been notorious for being believed to be conceptually mysterious. Modern textbooks will still talk about “divergencies that plague the theory” and, worse, appeal to the folklore of the “path integral” without offering precise clues as to its nature, thereby disconnecting the theory from the mathematically informed discourse that distinguishes modern physics from the “natural philosophy” of the ages before Newton. This is a historical remnant of the early days of the theory as conceived by Tomonaga, Schwinger, Feynman and Dyson, when many steps still proceeded by educated guesswork

**Causal Perturbation Theory.**

However, a mathematically rigorous formulation of pQFT on Minkowski spacetime (describing processes seen in particle accelerators such as the LHC experiment) with precise well-defined concepts had been fully established already by 1975, as summarized in the seminal Erice summer school proceedings of Velo-Wightman 76. Among other contributions, this included the formalization of the theory due to

- Henri Epstein, Vladimir Glaser,

“The Role of locality in perturbation theory“,

Annales Poincaré Phys. Theor. A 19 (1973) 211 (Numdam)

which has come to be known as ** causal perturbation theory**.

The key idea of this approach is to *define *the perturbative *scattering matrix *of the pQFT by imposing (i.e. axiomatizing) how it *should* behave — in particular how it should behave with respect to spacetime causality, whence the name — instead of trying to define it by a path integral.

The scattering matrix of a pQFT is the collection of all probability amplitudes for a given set of field quanta (particles) coming in from the far past, then perturbatively interacting with each other and hence scattering off each other, to finally emerge in the far future as another set of field quanta. The corresponding scattering probabilities (“scattering cross sections”) are manifestly the kind of information that may be measured in the detector of a particle accelerator, where to good approximation the incoming beams are the particles “from the far past” and the hits on the detectors around the point where the beams collide is the particles emerging “in the far future”. The theory has to make predictions for which of the many detector cells (at which angles from the colliding beams) is going to be triggered with which ratio given the incoming particle beam, and this is all encoded in the scattering matrix .

In traditional approaches of pQFT, the scattering matrix is written schematically as

$$

S(L_{\text{int}} )

\overset{\text{not really}}{=}

\int \left[D\phi\right] e^{ \tfrac{1}{i \hbar} \int_X L_{\text{free}}(\phi) } \, \exp\left( \tfrac{g}{i \hbar} \int_X\left( L_{\text{int}}(\phi) \right) \right)

$$

where the informal schematic right hand side expresses the idea that the probability amplitude for a scattering process is a sum (integral) over all spacetime field configurations ##\phi## (with the given asymptotic behaviour) of the complex phase determined by the free Lagrangian density ##L_{\text{free}}## and the interaction Lagrangian density ##L_{\text{int}}## evaluated at that field configuration and integrated over spacetime ##X##.

There is no known way to make sense of this integral, apart from toy examples. The reason that traditional pQFT textbooks nevertheles make some sense is that all that is really being used are some structural properties that such a would-be integral should have. To make such reasoning precise, one is to give up on the *idée fixe *of an actual path integration and simply state exactly which properties the expression ##S(L_{\text{int}})## is actually meant to have!

The key such property of the S-matrix is “causal additivity“. This essentially just says that all effects caused in some spacetime region must be in the causal future (and past) of that region.

The main result of causal perturbation theory is the *proof* that

- Causally additive perturbative S-matrices exist, hence pQFT exists, rigorously;
- at each order there is a finite-dimensional space of choices, the renormalization freedom;
- any two such choices are related by a unique re-definition of the Lagrangian densities (by “counter-terms”);
- these re-definitions form a group, the
*Stückelberg-Peterson renormalization group.*

This is known as the *main theorem of perturbative renormalization*, and we will discuss this in detail later in this series.

A textbook account of QED in causal perturbation theory is

- Günter Scharf,

“Finite Quantum Electrodynamics – The Causal Approach“,

Springer 1995

and electroweak theory, QCD as well as pQG are discussed this way in

- Günter Scharf,

“Quantum Gauge Theories — A True Ghost Story“,

Wiley 2001

**Perturbative Algebraic Quantum Field Theory.
**

A key technical tool that allows pQFT in causal perturbation theory to be well-defined is that the interactions of the fields are considered “smoothly switched off outside a compact spacetime region” (called “adiabatic switching“).

Originally this was considered just an intermediate technical step to separate the issue of “UV-divergences” (the definition of the S-matrix at coinciding interaction points) from the “IR-divergences”, namely from the issue of taking the “adiabatic limit” of the S-matrix in which the adiabatic switching is removed and interactionse are considered over all of spacetime.

But it had been observed already in Il’in-Slavnov 78 that for realistic quantum observables which are supported in a compact region of spacetime (corresponding to an experimental setup of finite extension in space and time) all that matters is that the interaction is “switched on” in the causal closure of the support of the observable, while outside its support it may be “adiabatically switched off” at will without actually changing the value of the observables (up to canonical unitary equivalence, see here). Moreover, the system of spacetime localized perturbative quantum observables obtained this way from the causal S-matrix turns out to satisfy axioms that had earlier been proposed in Haag-Kastler 64 to provide a complete mathematical characterization of the physical content of a pQFT: they form a *local net of observables*. This will be explained in detail in the next part of this series.

Haag-Kastler originally aimed, ambitiously, for axiomatization of the non-perturbative quantum field theory, and hence required the algebras of observables in the local net to be ##C^\ast##-algebras. Their formulation of non-perturbative quantum field theory via local nets of ##C^\ast##-algebras came to be known as algebraic quantum field theory (AQFT). Here in perturbation theory these algebras are just formal power series algebras (in the coupling constant and in Plancks’s constant), but otherwise they satisfy the original Haag-Kastler axioms. This way pQFT in the rigorous guise of causal perturbation theory came to be called **perturbative algebraic quantum field theory** (pAQFT, Brunetti-Dütsch-Fredenhagen 09).

The terminology overlaps a bit. It may be useful to think of it as follows:

- causal perturbation theory elegantly deals with the would-be “UV-divergencies” in pQFT by the simple axiom of the causally additivity S-matrix;
- perturbative AQFT in addition elegantly deals with the “decoupling of the IR-divergences” in pQFT by organizing the system of spacetime localized quantum observables into a local net of observables and thereby proving that the adiabatically switched S-matrix yields correct physical localized observables even without taking the problematic adiabatic limit (i.e. even without defining the theory in the infrared).

**Locally covariant pAQFT.**

While there are other equivalent rigorous formulations of pQFT on Minkowski spacetime, causal perturbation theory is singled out as being the one that generalizes well to QFT on curved spacetimes (Brunetti-Fredenhagen 99), hence to quantum field theory in the presence of a background field of gravity. This is important: For example pQFT on cosmological spacetime backgrounds describes the processes whose remnant is seen in the cosmic microwave background, while pQFT on black hole spacetime backgrounds describes black hole radiation.

One reason this works so well is that the axiom of causal additivity, which essentially defines the perturbative S-matrix, manifestly makes sense on general time-oriented spacetimes. But moreover there is some hard analysis which guarantees that the construction proof of the perturbative S-matrix does generalize from Minkowski spacetime to general time-oriented globally hyperbolic spacetimes: This requires finding

- generalizations of the Minkowski vacuum state to curved spacetimes to define the free quantum field theory via its Wick algebra (the “normal-ordered product”);
- corresponding Feynman propagators on curved spacetimes to define the perturbative interacting field theory via its time-ordered product.

This is non-trivial, because on general (even globally hyperbolic) spacetimes there exists no vacuum state, since there does not even exist a global concept of particles. But it turns out that time-ordered globally hyperbolic spacetimes do admit quantum states that, while not being vacuum states in general, do satisfy all the properties that are needed for the definition of a free field quantization, these are known as the Hadamard states, essentially unique up to addition of a regular term (Radzikowski 96). Moreover, each Hadamard state induces a corresponding Feynman propagator on the curved spacetime. With this in hand, the construction of the pQFT on curved spacetime may be obtained closely following the causal perturbation theory on Minkowski spacetime (Brunetti-Fredenhagen 00).

This then allows to generalize causal perturbation theory to construct pQFTs “general covariantly” on all time-oriented globally hyperbolic spacetimes, it has come to be called locally covariant algebraic quantum field theory (lcpAQFT).

**The traditional toolbox made rigorous.**

Eventually all the traditional lore and tools of pQFT have been (re-)obtained in precise form in the context of pAQFT. For instance:

- the Feynman perturbation series of the S-matrix in terms of Feynman diagrams and their dimensional regularization (Keller 10, Dütsch-Fredenhagen-Keller-Rejzner 14);
- the gauge fixed quantization of gauge theories via BRST-BV formalism (Fredenhagen-Rejzner 11, Rejzner 16);
- cosmological perturbation theory (Brunetti-Fredenhagen-Hack-Pinamonto-Rejzner 16)

A fairly comprehensive review of the theory as of 2016, with pointers to the research literature for further details, is in

- Katarzyna Rejzner,

“Perturbative Algebraic Quantum Field Theory“,

Springer 2016

In this series I will broadly follow this view of the subject, spelling out some more details here and there and maybe omitting other details at other places. I have a plan to follow, but will be happy to try to react to requests, comments and criticism from the PF-Insights readership.

**From first principles.**

Besides conceptual precision of our physical theories, we also want them to be conceptually coherent, preferably to follow from a small set of joint principles. While causal perturbation theory / perturbative AQFT is a mathematically precise formulation of traditional pQFT, many of its constructions appear somewhat *ad hoc,* even though well motivated and certainly right.

For instance the causal additivity axiom on the perturbative S-matrix was originally introduced as a really clever guess concerning the generalization to higher dimensional Lorentzian spacetimes of the simple 1-dimenional “path-ordering” in the Dyson formula (known as iterated integrals to mathematicians), and the construction of the interacting quantum observables from the S-matrix by Bogoliubov’s formula was mainly motivated from the fact that Bogoliubov gave that formula.

Of course, this being physics, all these constructions are physically justified by the fact that they do yield a precise formulation of traditional pQFT, and that traditional pQFT receives excellent confirmation in scattering experiments.

But even better than fitting our physical theory to observation in nature would be if we could derive the physical theory from deeper first theoretical principles, and then still match it with nature.

Here we should ask (at least): What does it mean to quantize any classical theory? And is pQFT the result of applying a general quantization prescription to classical field theory?

For ages people have chanted “The path integral does it!” in reply to this question. But as a matter of fact it does not — it does not even exist.

There are two general quantization prescriptions that do exist as mathematically well-defined concepts: geometric quantization and algebraic deformation quantization. Remarkably, it turns out that pAQFT does follow as a special case of “formal” (perturbative) algebraic deformation quantization (specifically Fedosov deformation quantization), and maybe yet more remarkable is that this was figured out only last year:

- Giovanni Collini,

“Fedosov Quantization and Perturbative Quantum Field Theory”

(arXiv:1603.09626) - Eli Hawkins, Kasia Rejzner,

“The Star Product in Interacting Quantum Field Theory”

(arXiv:1612.09157)

This may give some hints concerning the non-perturbative completion of the theory: A good concept of non-perturbative algebraic deformation quantization exists, called strict ##C^\ast##-algebraic deformation quantization.

Therefore it is suggestive that strict algebraic deformation quantization may be the right conceptual approach for attacking the non-perturbative quantization of Yang-Mills theory, as opposed, possibly, to the “constructive field theory” approach (which is trying to construct rigorous measure for the Wick rotated path integral) that is considered in the problem description by Jaffe-Witten.

**The unknown theory.**

This shows that despite the more than 40 years since Velo-Wightman 76, we may still be pretty much at the beginning of understanding the true conceptual nature of pQFT. There are various further hints that this is the case:

The available techniques for quantizing gauge theory in pQFT disregard the global topological sectors of the gauge field (instantons, argued to be crucial for the true vacuum of QCD). It follows on general grounds (Schreiber 14, Schenkel 14) that if these are to be included, then the space of local qantum observables can no longer be an ordinary algebra, but must become a “homotopical algebra” of sorts (“higher structure“). The principles of such “homotopical AQFT” are being explored (Benini-Schenkel 16, Benini-Schenkel-Schreiber 17), for review see Schenkel 17, but much remains to be done here.

Given that gauge theories and their instanton sectors are not some fringe topic in pQFT, but concern the core of the key application, the standard model of particle physics, much of the development of the theory may still lie ahead. And this is only pQFT. When this is finally really understood, mankind needs to look into non-perturbative QFT. Given the wealth of mathematical subtleties involved, this will only work with a conceptually clean rigorous formulation of the theory at hand. The following articles in this series will be an introduction to the the clean rigorous formulation of pQFT, as far as understood so far, in the guise of locally covariant perturbative AQFT.

This series on QFT continues here:

**A first idea of Quantum Field Theory**.

I am researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.

Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.

Right, the traditional lore highlights a would-be problem that does not actually arise because before it could, another problem kicks in (non-convergence of the perturbation series).

There is an interesting comment about this state of affairs in

Sure, but why do you say "only"? This is the point that the perturbative interacting observables, as long as they have bounded spacetime support, may consistently be computed in perturbation theory without passing to the adiabatic limit. This says that the perturbation theory is well defined, irrespective of infrared divergencies.

In this sense it seems correct to me to write that "pAQFT deals with the IR-divergencies by organizing the observables into a local net". Or maybe instead of "deals with" it would be better to write "circumvents the problem of". (?)

That's why low-energy QCD, if not using lattice-QCD simulations (within their range of applicability), is usually treated in terms of various effective field theories. For the light (+strange) quark domain one uses chiral symmetry (ranging from strict chiral perturbation theory for the ultra-low-energy limit to more or less "phenomenological" Lagrangians constrained by chiral symmetry). Another example is heavy-quark effective theory (also combined with chiral models if it comes to light-heavy systems like D-mesons).

The naive phenomenological physicists approach is indeed that such effective non-renormalizable theories use some low-loop orders of the effective theory with the corresponding low-energy constants, and this provides also predictive power. Often one has to resum ("unitarization"). Another quite popular non-perturbative approach is the renormalization-group approach ("Wetterich equation").

I guess, these more or less handwaving methods are not subject to the mathematically more rigorous approach discussed here, or can the here discussed approaches like pAQFT provide deeper insight to understand, why such methods are sometimes amazingly successful?

Another somewhat related question in my field (relativistic heavy-ion collisions) is the amazing agreement between relativistic viscous hdyrodynamics, derived from relativistic transport theory via the method of moments, Chapman-Enskog, and the like and full relativistic transport theory in a domain (of, e.g., Knudsen numbers around 1), where naively neither of these methods should work. On the other hand the finding of agreement suggest that two methods which are valid in opposite extreme cases (transport theory for dilute gases a la Boltzmann, where the particles scatter only rarely and otherwise are "asymptotically free" most of the time, i.e., large mean-free path vs. ideal hydrodynamics which is exact in the limit of vanishing mean-free path, i.e., the dynamics is slow compared to the typical (local) thermalization time).

It avoids having to deal with it, just as standard renormalized perturbation theory does. The infrared divergences still show up (in both cases) when you try to calculate S-matrix elements. Indeed, the perturbatively constructed S-matrix elements cannot even have mathematical existence in case of QCD, because of confinement – there are no asymptotic quark states.

Indeed this

isstandard renormalized perturbation theory, just done right.Nothing in pAQFT is alternative to or speculation beyond traditional pQFT. It is traditional pQFT, but done cleanly. The observation that I have been highlighting, that the algebra of quantum observables localized in any compact spacetime region may be computed, up to canonical isomorphism, already with the adiabatically switched S-matrix supported on any neighbourhood of the causal closure of that spacetime region, is "just" the formal justification for why indeed it is possible to ignore the adiabatic limit in perturbation theory.

This is exactly like causal perturbation theory is "just" the formal justification for the standard informal construction of the perturbation series.

Anyway, we don't have a disagreement about the facts, maybe just about the wording.

Yes. pAQFT removes cleanly all UV problems but

noneof the IR problems. The latter are resolved only by performing the adiabatic limit in causal perturbation theory – and there sit the constructive problems.The problem to be dealt with is that in the absence of the adiabatic limit, the perturbative S-matrix only exists in adiabatically switched form, which, taken at face value, does not make physical sense.

To make sense of causal perturbation theory in the absence of the adiabatic limit one needs to prove that the adiabatically switched S-matrix does, despite superficial appearance, serve to define the correct physical observables.

That proof is not completely trivial. It's result shows that the adiabatically switched S-matrix, while unable to define the global (IR) observables in the adiabatic limit, does, despite superficial appearance, induce the correct local net of localized physical perturbative observables. What is called pAQFT is just the name given to the result of this proof, the well-defined local net of perturbative observables obtained from unphysical switched S-matrices in absence of an adiabatic limit. This way pAQFT deals with the problem.

Without an argument like this you would have to make sense of the adiabatic limit in order to even define the perturbation theory. Which would essentially mean that you'd have to define the non-perturbative theory in order to define the perturbative theory. Which would be pointless.

I suppose the reason why we keep talking past each other is that you keep reading "deal with the IR problem" as "define the theory in the IR". But even before it gets to this ambitious and wide open goal, there is the problem of even defining the perturbation theory without taking the adiabatic limit.

But isn't the real solution of the IR problem in pQFT to use the correct asymptotic free states a la Kulish and Faddeev,

P. Kulish and L. Faddeev, Asymptotic conditions and infrared divergences in quantum electrodynamics, Theor. Math. Phys., 4 (1970), p. 745.

http://dx.doi.org/10.1007/BF01066485

and many other authors like Kibble?

In the standard treatment one uses arguments a la Bloch&Nordsieck, Kinoshita&Lee&Nauenberg and soft-photon/gluon resummation to resolve the IR problems. It's of course far from being rigorous.

I've also no clue, how you can define proper S-matrix elements without adiabatic switching (in both the remote past and the remote present). Forgetting this leads to pretty confusing fights in the literature. See, e.g.,

F. Michler, H. van Hees, D. D. Dietrich, S. Leupold, and C. Greiner, Off-equilibrium photon production during the chiral phase transition, Annals Phys., 336 (2013), p. 331–393.

http://dx.doi.org/10.1016/j.aop.2013.05.021

http://arxiv.org/abs/1310.5019

All this is, of coarse, far from being mathematically rigorous, but maybe it's possible to make it rigorous in the sense of pAQFT?

but this has nothing to do with the infrared (i.e., low energy) behavior, so you shouldn't use the term IR in this connection.

The basic conflict in QCD (or quantum Yang-Mills) is that there are no physical quark fields although there are perturbative quark fields.

In QED, the conflict is less obvious but you may look at Weinberg's Volume 1, Chapter 13 for a discussion of IR effects in QED.

These effects appear although the renormalized perturbative asymptotic series is already completely well-defined!The reason is that at a given energy the number of massless particles produced is unbounded, and to get physical results one must integrate over all these soft photon degrees of freedom. This is most correctly (but still in a mathematically nonrigorous way) handled by using coherent state techniques, as in the references given by Handrik van Hees.Exactly, and so one needs to prove that this may indeed be ignored in the perturbation theory. It is commonly said that causal perturbation theory disentangles the UV from the IR effects, but this only becomes completely true once one proves that the adibatically switched S-matrix produces correct physical observables even without taking its adiabatic limit.

I feel like we have exchanged this same point a couple of times now. And we still don't disagree about any facts, the only disagreement you have seems to be against the words by which I referred to the issue of proving that causal perturbation theory makes physical sense without taking the adiabatic limit. I called this "deal with the IR divergences". You seem to be saying that "deal with the IR divergences" sounds to you like "define the theory in the IR". Maybe a resolution would be if I changed the wording to "deal with the decoupling of the IR divergences"?

I am open for suggestions of the rewording, if it gets us past this impasse. You have so many interesting things to say, it is a pity that we seem to be stuck on a factual non-issue.

By the way, the next article in the series is ready, but it is being delayed by some formatting problems.

I have prepared my code for the next article in the "Instiki"-markup language, on an nLab page here

pAQFT 1: A first idea of quantum fieldsMy plan had been to simply port this code here to Physics Forums. Unfortunately, this turns out to be impractical, due to numerous syntax changes that would need to be made.

With Greg we are looking for a solution now. A technically simple solution would be to simply include that webpage inside an "iframe" within the PF-Insights article. But maybe this won't be well received with the readership here? If anyone with experience in such matters has a suggestion, please let me know.

Saying something like ''cleanly decouples the fully resolved UV issues from the (in causal perturbation theory still unresolved) IR issues'' would be fine with me.

Seemingly being stuck is also a factual non-issue. As you can see from my contributions, even when I discuss terminology, I enrich it with interesting information for other readers….

That's interesting. I always thought the IR divergences of standard PT are easier cured than the UV problems. It's just the soft-photon/gluon (or whatever is soft in some model with massless quanta) resummation, and then there's "theorems" like Bloch/Nordsieck and/or Kinoshita/Lee/Nauenberg:

https://en.wikipedia.org/wiki/Kinoshita-Lee-Nauenberg_theorem

What are the issues that you call them "still unresolved" in pAQFT?

The IR problem in QED is well understood only in the absence of nuclei (i.e., if only external fields are present beyond photons, electrons and positrons). If there are nuclei (whether assumed pointlike or with appropriate assumed form factors doesn't matter much) there are many bound states, and their treatment is very poorly understood.

Symptomatic for the state of affairs is the remark in Weinberg's QFT book, Vol.1, p.560:

''It must be said that the theory of relativistic effects and radiative corrections in bound states is not yet in entirely satisfactory state.''This is a very euphemistic description of what in reality is a complete and ill-understood mess.In QCD

alllow energy phenomena involve bound states – due to confinement, and these problems permeate everything.The Lee-Nauenberg theorem is flawed when analyzed carefully:

https://arxiv.org/abs/hep-ph/0511314

I need to digest the concept of a smooth set employed in your setting. Are there relations to the Conceptual Differential Calculus of Wolfgang Bertram?

(This exists in a number of variants, one of them being in https://arxiv.org/abs/1503.04623 .)

I always converted by hand, though it takes a considerable amount of time.

Interesting article!

You say that pQFT is a perturbational expansion not only in coupling constant but also in Plancks constant. The latter point is not immediately clear to me.

It is since the number of loops counts the powers of ##hbar##. This is clear from the path-integral formalism since you can understand the Dyson series also as saddle-point approximation of the path integral. See, e.g., Sect. 4.6.6 in

https://th.physik.uni-frankfurt.de/~hees/publ/lect.pdf

Yes, but the free theory which forms the starting point of the perturbation expansion contains already quantized electrons, photons, etc. How comes we consider this to be a classical theory?

Happily, no experiment occurs in an infinite laboratory, so IR divergences are a mere calculation inconvenience (it is not very practical to perform analytic calculations with big but finite IR cutoffs), not a genuine physical problem.

Here is how to see it:

The explicit ##hbar##-dependence of the perturbative S-matrix is

$$

S(g_{sw} L_{int} + j_{sw} A)

=

T expleft(

tfrac{1}{i hbar}

left(

g_{sw} L_{int} + j_{sw} A

right)

right)

,,

$$

where ##T(-)## denotes time-ordered products. The generating function

$$

Z_{g_{sw}L_{int}}(j_{sw} A)

;:=;

S(g_{sw}L_{int})^{-1} star S(g_{sw}L_{int} + j_{sw} A)

$$

involves the star product of the free theory (the normal-ordered product of the Wick algebra). This is a formal deformation quantization of the Peierls-Poisson bracket, and therefore the commutator in this algebra is a formal power series in ##hbar## that, however has no constant term in ##hbar## (but starts out with ##hbar## times the Poisson bracket, followed by possibly higher order terms in ##hbar##):

$$

[L_{int},A] ;=; hbar(cdots)

,.

$$

Now by Bogoliubov's formula the quantum observables are the derivatives of the generating function

$$

hat A

:=

tfrac{1}{i hbar} frac{d}{d epsilon}

Z_{g_{sw}L_{int}}(epsilon j A)vert_{epsilon = 0}

$$

Schematically the derivative of the generating function is of the form

$$

begin{aligned}

hat A

& :=

tfrac{1}{i hbar} frac{d}{d epsilon}

Z_{g_{sw}L_{int}}(epsilon j A)vert_{epsilon = 0}

\

& =

expleft(

tfrac{1}{i hbar}[g_{sw}L_{int}, -]

right)

(j A)

end{aligned}

,.

$$

(The precise expression is given by the "retarded products", see (Rejzner 16, prop. 6.1).)

By the above, the exponent ##tfrac{1}{hbar} [L_{int},-]## here yields a formal power series in ##hbar##,

and hence so does the full exponential.

Here is how this relates to loop order in the Feynman perturbation series:

Each Feynman diagram ##Gamma## is a finite labeled graph, and the order in ##hbar## to which this graph contributes is

$$

hbar^{ E(Gamma) – V(Gamma) }

$$

where

This comes about (see at

S-matrix — Feynman diagrams and Renormalizationfor details) because1) the explicit ##hbar##-dependence of the S-matrix is

$$

Sleft(tfrac{g}{hbar} L_{int} right)

=

underset{k in mathbb{N}}{sum} frac{g^n}{hbar^n n!} T( underset{k , text{factors}}{underbrace{L_{int} cdots L_{int}}} )

$$

2) the further ##hbar##-dependence of the time-ordered product ##T(cdots)## is

$$

T(L_{int} L_{int}) = prod circ expleft( hbar int omega_{F}(x,y) frac{delta}{delta phi(x)} frac{delta}{delta phi(y)} otimes right) ( L_{int} otimes L_{int} )

,,

$$

where ##omega_F## denotes the Feynman propagator and ##phi(x)## the (generic) field observable at point ##x## (where we are notationally suppressing the internal degrees of freedom of the fields for simplicity, writing them as scalar fields, because this is all that affects the counting of the ##hbar## powers).

The resulting terms of the S-matrix series are thus labeled by

1. the number of factors of the interaction ##L_{int}##, these are the vertices of the corresponding Feynman diagram and hence each contibute with ##hbar^{-1}##

2. the number of integrals over the Feynman propagator ##omega_F##, which correspond to the edges of the Feynman diagram, and each contribute with ##hbar^1##.

Now the formula for the

Euler characteristic of planar graphssays that the number of regions in a plane that are encircled by edges, thefaces, here thought of as the number of "loops", is$$

L(Gamma) = 1 + E(Gamma) – V(Gamma)

,.

$$

Hence a planar Feynman diagram ##Gamma## contributes with

$$

hbar^{L(Gamma)-1}

,.

$$

So far this is the discussion for internal edges. An actual scattering matrix element is of the form

$$

langle psi_{out} vert Sleft(tfrac{g}{hbar} L_{int} right)

vert psi_{in} rangle

,,

$$

where

$$

vert psi_{in}rangle

propto

tfrac{1}{sqrt{hbar^{n_{in}}}}

phi^dagger(k_1) cdots phi^dagger(k_{n_{in}}) vert vac rangle

$$

is a state of ##n_{in}## free field quanta and similarly

$$

vert psi_{out}rangle

propto

tfrac{1}{sqrt{hbar^{n_{out}}}}

phi^dagger(k_1) cdots phi^dagger(k_{n_{out}}) vert vac rangle

$$

is a state of ##n_{out}## field quanta. The normalization of these states, in view of the commutation relation ##[phi(k), phi^dagger(q)] propto hbar##, yields the given powers of ##hbar##.

This means that an actual scattering amplitude given by a Feynman diagram ##Gamma## with ##E_{ext}(Gamma)## external vertices scales as

$$

hbar^{L(Gamma) – 1 + E_{ext}(Gamma)/2 }

,.

$$

in the formula you sum over ##k## but the factors have an ##n##-dependence!