Introduction to Perturbative Quantum Field Theory
Show Complete Series
Part 1: Higher Prequantum Geometry I: The Need for Prequantum Geometry
Part 2: Higher Prequantum Geometry II: The Principle of Extremal Action – Comonadically
Part 3: Higher Prequantum Geometry III: The Global Action Functional — Cohomologically
Part 4: Higher Prequantum Geometry IV: The Covariant Phase Space – Transgressively
Part 5: Higher Prequantum Geometry V: The Local Observables – Lie Theoretically
Part 6: Examples of Prequantum Field Theories I: Gauge Fields
Part 7: Examples of Prequantum Field Theories II: Higher Gauge Fields
Part 8: Examples of Prequantum Field Theories III: Chern-Simons-type Theories
Part 9: Examples of Prequantum Field Theories IV: Wess-Zumino-Witten-type Theories
Part 10: Introduction to Perturbative Quantum Field Theory
Next: Mathematical 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 a general discussion of a pivotal part of the theory: the “S-matrix” in causal perturbation theory, see below for a quick idea.
Table of Contents
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 the 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 describes only the infinitesimal neighborhood 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 the 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 gave 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 behavior) 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 nevertheless 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 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 interactions 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 the 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 causal 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 that 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 free field quantization, these are known as the Hadamard states, essentially unique up to the 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 rigorously
Eventually, all the traditional lore and tools of pQFT have been (re-)obtained in the 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 the 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-dimensional “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 a 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 quantum 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 theory 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 the 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 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 a 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.
To everyone:
The terrific Leo1233783 above (who seems to like to remain anonymous) has now provided tremendous help with converting my source to PF-Insights format. A million thanks for that!!
This way we are finally set for the series on QFT to begin:
A First Idea of Quantum Field Theory – 20 Part Series
So far that page has just a few lead-in words and then a table of contents. In the next days and weeks, each chapter in that table of contents will appear as a separate PF-Insights article, and will be hyperlinked from that page.
Data received. A simple line transformer will be probably enough and extensible. Else I have other solutions. I'll come back to you in a private message :)
Did you find a solution ?No, I didn't.
if not, I can try.That would be great!!
What or where are exactly a typical input and a typical output ?Okay, my input source is here .
The output format that is needed for PF-Insights is mainly standard HTML, except for the maths formulas.
For the maths formulas
For the text decoration:
For the hyperlinks:
Other markup I use could just be stripped off. For instance
Please let me know if this is information that you can use. Otherwise I'll be happy to try to say more.
Thanks again for looking into this!
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.Did you found a solution ? if not, I can try. What or where are exactly a typical input and a typical output ?
Okay, I have touched the wording of these two remarks:
misprint: simimlarly
I'll try to rephrase these remarks a little to make sure that this becomes clear.Okay, I have touched the wording of these two remarks:
You refer to the path integral in Remarks 15.4 and 16.2 of
https://ncatlab.org/nlab/show/geometry+of+physics+–+A+first+idea+of+quantum+field+theory
So these are considered only as loose heuristics, not with a diffeological interpretation?Absolutely. I don't consider explicit path integrals. These remarks are meant for the reader who will have been exposed to the usual informal path integral lore and are meant to explain how the axiomatic construction of the S-matrix and of the interacting observables in causal perturbation theory are correspond to that informal lore.
I'll try to rephrase these remarks a little to make sure that this becomes clear.
Not sure what you want me to say. I won't be considering explicit path integrals. Maybe you could point to some concrete article and say something like: "How would you phrase that construction in terms of diffeological spaces!"?You refer to the path integral in Remarks 15.4 and 16.2 of
https://ncatlab.org/nlab/show/geometry+of+physics+–+A+first+idea+of+quantum+field+theory
So these are considered only as loose heuristics, not with a diffeological interpretation?
Actually limits of traces (e.g. over denser and denser lattice discretizations), and these limits are infinite-dimensional (i.e., functional) integrals.Not sure what you want me to say. I won't be considering explicit path integrals. Maybe you could point to some concrete article and say something like: "How would you phrase that construction in terms of diffeological spaces!"?
I suppose you are really thinking of taking the trace of a trace class operator?Actually limits of traces (e.g. over denser and denser lattice discretizations), and these limits are infinite-dimensional (i.e., functional) integrals.
The "microcausal local observables" out of which the Wick algebra and then the interacting field algebra are built want to be multilinear smooth functions on the diffeological space of field histories. Traditionally one ignores this and instead declares that they are given by distributions in the functional-analytic sense. But the statement is: it's the same! What looks like a functional-analytic definition of observables in fact is secretly the definition of smooth functionals on the diffeological space of field histories.I should add: To appreciate the usefulness, compare to the major trouble that Collini 16 has to go through with establishing the relevant smooth structure on observables (def. 15 and downwards).
How then do you define partition functions, which require infinite-dimensional integration!?I suppose you are really thinking of taking the trace of a trace class operator?
The link is blank.Sorry, here: ncatlab.org/nlab/show/distributions+are+the+smooth+linear+functionals
But surely functional analysis must enterYes, that's why I said "except for convenience": You want the traditional tools to reason about distributions, but the concept of distribution as such does not come externally onto the differential geometry of the space of field histories, but is part of it.
The "microcausal local observables" out of which the Wick algebra and then the interacting field algebra are built want to be multilinear smooth functions on the diffeological space of field histories. Traditionally one ignores this and instead declares that they are given by distributions in the functional-analytic sense. But the statement is: it's the same! What looks like a functional-analytic definition of observables in fact is secretly the definition of smooth functionals on the diffeological space of field histories.
Yes. The only point in the development that requires a diffeological space (or more generally a smooth set or super smooth set) to actually be a smooth manifold is if we want to integrate differential forms over it. For spacetime this is what we want to do in order to define local observables, and therefore it is required to be a smooth manifold..How then do you define partition functions, which require infinite-dimensional integration!?
By the way, another use of diffeological spaces is that in terms of these distribution theory becomes a topic native to differential geometry: see at
distributions are the smooth linear functionals .The link is blank.
This is very natural in pAQFT, as it makes the entire theory exist in differential geometry, with no fundamental recourse to functional analysis (except for convenience.)But surely functional analysis must enter once you have to show that solutions to differential equations exist!
It is also needed for defining the spectrum of the Hamiltonian!
By the way, another use of diffeological spaces is that in terms of these distribution theory becomes a topic native to differential geometry: see at
distributions are the smooth linear functionals .
This is very natural in pAQFT, as it makes the entire theory exist in differential geometry, with no fundamental recourse to functional analysis (except for convenience.)
So you require spacetime to be a manifold but the bigger objects (history spaces etc) only to be diffeological spaces?Yes. The only point in the development that requires a diffeological space (or more generally a smooth set or super smooth set) to actually be a smooth manifold is if we want to integrate differential forms over it. For spacetime this is what we want to do in order to define local observables, and therefore it is required to be a smooth manifold.
From the broader perspective of algebraic topology this is a familiar phenomenon: The theory lives on very general kinds of spaces, but as soon as one requires fiber integration to exist one gets that the fibers need to be manifolds equipped with suitable tangential structure.
Prodded by the feedback here, especially that by Arnold Neumaier, I have been restructuring the geometry background: Now it's being developed incrementally with the rest of the theory all in the one text here. I also brought in a skeleton of the remaining material including quantization. But not done typing yet…So you require spacetime to be a manifold but the bigger objects (history spaces etc) only to be diffeological spaces?
Mh? The lambshift is among the great successes of perturbative QED. What's deficient there?Well, the mathematical basis is deficient, as in most discussion of anything involving infrared problems. (Note that this is a thread about rigorous QFT!)
The usual discussions (e.g., Weinberg, Vol. 1, Section 14.3) involve a significant amount of handwaving that is hard to make rigorous, even from a perturbative point of view.
Even the Faddeev-Kulish procedure for treating dressed electrons (the simplest infrared problem) is at present not really rigorous; see https://www.physicsforums.com/posts/5863748 .
Mh? The lambshift is among the great successes of perturbative QED. What's deficient there?
Prodded by the feedback here, especially that by Arnold Neumaier, I have been restructuring the geometry background: Now it's being developed incrementally with the rest of the theory all in the one text here. I also brought in a skeleton of the remaining material including quantization. But not done typing yet…Thanks!
In the yet missing discussion of QED, you might also want to discuss the Lamb shift. Then you'll see that the perturbative approach (algebraic or not) is still severely deficient in the infrared and cures nothing….
Prodded by the feedback here, especially that by Arnold Neumaier, I have been restructuring the geometry background: Now it's being developed incrementally with the rest of the theory all in the one text here. I also brought in a skeleton of the remaining material including quantization. But not done typing yet…
What do you think of the following paper about QED?You should open a new thread fro discussing this!
Thanks for the feedback. I have edited a little in reaction, bringing up the examples further up front.
And I fixed the terminology overloading: "smooth space" is synonymous to "smooth set" (At some point I thought it would better match with "diffeological space", but then changed my mind). I made it read "smooth set" throughout now. Thanks for catching this.Thanks. A misprint: diffetrential
What do you think of the following paper about QED?
https://journals.aps.org/prd/pdf/10.1103/PhysRevD.96.085002
Here is the ansatz.
Infrared divergences in QED revisited
Daniel Kapec, Malcolm Perry, Ana-Maria Raclariu, and Andrew Strominger
Phys. Rev. D 96, 085002 (2017) – Published 10 October 2017
View attachment 212813
It has been found recently that the vacuum state of quantum electrodynamics (QED) is infinitely degenerate. The authors exploit this fact and show that any non-trivial scattering process in QED is necessarily accompanied by a transition among the degenerate vacua, making the scattering amplitude finite at low energy scales (infrared finite).
Recently, it has been shown that the vacuum state in QED is infinitely degenerate. Moreover, a transition among the degenerate vacua is induced in any nontrivial scattering process and determined from the associated soft factor. Conventional computations of scattering amplitudes in QED do not account for this vacuum degeneracy and therefore always give zero. This vanishing of all conventional QED amplitudes is usually attributed to infrared divergences. Here, we show that if these vacuum transitions are properly accounted for, the resulting amplitudes are nonzero and infrared finite. Our construction of finite amplitudes is mathematically equivalent to, and amounts to a physical reinterpretation of, the 1970 construction of Faddeev and Kulish.
Also, I am hoping that towards the end of your series that you will also talk about conformal field theory (CFT).
Well, I didn't know this page ,Thanks for the feedback. I have edited a little in reaction, bringing up the examples further up front.
And I fixed the terminology overloading: "smooth space" is synonymous to "smooth set" (At some point I thought it would better match with "diffeological space", but then changed my mind). I made it read "smooth set" throughout now. Thanks for catching this.
To repeat, there is a detailed pedestrian introduction at geometry of physics — smooth sets . I am getting the impression you have not actually looked at that yet. If you get distracted by the top level "Idea"-section you should jump right to where the discussion starts at "Model layer" (meaning: the down-to-earth version of the stroy) which starts out with the inviting sentence:
In this Model Layer we discuss concretely the definition of smooth spaces and of homomorphisms between them, together with basic examples and properties. Well, I didn't know this page. In pAQFT 1: A first idea of quantum fields, you referred at the first mention to smooth sets, which is quite abstract. You should have referred instead to the page you just mentioned, and you should add your present comment there at the top.
In geometry of physics — smooth sets, Definition 2.1 is still unmathematical and hence empty. It doesn't tell what sort of formal object a plot is, and it is not explained afterwards either. I guess you mean ''The elements of ##X(R^n)## are referred to as plots of ##X##'? This should then be part of Definition 2.2.1.
In Definition 2.2.2 it is clearer to write ''for each smooth function ##f## (called in the present context an abstract coordinate transformation)'' in place of ''for each abstract coordinate transformation, hence for each smooth function ##f##…'' and property 2.2.2 would read clearer if you wouldn't talk informally about change but only about composition. The informal interpretation (''to be thought of'') should not be part of the definition (which should be pure mathematics, introducing concepts, names, notation and properties) but a comment afterwards that adds intuition to the stuff introduced.
''But there is one more consistency condition'' – Is this still part of the definition, or is this a preamble to the definition of a smooth space in Definition 2.6?
And at that point (or later) I still don't know what a smooth set is! Is it just another word for a smooth space? Then why have two very similar names for it?
Nowhere the connection is made to diffeological spaces and to manifolds (except in a introductory sentence superficially justified very late in Remark 2.29, which is again quite abstract and does not make the connection transparent). But these should be the prime examples and hence figure prominently directly after Definition 2.2, to connect the general abstract concept to traditional objects more likely to be familiar to the reader. The example of the irrational torus as a diffeological space which is not a manifold would be instructive.
Our admin is fiddling with the installation right now.He brought it back now.
I think that giving in your proposed next insight article a category-free definition of a diffeological space such as that given in Wikipedia (rather than just saying ''information with the natural properties expected'') would be helpful, together with saying how a smooth set extends that notion.To repeat, there is a detailed pedestrian introduction at geometry of physics — smooth sets . I am getting the impression you have not actually looked at that yet. If you get distracted by the top level "Idea"-section you should jump right to where the discussion starts at "Model layer" (meaning: the down-to-earth version of the stroy) which starts out with the inviting sentence:
In this Model Layer we discuss concretely the definition of smooth spaces and of homomorphisms between them, together with basic examples and properties.
The link gives an error.Only as of a few minutes back, sorry for that. Our admin is fiddling with the installation right now.
There is also a useful book, Patrick Iglesias-Zemmour, Diffeology, 2013Yup, I have pointed that out before, last time in #63 .
A leisurely exposition of the grand idea behind this is at motivation for sheaves, cohomology and higher stacks.The link gives an error.
I think that giving in your proposed next insight article a category-free definition of a diffeological space such as that given in Wikipedia (rather than just saying ''information with the natural properties expected'') would be helpful, together with saying how a smooth set extends that notion.
There is also a useful book, Patrick Iglesias-Zemmour, Diffeology, 2013, and the author's diffeology blog.
Urs Schreiber wrote:
Here is how to see it: The explicit hbar-dependence of the perturbative S-matrix is
[tex]
S(g_{sw} L_{int} + j_{sw} A)
=
T \exp\left( t
\frac{1}{i \hbar}
\left(
g_{sw} L_{int} + j_{sw} A
\right)
\ right)
,
[/tex]
where T(−)T(−) denotes time-ordered products. The generating function
[tex]
Z_{g_{sw}L_{int}}(j_{sw} A)
S(g_{sw}L_{int})^{-1} \star S(g_{sw}L_{int} + j_{sw} A)
[/tex]
Schematically the derivative of the generating function is of the form
[tex]
hat A
:=
t \frac{1}{i \hbar} \frac{d}{d \epsilon}
Z_{g_{sw}L_{int}}(\epsilon j A)\vert_{\epsilon = 0}
=
\exp\left(
t\frac{1}{i \hbar}[g_{sw}L_{int}, -]
right)
(j A)
[/tex]
Darn, why is there no preview and edit in the insights comments?
That makes me sad. I carefully avoid all category theory in the notes and give down-to-earth discussion of everything. If you find any statement that requires mastery of category theory, I'll remove it.I wasn't referring to your explanations but rather to the other articles you were citing.
Wonderful!Glad you like it
Does not even require a master in category theory! :-)That makes me sad. I carefully avoid all category theory in the notes and give down-to-earth discussion of everything. If you find any statement that requires mastery of category theory, I'll remove it.