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.
I vaguely remember that all Feynman graphs which contain no loops are lowest order in hbar. Nevertheless we speak of the scattering of a single photon from a single electron which already implies quantization of particle number. So it is not a classical field we are describing.See also https://physics.stackexchange.com/questions/348942
Urs Schreiber
Solving classical field equations
(pdf)
Wonderful! Does not even require a master in category theory! :-)
An account that makes explicit how the tree level perturbation series is just the perturbation series for the classical field equations is in
Robert Helling,
Solving classical field equations
(pdf)
One should also note that in relativistic QFT particle number is only well defined for asymptotic free states. That's why cross sections and related quantities are defined via the S-matrix which gives transition rates between asymptotic free states.
DrDuI know what you mean but I don't see how this fits with my naive view of e.g. QED where, when describing the scattering of, say, an electron from a photon. I vaguely remember that all Feynman graphs which contain no loops are lowest order in hbar. Nevertheless we speak of the scattering of a single photon from a single electron which already implies quantization of particle number. So it is not a classical field we are describing.The collection of all tree diagrams really describes perturbation theory of a classical field theory in terms of powers of the coupling constant (one power per vertex)! This shows that Feynman diagrams have nothing to do with particles, except as a suggestive way of talking!
Wow, you never stop learning! So all this Feynman stuff in tree order is basically only first quantization?
So if I want count photons and electrons, I have to go beyond tree level. Can you show me how to see this?
It needs not even be Grassmann. The original paper dealt with the Dirac equation as if you could use it in the same way as the non-relativistic Schrödinger wave function.
Klein, O. & Nishina, Y. Z. Physik (1929) 52: 853. https://doi.org/10.1007/BF01366453
English translation
O. Klein and Y. Nishina, "On the Scattering of Radiation by Free Electrons According to Dirac's New Relativistic Quantum Dynamics", The Oskar Klein Memorial Lectures, Vol. 2: Lectures by Hans A. Bethe and Alan H. Guth with Translated Reprints by Oskar Klein, Ed. Gösta Ekspong, World Scientific Publishing Co. Pte. Ltd., Singapore, 1994, pp. 113–139.
But you need a quantized electron, or is it sufficient to use a classical Grassmann valued field for the electron?
The point is that you can get the Klein-Nishina formula for Compton scattering, i.e., in the modern way by just evaluating the tree-level Feynman diagrams, by investigating scattering of a classical electromagnetic wave on an electron using the Dirac equation of the electron. The same holds for the photoeffect. You come quite far with the semiclassical approximation in QED, i.e., quantizing only the electron and keep the em. field as classical. The most simple argument for the necessity to quantize also the em. field is the existence of spontaneous emission, which afaik cannot be derived from the semiclassical theory.
A. NeumaierIn an ##hbar## expansion, the theroy is expanded around the interacting classical limit, not around a free quantum field theory.I know what you mean but I don't see how this fits with my naive view of e.g. QED where, when describing the scattering of, say, an electron from a photon. I vaguely remember that all Feynman graphs which contain no loops are lowest order in hbar. Nevertheless we speak of the scattering of a single photon from a single electron which already implies quantization of particle number. So it is not a classical field we are describing.
DrDuYes, 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?In an ##hbar## expansion, the theroy is expanded around the interacting classical limit, not around a free quantum field theory.
dexterciobyUrs, can you write in no more than 2 lines the connection (if any) between diffeologic spaces and topological spaces?Good question.
A topological space is a set ##X## equipped with information which functions ##mathbb{R}^n longrightarrow X## are continuous.
A diffeological space is a set ##X## equipped with information which functions ##mathbb{R}^n longrightarrow X## are smooth.
A leisurely exposition of the grand idea behind this is at motivation for sheaves, cohomology and higher stacks.
Urs, can you write in no more than 2 lines the connection (if any) between diffeologic spaces and topological spaces?
Maybe I should re-amplify the point about diffeology:
The concept of "smooth sets" subsumes that of diffeological spaces , and essentially all examples of relevance in field theory fall in the class of diffeological space.
The diffeological spaces are the "concrete smooth sets". So the concept of diffeological spaces is a generalization of that of smooth manifolds, and the concept of smooth sets is yet a further generalization of diffeological spaces.
As far as the formalism is concerned, it is no harder to work in the generality of smooth sets than it is to work in the intermediate generality of diffeological spaces. But essentially all examples of smooth sets that appear in the context of field theory are actually diffeological spaces, and so if you are looking for literature on the subject, you should look for the keywords "diffeological spaces".
In particular, to highlight this once more, there is a down-to-earth non-categorical completely introductory and detailed textbook introducing all the standard material of differential geometry in terms of diffeological spaces.This is
Patrick Iglesias-Zemmour:
Diffeology
Mathematical Surveys and Monographs
Volume: 185; AMS 2013;
I don't think that studying this is necessary for following my notes, since the basic idea is really simple and really close to how physicists think anyway, but to all readers who do want to dig deeper into this differential geometric background to the theory I recommend looking at this textbook.
A. NeumaierI don't believe this. Already for classical existence proofs one needs to extend the infinitesimal description in terms of differential equations to a global description in terms of histories, and they are related as Lie algebras and Lie groups. In this relation the notion of tangent space is essential.But not the tangent space to the space of histories.
A. NeumaierTo be more concrete, can you show me a paper where Lie groups are studied from a smooth set point of view, and the typical difficulties in infinite dimensions are absent?The term to look for is "diffeological groups". For instance here
Urs Schreiberfor smooth sets which are not manifolds there appear different inequivalent concept of tangent spaces. But for applications to field theory, it turns out all one needs is differential forms on the spaces of field histories.I don't believe this. Already for classical existence proofs one needs to extend the infinitesimal description in terms of differential equations to a global description in terms of histories, and they are related as Lie algebras and Lie groups. In this relation the notion of tangent space is essential.
To be more concrete, can you show me a paper where Lie groups are studied from a smooth set point of view, and the typical difficulties in infinite dimensions are absent?
A. NeumaierApparently it means only that the standard examples are examples of the generalized concepts, not that the standard theorems also hold for the generalized concepts. I was asking for the latter. To get strong results for infinite-dimensional Lie groups, the latter probably need a full manifold structure, and not only a smooth set structure.One needs full manifold structure for surprisingly few things. Everything that involves only differential forms instead of vector fields generalizes to all smooth sets. If vector fields get involved one needs to be careful, as for smooth sets which are not manifolds there appear different inequivalent concept of tangent spaces. But for applications to field theory, it turns out all one needs is differential forms on the spaces of field histories.
A. NeumaierThe title is somewhat misleading. It keeps growing. In a few weeks the quantization will be covered, please have a little patience.
You see, this is my source from which I was going to successively produce Insights-articles here, one at a time. You are only seeing my master source only because the conversion to Insights-articles is running into technical difficulties.
A. NeumaierIn Section 2 of your nLab draft, you are unnecessarily onomatopoetic: ''expressions with repeated indicices''Thanks, fixed now.
A. NeumaierWell, I'd like to have a mathematically precise specification.A detailed introduction is here: geometry of physics — smooth sets . The quick way to state the definition is to say that a smooth set is a sheaf on the site whose objects are Cartesian spaces, whose morphisms are smooth functions between them, and whoe Grothendieck pre-topology is that coming from good open covers. But the introduction at geometry of physics — smooth sets spells this out in elementary terms, not assuming any sheaf-theoretic background (or any other background except the concept of smooth functions between Cartesian spaces).
A. NeumaierCan I replace Cartesian space by ##R^n##?Here "Cartesian space" means precisely : ##mathbb{R}^n##s.
A. NeumaierCan one use fields like the p-adic numbers in place of the reals? (Some people are interested in p-adic physics!)The analous definition work for any choice of test spaces with a concept of covering defined. If you take something like affinoid domains as in rigid analytic geometry you get somethng that deserves to be called "p-adic analytic sets" or the like. More relevant for physics is for instance the Choice of Stein spaces, in order to get "complex analytic sets". If you take affine schemes, you get ordinary algebraic spaces (among which ordinary schemes).
A. NeumaierDo the smooth functions have to be defined on all of ##R^n## or only on open subsets? What is the precise compatibility condition?One may equivalently take the site of open subsets of Cartesian spaces. Some authors do that. It does't change the resulting concept, though. The compatibility condition is gluing: The choice of what counts as a smooth function into your smooth set must be so that if you cover one Cartesian space by a set of other Cartesian spaces, then the smooth functions out of the former must be uniquely fixed by their restriction to those patches of the cover.
A. NeumaierI'd also like to be able to decide on the basis of the information provided questions such as whether any algebraic variety is a smooth set in an appropriate sense, or what restrictions apply.There is once you decide on what should count as a smooth function from a Cartesian space to the algebraic variety. In general there will not be a useful such choice, but if your algebraic variety happens to be complex-analytic, then of course there is, and you recover the underlying smooth manifold.
A. NeumaierBut one can fist define what a group is, and then define what a homomorphism between groups is, and one does not the second concept to understand the first and to play with examples.Same for smooth sets. To recall, a smooth set is defined to be a choice, for each ##n in mathbb{R}^n## of a set, regarded as the set of smooth functions from ##mathbb{R}^n## to the smooth set (called "plots"), such that this choice is compatible with smooth functions ##mathbb{R}^{n_1} to mathbb{R}^{n_2}## and respects gluing, as above.
That's the definition. Next, a homomorphism between smooth sets is a map that takes these plots to each other, again respectiing the evident compositions.
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
The title is somewhat misleading: 98% of the text is about classical field theory and only a few paragraphs at the end hint at quantum field theory through a sequence of remarks, without giving significant substance or interpretation. More appropriate would be something like ''The classical background needed for quantum field theory''.
The statement about the full embedding that I quoted means precisely that all standard theory embeds.Apparently it means only that the standard examples are examples of the generalized concepts, not that the standard theorems also hold for the generalized concepts. I was asking for the latter. To get strong results for infinite-dimensional Lie groups, the latter probably need a full manifold structure, and not only a smooth set structure.
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
In Section 2 of your nLab draft, you are unnecessarily onomatopoetic: ''expressions with repeated indicices''
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?I believe that the Wetterich equation can be described on a reasonably rigorous level, though still with some uncontrolled approximations. But I haven't seen any concrete work in this direction by mathematical physicists.
A smooth sets is, as I mentioned, a declaration for each Cartesian space (abstract coordinate system) of what counts as a smooth function out of that space into the smooth set, subject to the evident condition that this is compatible with smooth functions between Cartesian space (abstract coordinate changes).Well, I'd like to have a mathematically precise specification. Can I replace Cartesian space by ##R^n##? Are there other significantly different Cartesian spaces that need to be catered for? Can one use fields like the p-adic numbers in place of the reals? (Some people are interested in p-adic physics!) Do the smooth functions have to be defined on all of ##R^n## or only on open subsets? What is the precise compatibility condition?
I wonder whether after all these things have been spelled out, the definition is really simpler than that of a manifold over a convenient vector space (in the sense of Kriegl and Michor), say.
I'd also like to be able to decide on the basis of the information provided questions such as whether any algebraic variety is a smooth set in an appropriate sense, or what restrictions apply.
When you define what a group is, you also want to know what counts as a homomorphism between two groups. The high-brow term for this is "the category of groups" but it means nothing else but "groups and maps between them".But one can fist define what a group is, and then define what a homomorphism between groups is, and one does not the second concept to understand the first and to play with examples. I'd like to have a definition of smooth sets phrased in the same spirit. The categorial interpretation should be a second step that allows one to make certain universal constructions available, and not something already integrated into the definition.
But do you get the same strong existence results as one gets for infinite-dimensional Lie groups, say?The statement about the full embedding that I quoted means precisely that all standard theory embeds.
What I meant is: To understand the concept of a smooth set you apparently need the whole category.Not in any non-trivial sense, no. A smooth sets is, as I mentioned, a declaration for each Cartesian space (abstract coordinate system) of what counts as a smooth function out of that space into the smooth set, subject to the evident condition that this is compatible with smooth functions between Cartesian space (abstract coordinate changes).
That''s it.
Or can you define a smooth set without reference to a category, just as you can define a group without reference to a category?Sure. But this is a triviality unless you read some superficial scariness into the innocent word "category". When you define what a group is, you also want to know what counts as a homomorphism between two groups. The high-brow term for this is "the category of groups" but it means nothing else but "groups and maps between them".
Sure if you want to check full details from the foundations, you need to look at the full details starting from the foundations. But to just read the notes on QFT, you can take this as a black box. But the details are also easy to check, the category theory is minimal. It's really true that the theory of smooth sets is completely elementary certainly as compared to the heavy analytic machinery required for the usual infinite-dimensional manifold structures.But do you get the same strong existence results as one gets for infinite-dimensional Lie groups, say? It is not without reason that the machinery is heavy!
Nah. You need as many smooth sets as you would otherwise consider smooth manifolds. One for spacetime, one for the field bundle, one for its jet bundle, one for the space of sections, one for the phase space. Just the obvious spaces that are to be discussed in QFT.
That these objects will enjoy Cartesian closure is not something you need to take care of, that's guaranteed by them being smooth sets.What I meant is: To understand the concept of a smooth set you apparently need the whole category. Or can you define a smooth set without reference to a category, just as you can define a group without reference to a category?
I'd like to have a precise formulation of this theorem and what precisely is needed.Statement and pointers to proofs are given at nLab:manifold structure of mapping spaces in this section.
To turn it into rigorous math one needs to specify the properties demanded from a collection of smooth maps to deserve the name so that the foundation is sound.Sure if you want to check full details from the foundations, you need to look at the full details starting from the foundations. But to just read the notes on QFT, you can take this as a black box. But the details are also easy to check, the category theory is minimal. It's really true that the theory of smooth sets is completely elementary certianly as compared to the heavy analytic machinery required for the usual infinite-dimensional manifold structures.
Apparently you need not just one smooth set but a whole collection (or category?) of smooth sets closed under Cartesian products and who knows what else, to be able to formulate this.Nah. You need as many smooth sets as you would otherwise consider smooth manifolds. One for spacetime, one for the field bundle, one for its jet bundle, one for the space of sections, one for the phase space. Just the obvious spaces that are to be discussed in QFT.
That these objects will enjoy Cartesian closure is not something you need to take care of, that's guaranteed by them being smooth sets.
I'd like to have a precise formulation of this theorem and what precisely is needed.True, I should have provided this right away: Statement and pointers to the proofs are given here .
Here is all there is to it:
a "smooth set" (or "diffeological space") is defined simply by declaring which maps from RnRnmathbb{R}^ns into it are supposed to be smooth. There is some evident consistency conditions on such a declaration, but it just says what you intuitively expect to hold true anyway, so you need not bother on the first go.Yes, I looked at smooth sets since this was the first new term in the ''part to follow''. I didn't know it so I had to look it up – and the explanation given was mystifying. Your new explanation sounds simpler but it is not a mathematical definition but only handwaving. To turn it into rigorous math one needs to specify the properties demanded from a collection of smooth maps to deserve the name so that the foundation is sound. Apparently you need not just one smooth set but a whole collection (or category?) of smooth sets closed under Cartesian products and who knows what else, to be able to formulate this.
And it is a theorem that this is all we actually need.I'd like to have a precise formulation of this theorem and what precisely is needed.
Actually I find the category-theoretic language that you employ in your second part quite intimidating,Hm. There is intentionally zero category theory but only basic differential geometry in the part to follow !
I am guessing what you mean is that you clicked on the links to "smooth set" and found material not meant for your eyes. You don't need this.
The beginning of the part to follow , the section titled "Geometry" is meant to be all that you need to know for handling differential forms on spaces of field histories in a simple and yet precise manner. Maybe we could have some back and forth on it here, so that I see which words I should add.
Here is all there is to it:
a "smooth set" (or "diffeological space") is defined simply by declaring which maps from ##mathbb{R}^n##s into it are supposed to be smooth. There is some evident consistency conditions on such a declaration, but it just says what you intuitively expect to hold true anyway, so you need not bother on the first go.
For example the smooth structure on the set ##[Sigma,F]## of smooth functions from spacetime ##Sigma## to some field fiber ##F## is defined like so:
For ##U = mathbb{R}^n## some Cartesian space, then a function
$$ Phi_{(-)} : U longrightarrow [Sigma,F]$$
is declared to be smooth precisley if the cooresponding funcion in both variables
$$ Phi_{(-)}(-) : U times Sigma longrightarrow Y$$
is smooth in the ordinary sense.
That's it. Compare to what you need to go through to define Frechet manifold structure or similar on this space. This here is dead easy. And it is a theorem that this is all we actually need.
The only other thing to know is now how to define a differential form on a smooth set. Like so: it's a system of ordinary differential forms on Cartesian spaces, one for each map out of the latter into our "smooth set" that we declared to be a smooth function. The only condition is that under composition of smooth functions these forms pull along compatibly. That's it.
For example if ##[X,Y]## is the smooth mapping space as above, then a differential n-form ##omega in Omega^n([X,Y])## is for each Cartesian space ##U## and each smooth function ##Phi_{(-)}(-) colon U times X to Y## an ordinary differential n-form on $U$, suggestively denoted
##left(Phi_{(-)}right)^ast omega in Omega^n(U)##, such that for every ordinary smooth function ##V overset{f}{to} U## from another Cartesian space ##V##, we have the evident consistency relation ##(Phi_{f(-)})^ast omega = f^ast (Phi_{(-)})^ast omega ## between ordinary differential forms on ##V##.
That's precisely all background on differential geometry of mapping spaces that is needed for the notes . Nothing more. In particular no category theory is needed (the category theory is doing its work silently in the background, but you need not worry about that).
Right, that would be part of my plans for a later chapter of the series that is meant to be started here. But I would like to proceed systematically from the beginning and first have some articles on the basics.
I'll try to slowly-but-surely work through converting to PF-Insights-formatting the material that I have already prepared. It might be going slow, though. I am still hoping some kind soul could be found who would write a simple script that could autmatically convert my source code. This should be a triviality for experts versed in such matters…Actually I find the category-theoretic language that you employ in your second part quite intimidating, whereas the content of post #40 is of the same abstraction level as part 1 (which we discuss here) and hence accessible to a wider audience. I can understand thre categorial stuff only by translating item by item to something more concrete, whereas reading stuff at the abstraction level of standard quantum field theory is much easier to read. Thus putting the category-heavy stuff later, or accompanying it by a detailed interpretation key for nonexperts in categories, might enhance understandability a lot.
You could turn post #40 with little extra work into a self-contained Insight article.Right, that would be part of my plans for a later chapter of the series that is meant to be started here. But I would like to proceed systematically from the beginning and first have some articles on the basics.
I'll try to slowly-but-surely work through converting to PF-Insights-formatting the material that I have already prepared. It might be going slow, though. I am still hoping some kind soul could be found who would write a simple script that could autmatically convert my source code. This should be a triviality for experts versed in such matters…
Thanks! Fixed now.You could turn post #40 with little extra work into a self-contained Insight article. It would display the information more prominently and easier to find again and to reference.
in the formula you sum over ##k## but the factors have an ##n##-dependence!Thanks! Fixed now.
1) the explicit ##hbar##-dependence of the S-matrix isin the formula you sum over ##k## but the factors have an ##n##-dependence!
You say that pQFT is a perturbational expansion not only in coupling constant but also in Planck's constant. The latter point is not immediately clear to me.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 Renormalization for details) because
1) 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 graphs says that the number of regions in a plane that are encircled by edges, the faces, 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
$$
hbar^{L(Gamma) – 1 + E_{ext}(Gamma)/2 }
,.
$$
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.
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?
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.
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
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 .)
My 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.I always converted by hand, though it takes a considerable amount of time.
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 all low 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
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?
Maybe a resolution would be if I changed the wording to "deal with the decoupling of the IR divergences"?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.
You have so many interesting things to say, it is a pity that we seem to be stuck on a factual non-issue.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….
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
My 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.
but this has nothing to do with the infrared (i.e., low energy) behaviorExactly, 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.
induce the correct local net of localized physical perturbative observables.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.
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?
Yes. pAQFT removes cleanly all UV problems but none of the IR 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.
Anyway, we don't have a disagreement about the facts, maybe just about the wording.Yes. pAQFT removes cleanly all UV problems but none of the IR problems. The latter are resolved only by performing the adiabatic limit in causal perturbation theory – and there sit the constructive problems.
It avoids having to deal with it, just as standard renormalized perturbation theory does.Indeed this is standard 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.
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". (?)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.
The former only shows how to construct approximate observables at each order – This has nothing to do with the infrared limit. The latter is precisely the adiabatic limit. It is there (and only there) where the particle content of the theory (and hence issues such as confinement) would appear. For example, in QCD, the perturbative theory is in terms of quarks, but the infrared completed theory has no quarks (due to confinement) but only hadrons.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).
The former only shows how to construct approximate observables at each order – This has nothing to do with the infrared limit.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". (?)
Yes. And as you hint at, this is a totally unfounded argument. In an asymptotic power series in two variables there are an infinite number of terms to be chosen (at each order a growing number more), but nobody concludes that therefore power series at low order are not predictive. They are highly predictive as long as one is in the range of validity of the asymptotic expansion at this order (i.e., typically as long as the first neglected order contributes very little).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
[…]
Even though Dyson’s argument is unquestionable, it was hushed up or decried for many years: the scientific community was not ready to face the problem of the hopeless divergency of perturbation series
[…]
The modern status of divergent series suggests that techniques for manipulating them should be included in a minimum syllabus for graduate students in theoretical physics. However, the theory of divergent series is almost unknown to physicists, because the corresponding parts of standard university courses in calculus date back to the mid-nineteenth century, when divergent series were virtually banished from mathematics.
I believe I did provide a pointer, to the section here , […]
What you are referring to, and what remains unsolved in generality, is taking the adiabatic limit of the coupling constant.The former only shows how to construct approximate observables at each order – This has nothing to do with the infrared limit. The latter is precisely the adiabatic limit. It is there (and only there) where the particle content of the theory (and hence issues such as confinement) would appear. For example, in QCD, the perturbative theory is in terms of quarks, but the infrared completed theory has no quarks (due to confinement) but only hadrons.
The usual argument is that if there is an infinite number of terms to be chosen, then the theory is not predictive.Yes. And as you hint at, this is a totally unfounded argument. In an asymptotic power series in two variables there are an infinite number of terms to be chosen (at each order a growing number more), but nobody concludes that therefore power series at low order are not predictive. They are highly predictive as long as one is in the range of validity of the asymptotic expansion at this order (i.e., typically as long as the first neglected order contributes very little).
Of course, for gravity at the Planck scale (and for QCD at low energies, etc.) one expects that one is outside this domain, so that the value of the expansion becomes questionable at each order. Thus a perturbative theory is in many respects not a substitute for a nonperturbative version of the theory.
Very interesting. I subscribe to this thread. Thank you very much for sharing, Urs.
Note to those that like me do not understand all the mathematical detail. Of course try to correct that, but still read it and try to get a feel for the issues at the frontiers of current physics.That's the right attitude! Learning by osmosis.
And by asking questions! Feel invited to ask the most basic questions that come to mind.
How are pQFT and pAQFT related to lattice gauge theory? Would you accept lattice gauge theory, at least Hamiltonian lattice gauge theory, as a non-perturbative and physically relevant version of QED?Sure, I was briefly referring to this in the paragraph starting with "Hence we will eventually need to understand non-perturbative quantum field theory."
I suppose the point is that Monte-Carlo evaluation of lattice gauge theory is more like computer–simulated experiment than like theory. It allows us to "see" various effects, such as confinement, but it still does not "explain" them in the sense that we could derive these effects structurally.
Another problem is that lattice gauge theory relies on Wick rotation, so it does not help with pQFT on general curved spacetimes.
Also, why do you say the path integral doesn't exist? At least in 2D and 3D, doesn't constructive field theory, which you mention at the end, show that the path integral exists?Yes, that's what I meant by "toy examples" where I wrote "There is no known way to make sense of this integral, apart from toy examples"
Now of course it may be unfair to refer as a "toy example" to all the great effort that went into "constructive QFT". Mathematically it is a highly sophisticated achievement. But it remains a matter of fact that as far as the physical problem description is concerned, the real thing is interacting Lorentzian QFT in dimensions four or larger.
I should be careful with saying "the path integral does not exist in general", because there is no proof besides experience, that it does not. Maybe at one point people can make sense of it. But even so, it seems to me that the results of "constructive QFT" show one thing: even if one can finally make sense of the path integral, it does not seem all too useful. Very little followup results seem to have come out of the construction of interacting scalar field theory in 3d via a rigorous Euclidean path integral. If we follow the tao of mathematics, the path integral just does not seem to be the right perspective. Or so I think.
Could you please make a printable version of your slides https://ncatlab.org/schreiber/files/SchreiberTrento14.pdf, with the repetitions removed? (This is just an additional line in the latex before compilation.)Ah, I didn't code this with the "beamer" package, but "by hand". Is there a tool that could extract from the pdf just those pages that have the screen completed, and put these together to a smaller file? Sorry for the trouble
The link (web) to Schenkel in the nlab article https://ncatlab.org/schreiber/show/Higher+field+bundles+for+gauge+fields is not working.Thanks for the alert! I have fixed it now. The working link is here:
I recommend also Alexander's more recent exposition:
You stated in the article, ''perturbative AQFT in addition elegantly deals with the would-be “IR-divergencies” in pQFT by organizing the system of spacetime localized quantum observables into a local net of observables.''
I don't agree. The infrared problem remains unsolved in perturbative AQFT. You haven't even given a link to a reference where your claim would be addressed.I believe I did provide a pointer, to the section here , but I could have emphasized this further. This will be the topic of the next (or next to next) installment.
What you are referring to, and what remains unsolved in generality, is taking the adiabatic limit of the coupling constant. But the insight of pAQFT is that this limit need not even be taken in order to obtain a well defined (perturbative) quantum field theory!
Namely the observation is that
Sorry for the slow replies, I am seeing the further comments only now for some reason.
Note that there is already an insight article on causal perturbation theory complementing the present one.Right, sorry, I should have pointed to that. I do have pointers to your FAQ on the nLab here .
Your point 2 for causal perturabtion theory (finitely many free constants) holds of course only for renormalizable theories.I was careful to write "at each order there is a finite-dimensional space of choices" (emphasis added).
As you hint at, this is an important subtlety that is usually glossed over in public discussion: At each order of perturbation theory, there is a finite dimensional space of counter-terms to be fixed. As the order increases, the total number of counterterms may grow without bound, and then people say the theory is "non-renormalizable". But this is misleading terminology: The theory is still renormalizable in the sense that one may choose all counterterms consistently, even f there are infinitely many. What the traditional use of "non-renormalizable" really means to convey is some idea of predictivity: The usual argument is that if there is an infinite number of terms to be chosen, then the theory is not predictive. Of course a moment of reflection shows that it is not quite that black-and-white. The true answer is popular under the term "effective field theory": If we specify the counterterms up to a fixed oder (and there are only finitely many of these for any order) then the remaining observables of the theory are its predictions up to that order . As more fine-grained experimental input comes in, we can possibly determine counterterms to the next order by experiment, and then again the remaining observables of the theory are its predictions up to that next higher order. And so ever on.
How are pQFT and pAQFT related to lattice gauge theory?
As I said elsewhere, stunning, simply stunning and I have never said that about an insights article before.
I look forward to the whole series.
Note to those that like me do not understand all the mathematical detail. Of course try to correct that, but still read it and try to get a feel for the issues at the frontiers of current physics.
Feel free if interested to start a thread on, what for example, an instanton sea is, don't know that one myself – much food for thought here.
Thanks
Bill
Could you please make a printable version of your slides https://ncatlab.org/schreiber/files/SchreiberTrento14.pdf, with the repetitions removed? (This is just an additional line in the latex before compilation.)
The link (web) to Schenkel in the nlab article https://ncatlab.org/schreiber/show/Higher+field+bundles+for+gauge+fields
is not working.
You stated in the article, ''perturbative AQFT in addition elegantly deals with the would-be “IR-divergencies” in pQFT by organizing the system of spacetime localized quantum observables into a local net of observables.''
I don't agree. The infrared problem remains unsolved in perturbative AQFT. You haven't even given a link to a reference where your claim would be addressed.
Note that there is already an insight article on causal perturbation theory complementing the present one.
Your point 2 for causal perturabtion theory (finitely many free constants) holds of course only for renormalizable theories. Scharf also constructs (low order) perturbative gravity in the causal framework, but there the number of free constants proliferates with the order. (Mathematically, this is not a problem since the same happens for multivariate power series, but physicists used to think of this as non-renormalizability.)
How much of causal perturbation theory has been shown to match SM physics/ordinary qft?It's true that the literature on this topic is still comparatively small, but everything comes out:
Scharf's two books cover much standard basic material of QED, EW, QCD and pQG.
Feynman diagrammatics and dimensional regularization was realized in in Keller 10, Dütsch-Fredenhagen-Keller-Rejzner 14. (These authors speak in terms of scalar fields, but, as with Epstein-Glaser's original article, this is a notational convenience, the generalization is immediate.)
BV-BRST methods were realized in Fredenhagen-Rejner 11b.
One step further, do all the successes of causal perturbation theory match on to the new covariant algebraic qft?Yes, that starts with Brunetti-Fredenhagen 00, Hollands-Wald 01 and culminates in the construction of renormalized Yang-Mills on curved spacetimes in Hollands 07.
My impressions at the time was that this was a little bit like the theory of distributions vs Dirac's delta function formalism. The former is rigorous and nice, but just clutters up notation when you sit down to calculate things.Sure, once the dust of the theory has settled we want to compute leisurely, but we do want to understand what it is our computations are doing. Distribution theory is a good example for how it pays to spend a moment on sorting out the theoretical underpinning before doing computation. Causal perturbation theory shows that all that used to be mysterious about divergencies in pQFT is clarified by microlocal analysis of distributions: Properly treating the product of distributions with attention to their wave front set is what defines the normal-order product of free fields, and then properly treating the extension of distributions to coinciding interaction points is what defines the renormalized time-ordered products. That gives a solid background explaining what's actually going on in the theory. Not every kind of computation will be affected by this, but given that there remain open theoretical questions in pQFT, it will help to have the foundations sorted out.
Nice article. How much of causal perturbation theory has been shown to match SM physics/ordinary qft? When I looked at this (admittedly many years ago), people had successfully constructed the scalar field and there was work being done on spin 1/2, and some sketchy and complicated proof of concepts, but has it really been shown to be completely isomorphic? One step further, do all the successes of causal perturbation theory match on to the new covariant algebraic qft?
My impressions at the time was that this was a little bit like the theory of distributions vs Dirac's delta function formalism. The former is rigorous and nice, but just clutters up notation when you sit down to calculate things.
Great job Urs, looking forward to the next one!
I only wonder, why you only quote QED, QCD, and quantum gravity but not the full Standard Model, including weak interactions, i.e., quantum flavor dynamics (aka Glashow-Salam-Weinberg model of the electroweak interaction) ##otimes## QCD.True, I should mention electroweak theory, too, have edited the entry a little to reflect this. (It will take a bit until I get to these applications, I will first consider laying some groundwork.)
Great work (to come), Urs!
Well, once you're done writing it (all articles), I will go print it and store it in my physical library.
That's a great article, I've to study in closer detail later. I only wonder, why you only quoate QED, QCD, and quantum gravity but not the full Standard Model, including weak interactions, i.e., quantum flavor dynamics (aka Glashow-Salam-Weinberg model of the electroweak interaction) ##otimes## QCD.
As someone who has only learned QFT via the path integral approach so far (and mostly with applications in condensed matter theory in mind), all this is crazily interesting, and i look forward to the rest of this series!