Introduction to Perturbative Quantum Field Theory

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

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

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

Perturbation and Non-perturbation.

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

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

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

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

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

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


Causal Perturbation Theory.

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

which has come to be known as causal perturbation theory.

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

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

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

S(L_{\text{int}} )
\overset{\text{not really}}{=}
\int \left[D\phi\right] e^{ \tfrac{1}{i \hbar} \int_X L_{\text{free}}(\phi) } \, \exp\left(  \tfrac{g}{i \hbar} \int_X\left( L_{\text{int}}(\phi)  \right) \right)

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

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

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

The main result of causal perturbation theory is the proof that

  1. Causally additive perturbative S-matrices exist, hence pQFT exists, rigorously;
  2. at each order there is a finite-dimensional space of choices, the renormalization freedom;
  3. any two such choices are related by a unique re-definition of the Lagrangian densities (by “counter-terms”);
  4. 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

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

Perturbative Algebraic Quantum Field Theory.

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

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

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

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

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

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

Locally covariant pAQFT.

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

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

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

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

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

The traditional toolbox made rigorous.

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

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

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

From first principles.

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

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

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

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

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

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

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

  • Giovanni Collini,
    “Fedosov Quantization and Perturbative Quantum Field Theory”
  • Eli Hawkins, Kasia Rejzner,
    “The Star Product in Interacting Quantum Field Theory”

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

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


The unknown theory.

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

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

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

This series on QFT continues here:
A first idea of Quantum Field Theory.



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

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

108 replies
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  1. bhobba
    bhobba says:

    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.


  2. Urs Schreiber
    Urs Schreiber says:

    Sorry for the slow replies, I am seeing the further comments only now for some reason.

    A. Neumaier

    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 .

    A. Neumaier

    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.

  3. Urs Schreiber
    Urs Schreiber says:
    A. Neumaier

    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

    1. the algebra of quantum observables localized in any spacetime region may be computed, up to canonical isomorphism, already with any adiabatic switching function that is constant on a neighbourhood of that region of support
    2. as the region of support varies arbitrarily, the system of algebras of localized quantum observables obtained this way do form a causally local net in the sense of the Haag-Kastler axioms (this prop, the only difference to the original axioms being that here they are formal power series algebras instead of C-star algebra, reflecting the perturbation theory)
    3. AQFT lore implies that this causally local net of observables is sufficient to fully define the quantum field theory.
    4. ibut f desired, we may still take the limit now, not of the S-matrix, but of the local net of observables it induces, in the sense of limits over the functor assigning observable algebras. In the pAQFT literature they call this the "algebraic adiabatic limit" or similar. It may be used to construct operator representations of the quantum observables, but the main point of pAQFT is really that by and large it is not actually necessary to consider this.
  4. Urs Schreiber
    Urs Schreiber says:
    A. Neumaier

    Could you please make a printable version of your slides, 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

    A. Neumaier

    The link (web) to Schenkel in the nlab article 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:

    • Alexander Schenkel, "Towards homotopical AQFT" (web , pdf)
  5. Urs Schreiber
    Urs Schreiber says:

    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.

  6. Urs Schreiber
    Urs Schreiber says:

    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.

  7. A. Neumaier
    A. Neumaier says:
    Urs Schreiber

    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.

  8. A. Neumaier
    A. Neumaier says:
    Urs Schreiber

    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.

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