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One way to approach QFT in mathematical terms is provided by the so-called Gårding-Wightman axioms, which defines in rigorous mathematical terms what a quantum field theory is supposed to be. If I'm not mistaken, this is what is called axiomatic QFT. Although it is a precise mathematical theory, it is widely known that to construct examples of quantum field theories which are proved to satisfy these axioms is no easy task and, as far as I know, only a few such theories were constructed by now.

Now, to better pose my question, let me use a specific theory in QFT, which is the Klein-Gordon theory, as an example. We can, for instance, follow the exposition of Folland's book and give mathematical meaning to such theory by defining the correct (single particle) Hilbert space $\mathscr{H}$, giving meaning to creation and annihilation operators as operator-valued distributions in the associated Fock space and obtaining the expressions for the associated quantum fields.

Although this construction is, in practice, using some ingredients of what's expected in the axiomatic construction, it seems to me that these two approaches have different phylosophies: the axiomatic approach tries to construct a theory which fulfills every one of its axioms while the second one seems more pragmatic and tries to take what is written in the physics literature and convert it to a mathematical precise language.

The non-axiomatic approach does not intend to check if its objects satisfy any kind of axioms; it is almost like it was a 'dictionary' that translates the physical theory to a mathematical audience. As a consequence, it seems less technical and more compatible with what is written in the physics literature.

Question: So, my question is: why do we need an axiomatic approach to formalize QFT? Is there any limitation with the non-axiomaic approach? Are both approaches research material, i.e. different practices of the current attempts to give mathematical meaning to QFT?

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    $\begingroup$ "axiomatic construction" here is an oxymoron. That's because the axiomatic approach does not construct anything. It just sits there waiting for you to construct some mathematical object and when you do, it will say yes this is a QFT, or no this is not a QFT. Likewise "non-axiomatic approach" is a non-approach by any mathematical standard. $\endgroup$ – Abdelmalek Abdesselam May 26 at 22:12
  • $\begingroup$ @Abdelmalek hahaha that's true. $\endgroup$ – IamWill May 26 at 22:16
  • $\begingroup$ Btw, "Garding" (with a "d"), apart from the diacritical mark above the "a", ... ? $\endgroup$ – paul garrett May 26 at 22:33
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    $\begingroup$ When all you have to work with are only a few concrete examples of "QFT", then you can't prove or disprove any statement that starts with $\forall$"QFT". To do that, you need to take off the quotes and give an actual definition, and this is where axiom systems like Whitman's come in. Of course, the choice of axioms has to be flexible, to actually accommodate examples that people care about (as already point out in other comments/answers). $\endgroup$ – Igor Khavkine May 27 at 8:28
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    $\begingroup$ Well, it depends on what you call "new physics results". There are quite a few important conceptual loose ends on axiomatic QFT that may have an impact on physics. Two of them are: (a) how to rigorously define QFT scattering theory with both massive and massless particles, and (b) how to rigorously define resonances in QFT. Both are directly linked to experiment and still are very poorly understood. $\endgroup$ – Pedro Lauridsen Ribeiro May 27 at 14:10
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As Abdelmalek Abdesselam pointed in his comment to the OP, the axiomatic approach to QFT is rather concerned with answering the question "what is a quantum field?". This is stated right at the Preface of the book of Streater and Wightman, PCT, Spin and Statistics, and All That. More precisely, it lists a minimal set of desiderata for a reasonable concept of a "quantum field", and deals with the consequences of such requirements, which can be thought of as "structural" or "model-independent". It leaves open the question of which fields used in physics actually satisfy these requirements, apart from checking that free fields do. This is important as a first relevance check, since free fields are the easiest to mathematically control - if not even a free field complies with a list of axioms for QFT, then these axioms are useless as they stand and should be modified. More generally, any list of mathematical axioms applied to a class of physical theories is bound not to be set in stone, since our knowledge of Nature is always approximate due to our limited experimental precision. One should rather see such axioms (and their limitations in the given context) as general physical principles whose robustness has to be constantly tested, in addition to forming a well-defined mathematical framework by themselves. Such is the way of natural sciences.

In that respect, it must be remarked that indeed not all physically relevant fields satisfy the Garding-Wightman axioms - most notably, fields acting in Krein spaces (i.e. "Hilbert" spaces with a possibly indefinite scalar product), such as the electromagnetic potential in a covariant gauge, do not. The corresponding axiom for vacuum expectation values that fails is that of positivity. There are ways to extend the Wightman formalism to such fields, but the results are nowhere near as mighty of even as rigorous, since positive definiteness of the scalar product is a powerful constraint. Another tricky example are perturbative (renormalized) quantum fields, since these are formal power series in the coupling constant (convergence of the renormalized perturbative series is usually expected to fail). One must in this case keep track of the order-by-order structure of all series involved, and to define certain concepts invoked by the Garding-Wightman axioms such as positivity is far from trivial. Regarding renormalization, the main conceptual challenge is not so much the ultraviolet problem (which is rigorously well understood on a formal perturbative level), but the infrared problem, which plagues all interacting QFT models with massless fields and is not completely understood on a rigorous level, even in formal perturbation theory. And all that before even considering how to extract some non-perturbative definition of the model from perturbative data by employing e.g. some generalized summability concept for the perturbative series.

This order of doing things - namely, stating a list of general desiderata and then checking if models comply with it or not - is what is often called a "top-down" approach to QFT. What people such as Folland try to do is rather a "bottom-up" approach: trying to make sense of the formal procedures physicists actually use in (perturbative) QFT. A more successful way of doing what Folland tries to do is achieved by the so-called perturbative algebraic QFT, which draws ideas from the Haag-Kastler algebraic approach to QFT (which is a different axiomatic scheme from that of Garding and Wightman, aiming at aspects which do not depend on the particular Hilbert space the fields live in) in order to mathematically understand formal renormalized perturbative QFT (summability of the perturbative series has not been addressed yet in this approach). The so-called constructive approach to QFT also fits in such a "bottom-up" philosophy: it tries to build QFT models by means other than perturbation theory, in view of its expected divergence in relevant cases (Abdelmalek can tell you far more than me on that). Once one has obtained the model, one may try to e.g. check the Wightman axioms (or some suitable modification thereof) in order to see if the model obtained complies with that particular notion of QFT, which is undoubtedly relevant.

To sum up, I would say that both approaches are important and complement each other.

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    $\begingroup$ @JulesLamers $\phi^4$ models have been constructed for $d=2,3$ and $d>4$ space-time dimensions, and are known to satisfy the Wightman axioms. However, for $d>4$ the interaction doesn't survive after non-perturbative renormalization, i.e. the theory is free (trivial). The case $d=4$ is still open. Pure Abelian gauge theories are non-interacting and are not hard to constructed rigorously, apart from the fact that in covariant gauges the field acts on a Krein space, as mentioned above. Pure non-Abelian Yang-Mills is known (I think) for $d=2$, but for $d=4$ only in finite volume. $\endgroup$ – Pedro Lauridsen Ribeiro May 27 at 5:49
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    $\begingroup$ (continued) Massless QED in $d=2$ (Schwinger model) is equivalent to a free field theory and hence is not difficult to construct. Massive QED in $d=3$ has been recently constructed (2020) also in finite volume by Dimock. The case of $d=4$ QED is also open - both QED and $\phi^4$ in $d=4$ are believed to be trivial, but this is based on information coming from partial resummation of the perturbative series, so it's hard to tell for sure at our current state of knowledge. $\endgroup$ – Pedro Lauridsen Ribeiro May 27 at 5:56
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    $\begingroup$ @JulesLamers As a rule, if the theory has been completely constructed and satisfies the Euclidean version of the Wightman axioms - the so-called Osterwalder-Schrader axioms - one can recover the theory in Lorentzian signature, which will then satisfy the Wightman axioms. This cannot be done if one only knows the Euclidean theory in finite volume, as it's the case for QED in $d=3$ and pure Yang-Mills in $d=4$. In the case of non-interacting models, the construction is done directly in Lorentz signature if the lack of interaction is know a priori. This was not the case for $\phi^4$ in $d>4$. $\endgroup$ – Pedro Lauridsen Ribeiro May 27 at 6:01
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    $\begingroup$ About $\phi^4$ in $d=4$, the problem is not that open anymore. Confirming RG based predictions by physicists, Aizenman and Duminil-Copin have proved triviality in an article soon to appear in Annals of Math annals.math.princeton.edu/articles/17711 $\endgroup$ – Abdelmalek Abdesselam May 27 at 14:35
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    $\begingroup$ @Blazej: See this article by Magnen, Rivasseau and Sénéor projecteuclid.org/journals/… and references therein to work of Balaban and Federbush. $\endgroup$ – Abdelmalek Abdesselam May 27 at 17:42
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Let me enter the arena with the statement that there is a basic problem with the Wightman-Gårding system. To put it baldly the central concept, that of an observable- valued distribution, is defined(implicitly) in a way that is simultaneously too strong (leads to difficulties in constructing models) and too week (difficulties in proving theorems). At the heart of quantum theory lies the fact that the observables are represented by unbounded s.a. or normal operators. Since these play the role of the real and complex numbers in this context, it would seem to be imperative to develop a parallel theory there, including the basic ideas of real and complex analysis—regularity conditions (continuity, measurability, smoothness) of observable-valed functions and functions between observables, differentiation of such functions, series expansions (Taylor, Hermite, Fourier) and, most importantly for QFT, observable-valued distributions. It seems that I need to emphasise that these are a priori purely mathematical questions—in my opinion, highly interesting ones, completely independent of any relations to physics of any sort. Of course, the fact that there are such relations (not just in quantum theory, by the way) increases this interest.

The basic mathematical structures required for a rigorous development of QFT are the concepts of holomorphic functions and distributions with values in in the space of observables, i.e., the self-adjoint, resp., normal unbounded operators on a (say, separable) Hilbert space. I am arguing that a major stumbling block for a construction of the former is the lack of a suitable definition of these concepts.

Before proceeding to the constructive part of my posting, let me note that the Wightman-Gårding axioms implicitly give a definition of observable-valued distributions which assumes that all of the operators which occur have the same domain of definition. I would claim that this is much too strong as simple examples show—consider (in the holomorphic case) the family $T(z)=(z^n)$. This can be regarded as a family of complex sequences and hence of normal operators (by multiplication on $\ell^2$) dependent on $z$. It should plainly be, by definition, an observable-valued entire function but fails the stringent conditions imposed by the axioms on the domains of definition. A further example is the function $z\mapsto I+zA$ for an unbounded normal $A$. This should plainly be holomorphic but fails the condition on the domains of definition.

The axioms are often stated in a weaker form, demanding only that there be a dense subspace contained in the domains of the operators but this is hopelessly inadequate—if you know a s.a. operator on a dense subspace, you know nothing. Think of the Laplace operator on some domain in euclidean space regarded as an operator on the smooth functions with compact support. Without boundary conditions, and there are many candidates, this is not s.a. In fact, this operator defined, say, on the test functions has uncountably many self adjoint extensions and if we label them with a complex variable $z$ we get a wildly pathological observable which is holomorphic since it is constant on a dense subspace.

We remark that we have displayed these functions in the context of holomorphicity for simplicity but the ideas apply equally in other contexts.

Given the inadequacy of this definition, it can be no surprise that there are no indications in the literature of a general theory of observable-valued distributions or holomorphic functions (the usual basic things—functoriality, differentiation, series developments—Taylor or Hermite, for example, and so on). I am prepared to place my head on the block and claim that there never can be such a mathematically rigorous theory based on the above definition.

Now for the positive remarks. I will concentrate on the case of holomorphic functions since this is the most transparent one but many other spaces of functions and distributions can be handled analogously. We begin with the remark that the extension of the concepts of smoothness of functions to the vector case (Banach spaces, locally convex spaces, even topological vector spaces, although the latter is rather delicate) was carried out over 50 years ago, for example by Schwartz and Grothendieck). in carrying this over the the case of observables, there are a number of stmbling blocks—the natural ingredients—topological and algebraic structures (addition and multiplication of observables) are no longer available, or rather they are but in much more delicate forms). The starting point is that the space of observables has a natural topology for which it is a polish space.

If $U$ is a complex domain (to be specific), the we denote by $H(U)$ the space of holomorphic functions on $U$ and its dual by $H(U)‘$. It helps, but is not essential, that the former is a nuclear Fréchet space and the latter has a concrete representation as a space of holomorphic functions on the complement of $U$ in the Riemann sphere (Sebastião e Silva, Köthe).

Grothendieck extended the work of the latter by showing that the space of holomorphic functions on $U$ with values say in a Banach space $E$ can naturally be identified with $L(H(U)‘,E)$, the continuous linear operators on the dual into $E$.

This makes it natural to define the observable-valued holomorphic functions on $U$ to be the space of continuous linear operators from $H(U)‘$ into the space of observables. Since the latter is not a tvs (or even a vector space), the latter requires some explication. The continuity creates no problems (the space of observables has a topology). Linearity uses the rather subtle concept of addition for unbounded s.a. or normal operators—for which see the standard literature, e.g., Reed and Simon. We say that a mapping with values in the space of observables is linear if $T(\alpha f)=\alpha T(f)$ for all $\alpha$ and $T$ and for each $f_1$ and $f_2$ the operators $T(f_1)$ and $T(f_2)$ have a sum, with the usual equality holding.

In the case of distributions, say tempered—the ones most used by physicists, the same definition applies, using operators on the Fréchet space of rapidly decreasing smooth functions.

We remark briefly that these spaces have the basic properties that one uses , e.g., for constructing and computing with models, in perturbation arguments, etc. and which fail under the Wightman-Gårding axioms.

Tho holomorphic functions contain the classical ones (i.e., with values in the bounded operators), they can be differentiated (I emphasise—not true in W-G—who ever heard of not being able to differentiate holomorphic functions?), have Taylor expansions and are functorial property in dependence of the domain of definition (which means that symmetry groups there lift to the fields).

With regard to distributions, similar remarks hold—they can de differentiated, expanded in terms of the Hermite basis, Fourier transformed, symmetries of the underlying euclidean space operate on them and they contain, as a special case, the classical ones (i.e., with bounded operators as values).

Since this posting has become uncomfortably long, I will close with the fact that these concepts have the properties which one would reasonably expect from a theory of observable-valued holomorphic functions or distributions and so could serve as the basis for a more rigorous approach to QFT.

Some quotes:

Kazhdan: Physics is very interesting. There are many, many interesting theorems. Unfortunately, there are no definitions.

Zeidler. In mathematics one never does calculations with quantities that do not exist.

Jost: The fact that intuition barely can guide us forces us to use standards of rigour usually frowned on in theoretical physics.

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  • $\begingroup$ Your initial statement that there is a "lack of a suitable definition of these concepts" is not really true, for several reasons. $\endgroup$ – Pedro Lauridsen Ribeiro Jun 1 at 4:57
  • $\begingroup$ (1) if one is worried how one defines quantum fields as "distributions valued in unbounded operators in a Hilbert space", specially regarding w.r.t. which topologies such distributions are continuous, one must remember that such operators can be defined in terms of *-representations of a nuclear locally convex *-algebra - to wit, the tensor algebra generated by the space of (tempered) test functions / vector / tensor / spinor fields, also called the Borchers-Uhlmann *-algebra. $\endgroup$ – Pedro Lauridsen Ribeiro Jun 1 at 4:58
  • $\begingroup$ (continuation of (1)) The ${}^*\!$-representation is built from a state in this ${}^*\!$-algebra (that is, a positive linear functional equaling one on the identity), which is nothing else than the sequence of Wightman n-point (tempered) distributions. The Wightman axioms can be equivalently formulated in terms of these. The recovery of the quantum field operators, the Hilbert space where they act and their common domain of definition therein is a consequence of the so-called Wightman reconstruction theorem, which is a version of the GNS representation for (non C)${}^*\!$-algebras. $\endgroup$ – Pedro Lauridsen Ribeiro Jun 1 at 5:02
  • $\begingroup$ (2) The analytic structure of the quantum fields which follows from the Wightman axioms is expressed by that of the n-point functions, viewed (as well as their Fourier transforms) as boundary values of holomorphic functions, as you suggested. However, if you want fo extend the Wightman axioms to other spacetimes (with the spectral condition replaced by possibly some suitable microlocal variant), this analytic continuation of the n-point functions is no longer possible is the space-time is not real-analytic (as many physical examples thereof indeed aren't). $\endgroup$ – Pedro Lauridsen Ribeiro Jun 1 at 5:06
  • $\begingroup$ (3) If you want to deal with the structure of quantum field theories in a representation-independent manner and focus on the observables - or, more appropriately, on the relative inclusions of the algebras of observables localized in each space-time region -, the Wightman formalism should be replaced by the Haag-Kastler algebraic approach to QFT, based on (C)*-algebras. This is needed because e.g. (a) quantum fields are often not directly observable, and (b) the whole state space of a QFT cannot fit into the Wightman-GNS vacuum Hilbert space, among other things. $\endgroup$ – Pedro Lauridsen Ribeiro Jun 1 at 5:12

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