1. Introduction

To date, philosophers of quantum field theory (QFT) have paid much attention to roughly two kinds of QFT: axiomatic approaches to algebraic QFT (Halvorson and Müger Reference Halvorson, Müger, Butterfield and Earman2006) and Lagrangian-based QFT as used by particle physicists (Wallace Reference Wallace2006). Comparatively less attention, however, has been paid to constructive QFT, an approach that aims to rigorously construct nontrivial solutions of QFT for Lagrangians and Hamiltonians that are important in particle physics, ensuring that such solutions satisfy certain axioms. In doing so, constructive QFT mediates between axiomatic approaches to QFT and physicists’ Lagrangian-based QFT. It ensures that the various axiom systems have a physically meaningful correspondence with the world. Since we usually take solutions in physical theories to describe the behavior of systems falling under the theory and constructive QFT aims to produce these solutions, constructive QFT deserves some philosophical attention.

Constructive QFT is rather different in approach and aims from what I call, borrowing terminology that stems from Wightman (Reference Wightman, Iverson, Perlmutter and Mintz1973), axiomatic QFT.Footnote ¹ Following Wightman and modern practitioners like Summers (Reference Summers2012), I define constructive QFT to be the attempt to rigorously construct interacting models of QFT that correspond to the repertoire of Lagrangians of interest to particle physics. In contrast, axiomatic QFT in this terminology aims to derive results about the structure of QFT independently of any particular Lagrangian. In this article I examine two aspects of constructive QFT in the functional integral tradition that are relevant to our understanding of the theoretical structure of QFT. The first is the question of what counts as a solution to a specific Lagrangian in constructive QFT. I argue that the criteria for what counts as a solution include some kind of correspondence with perturbation series derived in Lagrangian QFT. This correspondence relation is not part of any of the standard axioms of QFT. Thus, the constructed solutions may satisfy “physical criteria” that are not present in axiomatic QFT taken by itself.

The second aspect of constructive QFT in the functional integral tradition that I examine concerns the information that constructive field theorists use to construct solutions. Methods of construction rely heavily on perturbative Lagrangian QFT. The specifics of the regularization methods, counterterms, multiscale expansions, and so on, that are used affect the success of the construction. This suggests that a successful construction of physically interesting solutions relies on information that may not be just in the axioms of axiomatic QFT. This information, often derived from Lagrangian QFT, may thus be of philosophical interest.

The article proceeds as follows. In section 2, I clarify what I mean by perturbative Lagrangian QFT, constructive QFT, axiomatic QFT, and other key terms. In section 3, I provide a quick primer to perturbative Lagrangian QFT, which will help us understand some aspects of constructive QFT. In section 4, I make the two main points described above, then discuss their philosophical implications in section 5. I conclude with some reflections on general implications for the philosophy of applied mathematics.

2. Approaches toward Quantum Field Theory: A Primer

This primer is necessary for two reasons. First, there is a gulf between the methods that physicists take to constitute QFT and the mathematical structures that philosophers of QFT have often taken to constitute QFT. In the philosophy of QFT debate this has become simplified into a contrast between “Lagrangian” QFT and “algebraic” or “axiomatic” QFT (Fraser Reference Fraser2011; Wallace Reference Wallace2011), but more nuanced categories are required for my purposes.Footnote ² Second, the dominance of this terminology has led philosophers to gloss over a distinction, long recognized in mathematical physics, between what I will call axiomatic QFT and constructive QFT. This has led to much confusion about the role of perturbative Lagrangian methods in the foundations of QFT. Here, I lay out some terminology that will help clear up such confusions.

Most of the methods described in a standard QFT textbook for physicists involve calculations in perturbative Lagrangian QFT. To remain consistent with previous terminology used by Wallace (Reference Wallace2006), I will simply refer to this approach to QFT as Lagrangian QFT, even though physicists also use some methods that are Lagrangian but not perturbative. In Lagrangian QFT, one takes as a baseline an exactly solvable model of QFT in which there are no interactions. Having no interactions, this model is not of direct physical interest. But to solve models in which there are interactions, one can consider the interactions as small perturbations to the exactly solvable noninteracting model. This allows one to apply the apparatus of perturbation theory to obtain what are known as renormalized perturbation series. If we had a justification for the validity of perturbation theory in QFT, then these would be good approximate solutions for the interacting model.Footnote ³ However, perturbation theory is valid only under certain conditions, and it is hard to verify whether these conditions apply in the case of QFT. Furthermore, Lagrangian QFT uses mathematical tools known as Feynman path integrals. While the exact definition of path integrals is still in flux, physicists have devised ways to compute them without adhering to mathematical standards of rigor.

In the interests of clarity, it is worth noting that there are perturbative treatments of QFT that do not coincide with Lagrangian QFT as practiced by particle physicists. For example, there are programs to “rigorously” analyze perturbative Lagrangian QFT (Steinmann Reference Steinmann2000; Salmhofer Reference Salmhofer2007), but these form a very small proportion of the work on Lagrangian QFT. The “rigor” in these works concerns manipulating perturbative formalisms according to strict, clearly delineated syntactic rules, which may nonetheless lack nonperturbative justification. This lack means that the formalisms have an indeterminate semantics, since, roughly speaking, we do not know whether the perturbative expressions “refer” to anything.Footnote ⁴ The use of “rigor” to describe such programs is meant to contrast with the more common practice among physicists of using approximations and cancellations that may not even be syntactically consistent.

Apart from these syntactically rigorous perturbative approaches starting from Lagrangians, there are also perturbative approaches within algebraic QFT, which uses a different mathematical framework from Lagrangian QFT (Brunetti, Duetsch, and Fredenhagen Reference Brunetti, Duetsch and Fredenhagen2009). Lagrangian QFT has been helpful in providing what physicists regard as approximate solutions to problems in QFT. However, the lack of rigor has driven philosophers to turn their attention primarily to algebraic QFT. “Algebraic” here refers to the mathematics used, namely, the use of C* algebras to model local observables. The algebraic approach is based on axioms like the Haag-Kastler axioms. Algebraic QFT is not the only strain of QFT that philosophers regard as sufficiently rigorous. There are other sets of axioms that one can work with, such as the Wightman axioms and the Osterwalder-Schrader (OS) axioms. The former work in Minkowski space, while the latter work in Euclidean space. They are related to each other by the Osterwalder-Schrader Reconstruction Theorem, as I explain below. The Wightman axioms use a mixture of functional analysis and operator algebras. The OS axioms are the basis of rigorous functional integral approaches to QFT. As with the term “algebraic,” this term refers to the mathematics used—in the OS framework, Feynman path integrals used by physicists are reinterpreted in terms of rigorous functional integrals in Euclidean space.

Within both the Wightman and functional integral approaches to QFT, we can discern two broad ways of investigating QFT. The first way is what has often been called axiomatic QFT, where the idea is to investigate the consequences of the axioms without relying on the properties of any particular Lagrangians or Hamiltonians. The second way is a constructive approach, which has a very specific aim: to construct concrete, nontrivial examples of models of QFT that satisfy the relevant set of axioms.Footnote ⁵ Figure 1 displays the taxonomy I am describing. For example, one can take a constructive approach in the Wightman framework that aims to construct a ${ɸ}_2^4$ model and prove that it satisfies the Wightman axioms. This is done by defining a Hamiltonian and showing that it has the requisite algebraic properties (Glimm and Jaffe Reference Glimm and Jaffe1968). One could also have a corresponding project in the functional integral approach that aims to construct the same ${ɸ}_2^4$ model using Euclidean functional integrals (Glimm, Jaffe, and Spencer Reference Glimm, Jaffe and Spencer1974). Both these projects have the very specific aim of showing that specific models that are used in particle physics are consistent with axiomatic systems and are to be distinguished from methods that aim to derive general conclusions from axioms without reference to particular Hamiltonians or Lagrangians.

Figure 1. Rough taxonomy of rigorous approaches to QFT, which does not aim to be complete, leaving out perturbative algebraic QFT (Brunetti et al. Reference Brunetti, Duetsch and Fredenhagen2009) and “rigorous” perturbative Lagrangian QFT (Steinmann Reference Steinmann2000; Salmhofer Reference Salmhofer2007), for example. Embellished bubbles indicate those constructive approaches of central interest to this article, namely, those that seek to rigorously construct models of QFT corresponding to Lagrangians within the usual repertoire of particle physics. Constructive approaches using algebraic methods are as yet not obviously related to the Lagrangians of particle physics.

Finally, there is a different kind of constructive project in the algebraic framework, where one constructs models that, in contrast with the Wightman and functional integral constructions, are not guided by Lagrangian QFT (Summers Reference Summers2012, 38). I will not discuss this kind of constructive work in this article, as I am focusing on the significance of constructive field theory as a mediator between Lagrangian QFT and axiom systems. While algebraic approaches may one day hope to play such a mediating role, constructive field theory in the functional integral tradition is at present much more in contact with Lagrangian QFT.Footnote ⁶ The lack of this mediating role for such purely algebraic constructions makes their philosophical significance potentially very different from that of Wightman and functional integral constructions.

I call the project of constructing nontrivial solutions to models of QFT, where such models are derived from the usual repertoire of Lagrangian QFT as practiced by physicists, constructive QFT. In doing so I am merely using terminology that dates from Wightman and is still used by contemporary mathematical physicists. In an introductory article on constructive field theory, Wightman writes: “Constructive quantum field theory differs from axiomatic field theory in that it attempts to construct the solutions of specific concrete model theories, typically simplified analogues of self-coupled meson theories or Yukawa theories of meson-baryon couplings. But, axiomatic field theory customarily attempts to make statements about all theories satisfying certain quite general assumptions” (Reference Wightman, Iverson, Perlmutter and Mintz1973, 1). Wightman reiterates this distinction in other articles (Reference Wightman1969, Reference Iverson, Perlmutter, Mintz and Browder1976), emphasizing each time that the concrete models chosen are those from the repertoire of Lagrangian QFT.

A more modern review of QFT confirms that Wightman’s distinction still exists. Fredenhagen, Rehren, and Seiler (Reference Fredenhagen, Rehren, Seiler, Stamatescu and Seiler2006, 64) write that “the power of the axiomatic approach resides not least in the ability to derive structural relations among elements of the theory without the need to actually compute them in a model.” They then continue in the section on constructive QFT: “The axiomatic approach, on the other hand, does not answer the question whether the axioms are not empty, i.e., whether any nontrivial QFTs satisfy them. The constructive approach is in principle addressing both of these problems. On the one hand it attempts to show that the axiomatic framework of QFT is not empty, by mathematically constructing concrete nontrivial examples satisfying these axioms, and on the other hand it provides non-perturbative approximation schemes that are intimately related to the attempted mathematical constructions; the prime example [sic] are the lattice approximations to QFTs” (14). Constructive QFT is thus very different from axiomatic QFT in its aims and methodology. Philosophers have for the most part focused only on axiomatic QFT or perturbative Lagrangian QFT.Footnote ⁷ Looking at the rigorous approaches, when philosophers do consider models of QFT at all, these have been noninteracting models, which are not the focus of constructive QFT.Footnote ⁸ Constructive QFT contains the best attempts so far to construct rigorous solutions to Lagrangian models, having done this successfully for several systems with dimensions other than four.Footnote ⁹ Since the actual world is four-dimensional, the main aim is to extend these successes to the case of four dimensions, but this has not been achieved yet for Lagrangians resembling those used by particle physicists.

One can think of constructive QFT as performing the important task of ensuring that our axiom systems have a physically meaningful correspondence with the world. That is, since the confirmation of QFT proceeds via deriving the consequences of specific Lagrangian models, we need to show that our axiomatic systems can accommodate such models. That is why constructive QFT does not just attempt to construct models of random Lagrangians but focuses on specific ones, such as those in four dimensions, those of Yang-Mills theory, the Higgs interaction, and so on.Footnote ¹⁰ In philosophy of language terms, we can think of constructive QFT as attempting to show that our axiomatic systems are not just games of symbols played according to fixed syntactic rules, that they have external validity in addition to the internal consistency that their syntax provides. External validity lies in the connection to empirical success, which is why Wightman emphasizes the connection to heuristic Lagrangian QFT.Footnote ¹¹

A few caveats ought to be made at this point. First, my categorization of different approaches to QFT does not rule out borderline cases. There are constructive projects combining functional integral methods with Wightman-oriented methods and others combining algebraic and Wightman methods (Summers Reference Summers2012, 4, 8, 12). Second, there is some variation in terminology within the mathematical physics community. While I follow Wightman and others in classifying under “axiomatic QFT” those approaches that do not rely on the properties of specific Lagrangians or Hamiltonians, Jaffe (Reference Jaffe1969) has written an article titled “Whither Axiomatic Field Theory?” in which he discusses what I here call constructive field theory. Similarly, Araki (Reference Araki, Jackson and Roberts1972) discusses constructive QFT as an “area” of axiomatic field theory. However, both Jaffe and Araki are in agreement that constructive field theory involves constructing models of QFT, these models being concrete realizations of particular Lagrangians or Hamiltonians. Jaffe (Reference Jaffe1969, 576) writes: “Constructive field theorists have approached this problem with Lagrangian field theories and attempted to solve particular model Lagrangians.” Araki (Reference Araki, Jackson and Roberts1972, 1) writes: “[Constructive field theory] is an attempt to construct a quantum field theory for a given interaction such as ɸ⁴ and $\overline{\psi }\psi ɸ$ in a mathematically satisfactory manner.” Thus, there is a consensus in the literature that constructive field theory, whether one classifies it under the axiomatic program or not, is an approach that is defined by its attempt to construct models of particular interactions, represented by Lagrangians or Hamiltonians.

In short, we can clearly recognize, and the community does recognize, a distinction between approaches that proceed from the axioms alone and approaches that work with concrete models, where these models are taken from Lagrangian QFT. I have, following one convention, chosen to call the former axiomatic and the latter constructive. Terminological disputes aside, it still remains a fact that philosophers have paid little attention to the latter and that the latter affords many interesting connections with Lagrangian QFT that are easy to ignore if one concentrates on the former as the paradigm example of rigorous QFT.

Now that we have a clear terminology, I will move on to provide more details about the general approach of Lagrangian QFT. This is necessary to explain later how solutions in constructive QFT relate to Lagrangian QFT.

3. Perturbative Solutions in Lagrangian Quantum Field Theory

For physicists, the dynamics of QFT are derived from a quantity known as the action

$S\left[ɸ\right]=\int d^{4}xL\left(ɸ\left(x\right),\frac{\partial ɸ\left(x\right)}{\partial x^{\mu }}\right),$

where $L$ is the Lagrangian of the quantum field ɸ. The square brackets indicate that S[ɸ] is a functional, not a function. It depends on the values of ɸ everywhere, not just at some particular point of space-time. The form of the Lagrangian is based on considerations of the kind of interactions we expect in the system of interest, and on the symmetries we expect the system to obey. The action is closely related to the classical field equations by way of a variational principle. Essentially, postulating that S[ɸ] has to be at an extremum will lead us to the classical field equations. Thus, the action can be regarded as representing the dynamics of the theory.

From the action we get the partition function, which is defined as follows:

(1)$\begin{array}{}Z=\int Dɸe^{S\left[ɸ\right]}\mathrm{.}\end{array}$

Here, the integral is a functional integral, meaning that we are integrating over all possible combinations of ɸ’s values over space-time.Footnote ¹² Once we know Z, we can typically calculate all the empirical quantities associated with that particular Lagrangian, such as the scattering cross-sections found in particle physics experiments. Thus, finding an expression for Z is a primary goal of much of QFT. A successful evaluation of Z is considered to be a solution of Lagrangian QFT.

Equation (1) can be given a straightforward finite, analytic expression when the action involved is that of a free scalar field with no interactions, also known as a “Gaussian” field. In this case, $L=(1/2)\left({\left(\partial ɸ\right)}^2-m^2{ɸ}^2\right)$. For interacting fields, physicists typically use perturbation theory to evaluate the path integrals. Since the path integral for the free field has a known analytic expression, the perturbations are applied using the free field case as a reference—we consider the interaction as a small perturbation to the free field Lagrangian. The following example illustrates how this is done in a simple case.

Suppose a small interaction $-{(\lambda /4!)ɸ}^4$ is added to the free field Lagrangian, so that $L=(1/2)\left({\left(\partial ɸ\right)}^2-m^2{ɸ}^2\right)-(\lambda /4!){ɸ}^4$.Footnote ¹³ This is the Lagrangian of the so-called ɸ⁴ theory, which describes a self-interacting scalar field. The partition function is

(2)$Z=\int Dɸe^{\int d^{4}x\left(\left({\left(\partial ɸ\right)}^2-m^2{ɸ}^2\right)-(\lambda /4!){ɸ}^4\right)}.$

Assuming λ to be small, we then convert the $e^{-(\lambda /4!){ɸ}^4}$ factor into a Taylor series in λ:

$\begin{array}{}Z=\int Dɸ\left(1-\frac{\lambda }{4!}{\int }_{x_1}{ɸ}^2\left(x_1\right)d{x}_1+\frac{1}{2}{\left(\frac{\lambda }{4!}\right)}^2{\int }_{x_1,x_2}{ɸ}^4\left(x_1\right){ɸ}^4\left(x_2\right)d{x}_{1}d{x}_2+\cdots \right)e^{\int d^{4}x\left({\left(\partial ɸ\right)}^2-m^2{ɸ}^2\right)},\end{array}$

where I have included only the first two terms of the Taylor series to illustrate the general rule.

Unlike in the free field case, when evaluating path integrals such as the above, several mathematical problems arise. One is how to define the measure $Dɸ$. Physicists calculate Z without a precise definition of the measure. Constructive field theorists attempt to define it rigorously. The second class of problems surrounds infinities known as divergences that arise in one’s calculations. These divergences occur in two forms. First, individual terms in the perturbation series might diverge. Second, the perturbation series itself may not be a convergent series, although it may be an asymptotic series that can be summed by special summation methods.

To deal with the first type of divergence, physicists perform the following procedures:

1. 1. Regularization, which is the reduction of the number of degrees of freedom in the problem by adding momentum cutoffs, dimensional regularization, or moving to a lattice formulation;

2. 2. Addition of counterterms to compensate for the regularization.

A theory is said to be perturbatively renormalizable if after regularization and the adding of counterterms, the number of coupling parameters in the theory is finite and constant at each order in perturbation theory. Another way of putting this is that renormalization occurs when the effects of regularization and the adding of counterterms are absorbed entirely by changing only the values of a finite number of coupling parameters. For example, in the ɸ⁴ model, the Lagrangian we start with before regularization and renormalization is $L=(1/2){\left(\partial ɸ\right)}^2-(1/2)m^2{ɸ}^2-(\lambda /4!){ɸ}^4$. In carrying out dimensional regularization, we first go from four dimensions to 4 − ε dimensions. This introduces a new scale μ and an ε exponent into the Lagrangian, so that $L=(1/2){\left(\partial ɸ\right)}^2-(1/2)m^2{ɸ}^2-{\mu }^{2\epsilon }(\lambda /4!){ɸ}^4$ after regularization. After both dimensional regularization and renormalization, we obtain a renormalized Lagrangian $L_{\mathrm{ren}}$ with counterterms $L_{\mathrm{ct}}$ (Ramond Reference Ramond1981, 132):

$L_{\mathrm{ren}}=L+L_{\mathrm{ct}},$

where

$L_{\mathrm{ren}}=\frac{1}{2}{\left(\partial {ɸ}_0\right)}^2-\frac{1}{2}{m}_0^2{ɸ}_0^2-\frac{{\lambda }_0}{4!}{ɸ}_0^4,$

$L=\frac{1}{2}{\left(\partial ɸ\right)}^2-\frac{1}{2}{m}^2{ɸ}^2-{\mu }^{2\epsilon }\frac{\lambda }{4!}{ɸ}^4,$

and

$L_{\mathrm{ct}}=\frac{1}{2}A{\left(\partial ɸ\right)}^2-\frac{1}{2}{m}^{2}B{ɸ}^2-{\mu }^{2\epsilon }\frac{\lambda }{4!}{ɸ}^4.$

The coupling parameters ɸ, m, and λ have been renormalized as follows:

$\begin{array}{c}{ɸ}_0={\left(1+A\right)}^{1/2}ɸ,\\ m_0^2=m^2\frac{1+B}{1+A},\\ {\lambda }_0=\lambda {\mu }^{2\epsilon }\frac{1+C}{{\left(1+A\right)}^2},\end{array}$$$

where A, B, and C are constants that determine how the coupling parameters are renormalized. As one can see, the counterterms that are added preserve the form of the interactions in the original Lagrangian, so that one has to change only the existing finitely many coupling parameters in $L$ in order to obtain $L_{\mathrm{ren}}$. This is why the ɸ⁴ theory is said to be perturbatively renormalizable.

The perturbation series with the new coupling parameters is known as the renormalized perturbative series, which has the same terms as the original one but with the renormalized coupling parameters instead of “bare” coupling parameters. This series is the basis of many, if not most, successful empirical predictions in QFT. However, because it is unclear whether the series converges or whether it is asymptotic to some function we can derive in constructive field theory, the series is purely “formal,” in the sense of being a mere string of symbols that may have no mathematical referent.Footnote ¹⁴ We will see next that the renormalized perturbation series also plays an important role in constructive QFT, forming part of the success conditions for a constructive solution.

4. Constructive Solutions in Quantum Field Theory

Constructive QFT is the effort to construct, according to the usual standards of mathematical rigor, solutions to Lagrangian models in QFT and to prove that these solutions satisfy certain axioms that we expect to apply to all Lagrangian models. In the functional integral tradition, the usual method is to shift from Minkowski space-time to Euclidean space-time. This is done by a Wick rotation, where one replaces the time parameter in Minkowski space-time with an imaginary time variable −it. This turns out to be extremely helpful for constructing the functional integral measures needed for equation (1)—many constructions that are possible in Euclidean space-time have not been accomplished directly in Minkowski space-time.

In Euclidean field theory, the relevant axioms to be satisfied are the OS axioms. These axioms state the properties that the Schwinger functions of the model must satisfy. The Schwinger functions are as follows:

(3)$\begin{array}{}{S}_N\left(z_1,\dots ,z_N\right)=Z^{\mathrm{-1}}\int \prod _{j=1}^Nɸ\left(z_j\right)e^{-\int V\left(ɸ\left(x\right)\right)dx}d{\mu }_0\left(ɸ\right),\end{array}$

where

(4)$\begin{array}{}Z=\int e^{-\int V\left(ɸ\left(x\right)\right)dx}d{\mu }_0\left(ɸ\right).\end{array}$

The interaction part of the Lagrangian is V. The measure dμ₀(ɸ) is a Gaussian measure on the Schwartz space of rapidly decreasing functions. This measure accounts for the free field part of the Lagrangian, so that equation (4) is a translation of equation (1), even though the former does not contain the free-field portion in the exponential (Sénéor Reference Sénéor, Breitenlohner, Maison and Sibold1988, 23).

The OS axioms are related to the Wightman axioms by the Osterwalder-Schrader Reconstruction Theorem, which states that any set of functions satisfying the OS axioms determines a unique Wightman model whose Schwinger functions form that set (Rivasseau Reference Rivasseau1991, 22). This theorem allows constructive field theorists in the functional integral tradition to work in Euclidean space-time, knowing that any successful construction satisfying the OS axioms can be translated into a successful construction satisfying the Wightman axioms. This way, they can exploit the advantages of Euclidean space-time when it comes to defining a measure for equation (1), while ensuring that they are still effectively constructing models that can live in Minkowski space-time.

A solution in constructive QFT would be a construction of Schwinger functions that satisfy the OS axioms and some other conditions (to be described below). Before evaluating equation (3), however, we have to provide a meaning to it. We have to define what the integral over function space could possibly mean. This is not a straightforward task because for interacting fields, we encounter the same problems as in Lagrangian QFT with regularization and counterterms.Footnote ¹⁵ These questions of defining the functional integral will be discussed in section 4.2. Additionally, to make sure that the construction is indeed a solution that corresponds to an interaction of interest in Lagrangian QFT, we have to show that the construction has some kind of correspondence to the renormalized perturbation series that physicists derive for the corresponding Lagrangian in Lagrangian QFT.Footnote ¹⁶ Crucially, this latter property, which I will discuss next in section 4.1, is not part of the OS axioms.

5. Philosophical Implications

The features of constructive QFT I have discussed are of philosophical interest for the following reasons. First, while some philosophers attach little interpretive significance to perturbative solutions in Lagrangian QFT due to their lack of rigor, constructive QFT attempts to remedy this by providing rigorous nonperturbative analogues of these perturbative solutions.Footnote ²² Generally speaking, these solutions describe the behavior of QFT systems and thus are of interpretive interest to philosophers of QFT.

Second, the solutions of constructive QFT are a way to provide the axioms of QFT with physical content. Perturbative Lagrangian QFT is confirmed by its empirical success, but axiomatic QFT cannot be confirmed in as direct a manner. Constructive QFT attempts to ensure that axiomatic QFT can be linked to Lagrangian QFT and thus indirectly confirmed. It does this by trying to show that we can construct solutions that satisfy the relevant axioms and that these solutions are in some sense “the same” solutions as the perturbative solutions of Lagrangian QFT. Constructive QFT is thus important to those who are interested in confirming or improving extant axiom systems.

Third, critics of Lagrangian QFT, like Fraser (Reference Fraser2011), have dismissed the relevance of Lagrangian QFT to the “theoretical content” of QFT on the basis of Lagrangian QFT’s lack of rigor. Furthermore, Fraser marshals the successes of constructive QFT to defend rigorous approaches to QFT against Lagrangian QFT. Against this, I have argued that information from Lagrangian QFT is key to the success of constructive QFT. With the example of constructive QFT in the functional integral tradition, we can see that unrigorous Lagrangian QFT is extremely relevant to the rigorous program of constructive QFT, providing essential guidance to a correct construction. By Fraser’s own lights, constructive QFT might illuminate the theoretical content of QFT. But given how constructive QFT relies on information from Lagrangian QFT, perhaps Lagrangian QFT itself might contribute to theoretical content.

Fraser acknowledges that constructive QFT attempts to implement analogues of renormalization group (RG) methods that are used in Lagrangian QFT. However, she interprets this fact as showing that RG methods “concern the empirical structure of the theory rather than the theoretical content” (Fraser Reference Fraser2011, 132). In my view, it is unclear that the elements of Lagrangian QFT borrowed by constructive QFT fall on the “empirical” rather than “theoretical” side of the divide—even supposing there is such a dichotomy. The implementation of the RG concerns scaling properties of fields, which is not something that is merely empirical since it concerns more than just scattering matrices. Without further argument, it is also hard to see why issues of regularization, counterterms, sign of the coupling constant, and other issues mentioned in section 4.2 are merely empirical. The other main contribution of Lagrangian QFT to constructive QFT is the requirement of correspondence to the renormalized perturbation series. While this has some connection to empirical success, it is also something that goes beyond mere empirical success. If we wanted to capture only the empirical content of Lagrangian QFT, all we would ask for is equivalent scattering matrices. In addition, we saw that the requirement of correspondence is not merely an “adding” of content to constructive QFT but is constitutive of its success. It is not that constructive QFT on its own already has theoretical content without Lagrangian QFT, but to decide even what its theoretical content is, namely, its solutions, we need Lagrangian QFT for guidance. In other words, even if one insists on interpreting the contribution of Lagrangian QFT as merely “empirical,” it is still the case that the empirical content is guiding what counts as acceptable theoretical content. This should be unsurprising given the role of constructive QFT as essentially mediating between empirical success and as-yet-unconfirmed axiom systems.

Fourth, constructive QFT contributes some conceptual clarity to what is going on in Lagrangian QFT. The Wightman axioms, for example, have a clear conceptual content. They describe a Hilbert space, and they incorporate straightforward physical principles like microcausality. This kind of physical content may be somewhat obscured in the complicated techniques of Lagrangian QFT. Constructive QFT thus serves as a bridge between the physical content of the Wightman axioms and the empirically successful techniques of Lagrangian QFT.Footnote ²³ Admittedly, this picture is complicated by the fact that this bridge proceeds via a detour through the OS axioms, an issue that may deserve further philosophical attention.

Finally, given the crucial role of Lagrangian QFT in setting the success conditions for a solution in constructive QFT and in guiding us as to the details of the construction, it is possible that Lagrangian QFT adds physical content to QFT, over and above what the various axiomatic systems say. This might serve as guidance to future, better axiomatic systems. One particular source of physical content might be the contributions of the RG. The scaling properties of fields that the RG reveals are not something that can be obviously read off the axioms. There is also an open question whether the diagrammatic methods used to manage expansions in constructive QFT have any physical content.Footnote ²⁴ This is related to ongoing debates about whether Feynman diagrams in Lagrangian QFT are merely a convenient calculational tool or have real physical content (Wüthrich Reference Wüthrich2012). The lack of mathematical rigor of Lagrangian QFT may be a reason to dismiss the foundational importance of techniques associated with Lagrangian QFT, such as the RG and Feynman diagrams, but their continued importance even in rigorous enterprises like constructive QFT removes this reason.

6. Conclusion

In this article, I highlighted two aspects of constructive QFT that have been neglected by philosophers. The first is that a solution in constructive QFT must have some kind of correspondence relation to the renormalized perturbation series of Lagrangian QFT, even though the latter is not mathematically well defined. The second is that a successful construction depends on various choices of regularizations, counterterms, and so on, and these choices depend at least partly on information from Lagrangian QFT.

One perspective from which to view this situation is that the empirically successful formalism of Lagrangian QFT and its distance from axiomatic QFT creates a problem that constructive field theory tries to solve. The problem is to show that the “solutions” of Lagrangian QFT, which are not well defined, can be reproduced in some sense by something that is consistent with the axioms. However, the fact that we are trying to use mathematics to reproduce something that is not mathematically well defined suggests that the problem is not one of mathematics alone. Rather, it also concerns how we can best translate a semiformal language (i.e., Lagrangian QFT) into rigorous mathematics. A good translation ought to reproduce the important effects of the original language, and in this case, constructive field theorists have judged that the renormalized perturbation series are worth recovering. But this is a judgment that considers extramathematical values concerning what the “important effects” that the translation must reproduce are. There is an analogous situation in the case of partial differential equations (PDEs) in applied mathematics. Often, there might not be a strong solution to a given PDE—a solution that is a function that is sufficiently smooth to have all the derivatives that appear in the PDE. In such cases, we often look for generalized solutions, where we expand our search for solutions to include more discontinuous options. There are many ways to expand this search. Including distributions—entities of the same kind as delta functions—is only one option. Crucially, the PDE by itself does not tell us what kind of generalized solution is appropriate. Instead, the physical situation suggests how we should define a generalized solution.Footnote ²⁵ The formalism of the PDE can be conceived as applying to different types of solution spaces depending on the physical problem of interest. There is an extramathematical judgment of which solution space would best capture the effects of interest in that physical situation.

The point I am making has been noted elsewhere in the philosophy of mathematics. Davis (Reference Davis2009) has argued that in mathematics generally, it is not clear that there is a homogeneous conception of what it means to solve a problem. One could also view the issue as one in which the initial problem setting, as a mere formalism, contains multiple possible mathematical interpretations. Often, finding the correct solution involves first figuring out what the correct mathematical translation of the problem setting is. Thus, in PDEs one must first figure out which function space the derivative operators are acting on, and in QFT one must figure out what the functional integral means and how it should relate to perturbative series. Rivasseau, one of the leading figures in constructive QFT, endorses this view of the flexibility of mathematical interpretations: “in mathematics non-existence theorems, although quite common, rarely remain the last word on a subject. Often a problem with no solution is simply badly formulated and has to wait until the proper formalism in which it does have a solution is found” (Reference Rivasseau1991, 271).

Constructive QFT, as the best attempt to obtain solutions modeling particular interactions in QFT, tells us the possible behavior of QFT systems with those interactions. It thus ought to be interesting to philosophers who care about what QFT says the world is like. Yet the mathematical structure of solutions in constructive QFT is not simply given by the axioms they satisfy. These solutions also have to satisfy some kind of correspondence to perturbative solutions, and they can only be defined as limits of very specific kinds of approximations which appear to be indispensable. Thus, Lagrangian QFT might add constraints to the kinds of QFT systems of interest to philosophers, constraints not to be found in the axioms. Constructive QFT also offers an interesting example of the interaction between unrigorous and rigorous mathematics, complicating the debate about whether philosophers should take Lagrangian QFT seriously. For all these reasons, constructive QFT deserves more philosophical attention.