1. Introduction

Quantum field theory (QFT) is an example of a mature field within a mature science in which there are parallel research programs. Different communities of researchers (including experimentalists, theoreticians, and applied mathematicians) employ different variants of QFT. At one end of the spectrum of variants of QFT is the version of the theory that is found in the first part of most introductory textbooks and employed by most working physicists, namely, the version of QFT that introduces renormalization procedures to facilitate the calculation of scattering matrix elements. At the other end of the spectrum are axiomatic presentations of QFT, such as the Haag-Kastler algebraic formulation and the Wightman axiomatization, which are rigorous but remote from practical applications. There is no consensus on how philosophers should deal with this multitude of QFTs. Before the philosophical work of interpreting QFT can begin, a question must be addressed: which variant of QFT should be subject to interpretation? Arguments have been made in favor of interpreting variants of QFT on both ends of the spectrum. Teller treats the standard textbook variant of QFT in his book An Interpretive Introduction to Quantum Field Theory (Reference Teller1995); Wallace's article “In Defence of Naiveté: The Conceptual Status of Lagrangian Quantum Field Theory” (Reference Wallace2006) is devoted to defending the use of textbook QFT for foundational purposes; and, most recently, MacKinnon (Reference MacKinnon2008) has advanced arguments in favor of the same approach in this journal. In contrast, the recent work of Ruetsche, Earman, Halvorson, Clifton, and others examines the algebraic formulation of QFT.

This debate is an instance of a more general conflict of desiderata of philosophers of science, a conflict that seems particularly endemic to philosophy of physics. On the one hand, it is desirable to stick as close to actual scientific practice as possible. This means that philosophers should focus attention on the versions of theories that practicing scientists actually use and, in particular, how theories get applied. However, these theories are often messy. A theme of the work of Batterman and Cartwright (among others) is that the messy context of application is important for foundational and interpretive questions. A second desideratum of philosophers of science is to clarify the foundations of theories and to provide interpretations of theories, where necessary. These goals are often more easily achieved by focusing on cleaner versions of theories that are farther removed from actual applications. For instance, ‘toy’ models—models that represent idealized situations—have been used to investigate many foundational and interpretive questions.

The choice between these two desiderata is particularly stark in the QFT case because the desiderata are best fulfilled not by different aspects of a single theoretical framework, but by different theoretical frameworks. The textbook variant of QFT has a range of applications, and its predictions have been borne out to an impressive degree of accuracy, but it is not mathematically rigorous. In contrast, rigorous axiomatizations of QFT have been proposed, but to date no physically realistic model of any set of axioms has been found. More specifically, there is no known model for any interaction in four space-time dimensions.Footnote ¹ The arguments that have been offered for focusing on a particular variant of QFT track the general debate. For instance, Teller explains that he does not discuss “formal and rigorous work in axiomatic field theory” in his book because “although [axiomatic field theory] is a useful enterprise in the study of formal properties of quantum field theories, axiomatic quantum field theory as it exists today does not appear usefully to describe real physical phenomena” (Reference Teller1995, 146, note 5). In their survey article on the philosophical significance of algebraic QFT, Halvorson and Müger concede that the algebraic approach is “indeed idiosyncratic in the sense of demographics,” but argue that philosophers of physics should study algebraic QFT because “there remains an implicit working assumption among many philosophers that studying the foundations of a theory requires that the theory have a mathematical description” and, moreover, “whether or not having a mathematical description is mandatory, having such a description greatly facilitates our ability to draw inferences securely and efficiently” (Halvorson and Müger Reference Halvorson, Müger, Butterfield and Earman2007, 731).

I contend that an interpretation of QFT should be based on a rigorous axiomatic variant of QFT rather than any of the other variants. I will argue that there is more to recommend a rigorous variant of QFT than ease of interpretation and transparency of the foundations of the theory. A fortiori, the choice among formulations of QFT does not come down to a subjective preference for a certain methodology in philosophy of science. The content of QFT is the point at issue.

2. Setup: Three Variants of QFT, Empirical Indistinguishability, and Underdetermination

To set up the discussion, I will categorize the variants of QFT on the basis of the types of renormalization procedures that are invoked. In the course of the discussion, it will become clear why this is a natural categorization. This set of distinctions will also help to clarify the foundational significance of renormalization.

For the sake of concreteness, consider the admittedly physically unrealistic case of a $( \phi ^{4}) _{2}$ interaction (i.e., an interaction represented by a Lagrangian with a scalar φ⁴ self-interaction term on two-dimensional space-time). A φ⁴ interaction is the simplest nontrivial, physically meaningful interaction because a quadratic self-interaction term represents a free system and a cubic self-interaction term is not physically meaningful because even the classical wave equation has energy that is unbounded from below and has singular solutionsFootnote ² (Keller Reference Keller1957; Jaffe Reference Jaffe and Cao1999, 133). The restriction to two space-time dimensions allows the ultraviolet divergences to be completely removed by normal ordering (Glimm and Jaffe Reference Glimm and Jaffe1968, 1945). The Hamiltonian for the $( \phi ^{4}) _{2}$ interaction is defined as follows (Glimm and Jaffe Reference Glimm and Jaffe1968, 1945):

where : : denotes normal ordering. In classic introductory textbook presentations of QFT (e.g., Schweber Reference Schweber1961), this Hamiltonian acts on the Hilbert space of the interaction picture representation.Footnote ³ For the moment, the only relevant feature of the interaction picture representation is that, by assumption, there is some time $t_{0}$ at which the Hilbert space representation for the interaction coincides with the Fock space associated with a neutral scalar field of the same mass. (For further details, see Schweber Reference Schweber1961, 316–325, or Fraser Reference Fraser2006, Section 2.1.) In particular, at time $t_{0}$, the vacuum state for the interacting system coincides with the vacuum state for the associated free system, $\Omega _{0}$. At time $t_{0}$, applying the full interaction Hamiltonian H to the vacuum state $\Omega _{0}$ yields the following well-known result:

From a theoretical point of view, this result is problematic because the vacuum state is supposed to be invariant under the time translation operator $e^{-iHt}$. From a practical point of view, this is problematic because the formula for calculating scattering matrix elements involves applying H to $\Omega _{0}$, and the scattering matrix elements encode the experimental predictions. (Naturally, this traditional introductory textbook presentation lacks the sophistication of the modern approach to renormalization [i.e., renormalization group methods]; however, the modern refinements do not affect the arguments of this paper. See Section 6 below.)

The variants of QFT adopt different solutions to this problem. In the classic textbook presentation of QFT, this result is interpreted as a sign that the vacuum self-energy must be renormalized. That is, the infinite vacuum self-energy counterterm $E_{0}$ is introduced to make the lowest eigenvalue of the renormalized interaction Hamiltonian zero:

As a result of the restriction to two space-time dimensions, the vacuum self-energy $E_{0}$ is the only renormalization counterterm that is infinite in this case (Glimm and Jaffe Reference Glimm and Jaffe1970b, 205). The classic textbook approach is to accept the infinite counterterm $E_{0}$ and to introduce the Hamiltonian $H_{\mathrm{ren}\,}$. I will refer to the resulting formulation of QFT as the infinitely renormalized variant of QFT.

There are two other ways of responding to the result that application of the $( \phi ^{4}) _{2}$ interaction Hamiltonian H to the vacuum $\Omega _{0}$ yields infinity. One is to introduce a spatial cutoff function into the Hamiltonian that “cuts off” long-distance contributions:

The introduction of the long-distance spatial cutoff function renders the vacuum self-energy counterterm $E^{\prime }_{0}$ finite. This is the cutoff variant of QFT. In general (i.e., for interaction terms in general), short-distance cutoffs will also be needed. (Long-distance cutoffs alone are sufficient for the $( \phi ^{4}) _{2}$ interaction because the ultraviolet divergences are rendered finite by normal ordering.)Footnote ⁴

The third variant of QFT—the formal variant—responds by taking a more formal approach. In the context of formal, rigorous mathematics, the fact that applying the interaction Hamiltonian H to the interaction picture vacuum $\Omega _{0}$ yields infinity is an indication that the vector $\Omega _{0}$ is not in the domain of H (see, e.g., Glimm and Jaffe Reference Glimm and Jaffe1970a, 363). Furthermore, it turns out that the only vector in that is in the domain of H is the zero vector (Glimm Reference Glimm and Jost1969). From this more formal point of view, the proper response is thus to find another Hilbert space on which to represent the interaction Hamiltonian (i.e., a Hilbert space representation of the canonical commutation relations that is unitarily inequivalent to ). For the $( \phi ^{4}) _{2}$ interaction this has been achieved: Glimm and Jaffe's $( \phi ^{4}) _{2}$ model supplies a Hamiltonian operator for the interaction without an infinite counterterm that is well defined on a Hilbert space representation of the canonical commutation relations that is unitarily inequivalent to at all times. The Glimm-Jaffe $( \phi ^{4}) _{2}$ model satisfies all the Wightman and the Haag-Kastler axioms (Cannon and Jaffe Reference Cannon and Jaffe1970; Glimm and Jaffe Reference Glimm and Jaffe1970a; Reference Glimm and Jaffe1970b, 208; Glimm, Jaffe, and Spencer Reference Glimm, Jaffe and Spencer1974). This formal approach to QFT has been pursued by the mathematical physicists and mathematicians named in this paragraph. One strategy for pursuing this approach is the formulation of axioms for QFT. The complementary strategy is the construction of models for particular interactions, which can then be checked for their compatibility with different sets of axioms.

The infinitely renormalized, cutoff, and formal variants of QFT are alternative theories with the same intended domain of application. This raises the question, How should these variants of QFT be regarded? This is an important question for the purposes of interpretation, as mentioned above, but also for foundational purposes (e.g., for determining which variant should inform the development of future theories, perhaps including quantum gravity). There is an obvious line of response to the question: perhaps this matter can be settled empirically. Perhaps only one variant of QFT is empirically adequate. However, it turns out that this is not the case. The three variants are empirically indistinguishable. In QFT, the quantities that are subject to experimental test are scattering matrix elements. Scattering matrix elements encode the outcomes of scattering experiments. The three variants of QFT are empirically indistinguishable in the sense that the sets of scattering matrix elements generated by the three variants can be brought into arbitrarily close agreement. Consider the infinitely renormalized and cutoff variants. In the limit as the cutoffs are removed, the scattering matrix elements of the cutoff formulation approach the scattering matrix elements of the infinitely renormalized formulation (see Fraser Reference Fraser2006, Section 3.3.2, for details). Thus, the sets of scattering matrix elements generated by these two variants can be brought into arbitrarily close agreement by choosing large (or small) enough cutoff functions.Footnote ⁵ It has been verified that the sets of scattering matrix elements yielded by the rigorous Glimm-Jaffe $( \phi ^{4}) _{2}$ model also agree with those yielded by the infinitely renormalized variant, modulo the fact that in the infinitely renormalized framework, scattering matrix elements can be calculated only approximately to some order in a perturbative expansion (Wightman Reference Wightman, Boer, Dal and Ulfbeck1986, 213; Jaffe Reference Jaffe and Cao1999, 140). Since a model for a given interaction is required to calculate the scattering matrix elements for that interaction, one might worry that models that are constructed in the future will reveal disagreements between the predictions of the formal variant and those of the other variants. However, this worry is addressed by the process of model construction: one of the means of establishing that a constructed model is a model for a particular interaction is to verify that the scattering matrix elements calculated using the model agree with those calculated using other variants (Glimm Reference Glimm and Jost1969, 103). In general, then, one would expect agreement between the scattering matrix elements generated by the formal variant and the other variants.

Since experimental tests cannot settle the matter, one might wonder whether these three variants of QFT are genuinely distinct. Perhaps, for example, they are merely notational variants of one and the same theory. However, it is clear that proponents of these approaches to QFT believe that they differ significantly. As Wightman described the situation in a 1962 lecture, “the root-mean-square deviation from the mean of opinion on what is a sensible thing to try to do in elementary particle theory seems to be one of those unrenormalizable infinities one hears about” (Reference Wightman1963, 11). From the perspective of philosophy rather than sociology, there is also reason to regard the variants as genuinely distinct. One reason is that they yield different metaphysical interpretations. The infinitely renormalized and cutoff representations for the $( \phi ^{4}) _{2}$ interaction are genuinely distinct from the Glimm-Jaffe representation because they support different ontologies. For instance, they disagree about whether QFT describes quanta. (“Quanta” are entities that resemble classical particles insofar as they are countable and possess the same energies as classical, relativistic, noninteracting particles.) The complete argument for this claim that the variants of QFT disagree about whether QFT describes quanta is a paper in itself (see Fraser Reference Fraser2008), but it can be sketched in a few paragraphs.

Consider the cutoff and Glimm-Jaffe representations. The cutoff representation for an interaction admits an interpretation in terms of quanta, but the Glimm-Jaffe representation does not admit a quanta interpretation. In QFT, the quanta interpretation is derived from the Fock space representation for a system. A Fock space representation for a free system has a basis of state vectors—the n-particle state vectors—each of which can be interpreted as representing a state in which a definite number of quanta is present. In a cutoff representation for an interacting system, this quanta interpretation can be extended from free systems to interacting systems. Since the number of degrees of freedom is finite,Footnote ⁶ the Stone–von Neumann theorem applies and all standard representationsFootnote ⁷ of the equal time canonical commutation relations (ETCCRs) are unitarily equivalent. In particular, the representation for the interacting system is unitarily equivalent to the Fock representation for the corresponding free system. Therefore, any state vector describing the interacting system can be written in terms of the n-particle states for the corresponding free system. Thus, in the cutoff representation, the interacting system can also be given a quanta interpretation. In contrast, the Glimm-Jaffe representation does not admit a quanta representation. The Fock representation for the corresponding free system is not unitarily equivalent to the Glimm-Jaffe representation for the interacting system. (See Fraser Reference Fraser2008 for the argument.) Thus it is not possible to extend the quanta interpretation from the free system to the interacting system, and there are no other candidates for quanta states. As a result, the cutoff and the Glimm-Jaffe representations disagree about the existence of quanta.

It is less clear how to interpret the infinitely renormalized variant. The difficulties are a direct result of lack of mathematical rigor. The argument can be made that the infinitely renormalized representation can be given a quanta interpretation. By assumption, there is some time $t_{0}$ at which the Hilbert space representation for the interaction coincides with the Fock space for the corresponding free system. Also by assumption, time evolution is unitary; therefore, the Hilbert space representation for the interacting system is unitarily equivalent to at all times. That is, the state vector for the interacting system is always in Fock space, so it is always possible to write it in terms of the n-particle states for the free system. Thus, it seems that, in the context of the infinitely renormalized variant, the interacting system can be given a quanta interpretation. However, infinite renormalization complicates matters. The key assumption that time evolution is unitary is undermined by infinite renormalization: the renormalized Hamiltonian for the interaction $H_{\mathrm{ren}\,}$ is not a well-defined self-adjoint operator on the Hilbert space for the interacting system because it contains an infinite term; consequently, the time translation operator $e^{-iH_{\mathrm{ren}\,}t}$ is not unitary.Footnote ⁸ Deciding which interpretation the infinitely renormalized variant actually supports will require a judgment about which principles this variant actually includes. The standard criticism leveled against unrigorous theories—that they are difficult to analyze and interpret—certainly applies in this case.

The disagreement on matters of metaphysics among the three variants of QFT can also be made out in more general terms. The variants disagree about what is possible. According to the cutoff variant, it is possible for the interacting system to be in the same state as any other system, governed by any dynamics. The infinitely renormalized variant incorporates the assumption that it is possible for the interacting system to be in the same state as a free system. (Indeed, scattering theory in the interaction picture works on the basis of the assumption that the free and interacting systems are actually in the same state at $t=\pm \infty $.) This disagreement is the upshot of their respective assumptions about unitary equivalence. In contrast, according to the formal variant, it is not possible for the interacting system to be in the same state as a system governed by any other dynamics. In addition, Baker (Reference Baker2009) argues that the considerations that rule out a quanta interpretation of the formal variant also rule out a natural field interpretation of the formal variant. Thus, the disagreement between the variants is not limited to the existence of quanta. Even if we decide that QFT does not describe quanta, we can expect that the representations will still disagree on matters of metaphysics.

A second (and perhaps more decisive) reason for regarding the variants of QFT as genuinely distinct theories is that they adopt disparate sets of theoretical principles. The physical differences between the sets of principles will be described in the next section.

The three variants are empirically indistinguishable yet must be regarded as distinct theories because they embrace different sets of theoretical assumptions and carry different metaphysical implications. In other words, this is a genuine case of the classic problem of underdetermination of theory by all possible evidence. To emphasize, this is not a contrived example of underdetermination, but a real case of underdetermination that actually arises in a mature physical theory. This addresses a criticism leveled by skeptics about underdetermination: that the only examples of the phenomenon are ‘toy theories’ invented by philosophers. However, the antirealist should not be too quick to celebrate. I contend that this case of underdetermination does not provide support for scientific antirealism because there are good arguments for paying heed to the metaphysical implications of one variant of QFT and disregarding the others.

3. Inconsistency

The starting point for resolving the question of which variant of QFT to adopt is a better understanding of the point of departure for the variant formulations of QFT. What went wrong with the construction of the interaction picture?Footnote ⁹ Theorists adopted a set of seemingly plausible principles for a relativistic quantum theory but ended up with nonsensical results (i.e., a vacuum state with infinite energy and infinite probabilities for the outcomes of scattering experiments). Historically, this is precisely the difficulty that Feynman and others had to overcome in order to formulate quantum electrodynamics (QED). Renormalization methods were the remedy, but they do not directly address the question of what went wrong in the first place.

Haag's theorem provides some insight into what went wrong with the interaction picture. Haag's theorem can be viewed as a ‘no-go’ theorem for the interaction picture. The theorem establishes that if a theory adopts the specified set of assumptions {T}, then the theory necessarily describes a free system. The interaction picture goes wrong by adopting the complete set of assumptions {T}. The set {T} contains the following assumptions:Footnote ¹⁰

These assumptions are, taken individually, each plausible. However, Haag's theorem establishes that, when combined, this package of assumptions has the unwanted consequence that the field that was intended to describe an interacting system actually describes a free system. More specifically, what is proven is that all the vacuum expectation values (VEVs) of the two fields are equal:

This means that the ‘interacting’ representation makes exactly the same predictions as the representation for the free system. The set of all VEVs for a field contains the set of all its scattering matrix elements. Thus, the representation that was intended to describe an interacting system actually describes a system with trivial scattering matrix elements (i.e., the scattering matrix is the identity); that is, the representation actually describes a free system in which the initial state is identical to the final state. Assurance that theories that share a complete set of VEVs share the same dynamics in the sense of having the same Hamiltonian is provided by Wightman's reconstruction theorem (Streater and Wightman Reference Streater and Wightman2000, 117–126).

Put in slightly different terms, Haag's theorem demonstrates that the interaction picture adopts an inconsistent set of principles. Inconsistency is a straightforward consequence of the theorem. Let F be the statement that the system described is free. By Haag's theorem, $\{ T\} \Rightarrow F$. But, the interaction picture was introduced for the purpose of treating interacting systems; thus, by assumption, the system described by the interaction picture is not free. This sets up a reductio ad absurdum: $\{ T\} \& \urcorner F\Rightarrow F\& \urcorner F$. Thus, Haag's theorem informs us that the source of the problem with the interaction picture is that it is inconsistent. Furthermore, Haag's theorem establishes that this is an entirely generic problem; the theorem does not hinge on any assumptions about the specific form taken by the interaction.Footnote ¹³

The three variants of QFT espouse different responses to the reductio of the interaction picture. The infinitely renormalized variant modifies the principles of the interaction picture by inserting infinite renormalization counterterms into the Hamiltonian. The practical consequence is that the scattering matrix elements become nontrivial and finite; the theoretical consequence is that Haag's theorem is inapplicable to the interaction picture after the renormalization procedures have been carried out. Strictly speaking, the assumptions of Haag's theorem no longer hold. For example, the renormalized interaction Hamiltonian

is not a formally well-defined self-adjoint operatorFootnote ¹⁴ in virtue of the infinity of $E_{0}$. By Stone's theorem, since $H_{\mathrm{ren}\,}$ is not self-adjoint, the operator $e^{-iH_{\mathrm{ren}\,}t}$ is not unitary. Thus, time translations are represented by operators that are not unitary, contrary to assumption (ii) of Haag's theorem. Put another way, infinite renormalization introduces informal mathematical reasoning in the form of infinite subtractions and informally defined mathematical expressions. Hall and Wightman's proof of Haag's theorem relies on heavy machinery from advanced mathematics (e.g., the theory of analytic functions). The infinitely renormalized variant of QFT dodges Haag's theorem by shifting to the context of informal, unrigorous mathematics, in which these techniques of formal, rigorous mathematics are inapplicable and hence proofs of the theorem do not go through.

The cutoff variant of QFT responds to the reductio of the interaction picture by Haag's theorem by denying one of the common assumptions of the interaction picture and Haag's theorem: that the fields transform appropriately under Poincaré transformations (assumption (iii) above). The Hamiltonian $H( g) $ with the spatial cutoff function $g( x) $ ($g( x) =1$ in region R and $g( x) =0$ outside of R) is associated with the ‘cutoff’ field equation

The ‘cutoff field’ $\phi _{g}( x) $ that satisfies this field equation is clearly not invariant under spatial translations:Footnote ¹⁵

for x such that $g( x) =1$, $x^{\prime }$ such that $g( x^{\prime }) =0$. In general, the cutoff variant of QFT introduces both long- and short-distance spatial cutoff functions, which serve to reduce the theory from an infinite number of degrees of freedom to a finite number. Haag's theorem is applicable only in the presence of an infinite number of degrees of freedom. The cutoff variant also invalidates assumption (iv) of Haag's theorem, that there exists a unique normalizable state $\vert 0_{j}\rangle $ (i.e., the vacuum state) that is invariant under Poincaré transformations. For a finite number of degrees of freedom the assumption of uniqueness fails. (See Fraser Reference Fraser2006, Section 1.2.2, for further discussion.) Naturally, when the time comes to evaluate the cutoff variant of QFT, the physical motivations for rejecting these assumptions will be a central issue.

The formal variant of QFT also confronts the inconsistency of the interaction picture by seeking to modify or reject at least one of the assumptions of the interaction picture. The rigorous Glimm-Jaffe model for the $( \phi ^{4}) _{2}$ interaction rejects assumption (v), that the fields are related at some time t by a unitary transformation V. This is the natural assumption to reject because, unlike the other assumptions, it does not have a strong physical motivation. While it seems reasonable to guess that the representation for the interaction will coincide with the representation for the corresponding free system in the limit of infinitely early and late times, when the interaction is negligible, this is not an essential feature of a relativistic quantum theory. Historically, the motivation for introducing this assumption was not foundational, but practical; it made it possible to extract predictions from QFT via renormalization. From a mathematical perspective, the assumption of unitary equivalence of the free and interacting representations is dubious. In the context of ordinary nonrelativistic quantum mechanics with a finite number of degrees of freedom, the Stone–von Neumann theorem guarantees that all representations of the ETCCRs are unitarily equivalent. However, the Stone–von Neumann theorem fails in QFT because of the infinite number of degrees of freedom; in fact, there exist uncountably many unitarily inequivalent representations of the ETCCRs (Gårding and Wightman Reference Gårding and Wightman1954).

4. QFT = SR + QT

With a view toward evaluating them, these three variants of QFT can be characterized as providing either a principled response to the inconsistency of the interaction picture or a pragmatic response. The formal variant is a clear example of the principled response: the inconsistency of the interaction picture must be addressed by fixing the principles of the interaction picture because the inconsistency of the principles reflects a problem with the foundations of the theory. In contrast, the infinitely renormalized variant is a clear example of the pragmatic response: the expedient of introducing infinite renormalization is adequate as a means of getting around the problem of inconsistency because it allows predictions to be derived from the theory. (Which category the cutoff variant falls into depends on how the cutoffs are justified, as discussed below.) The principled approach is to cure the disease; the pragmatic approach is to treat the symptoms.

For scientific theories in general, it is not clear that the principled strategy of revising the principles of the theory to make it consistent is always preferable. The argument has been made that, even for mature scientific theories, it can be appropriate to retain an inconsistent theory and to merely treat the symptoms.Footnote ¹⁶ However, in the case of QFT, there is a compelling reason to demand a consistent formulation of the theory. Quantum field theory is by definition the theory that best unifies quantum theory (QT) and the special theory of relativity (SR).Footnote ¹⁷ Historically, it was clear by the mid-1920s, when physicists had obtained nonrelativistic quantum mechanics, that physicists should work on formulating a relativistic version of quantum theory.Footnote ¹⁸ Since QFT = QT + SR, the project of formulating quantum field theory cannot be considered successful until either a consistent theory that incorporates both relativistic and quantum principles has been obtained or a convincing argument has been made that such a theory is not possible. The big foundational question lying in the background is, of course, whether the principles of quantum theory and special relativity are consistent. I do not presume that this question has a positive answer. However, I do maintain that the project of developing QFT cannot be considered complete until this central foundational question has been answered. The pragmatic response to the inconsistency of the interaction picture is inadequate because it leaves the question of whether QFT is possible unanswered. To put the point provocatively, it is not clear that any variant of QFT that adopts a pragmatic response to the infinities is a theory that satisfies the definition of QFT. Either a principled response or a demonstration that a principled response is not possible is required.

The infinitely renormalized variant of QFT is unsatisfactory because it neither furnishes a manifestly consistent set of principles for relativistic quantum theory nor provides a reason to believe that a consistent relativistic quantum theory is impossible. On the latter score, the inconsistency of the interaction picture does not constitute evidence that it is not possible to consistently formulate a relativistic quantum theory because we have no reason to believe that the interaction picture is the only possible way to realize such a theory. For example, the formal variant furnishes examples of alternatives (e.g., the Wightman axiomatization and the Haag-Kastler axiomatization). With respect to the former point, after infinite renormalization procedures have been carried out, the resources of formal, rigorous mathematics cannot be brought to bear to test the consistency of the theory. Consequently, it would be difficult to determine whether the infinitely renormalized variant of QFT is consistent. Note that these criticisms have nothing to do with the efficacy of infinite renormalization procedures for the purpose of deriving predictions. To the best of my knowledge, no set of contradictory predictions has been derived in the more than 50-year history of applications of the theory. I grant that it is extremely unlikely that physicists will ever unearth a contradiction.Footnote ¹⁹ However, this fact should not be taken as evidence in favor of the consistency of the theory. It is possible that physicists are just reasoning very carefully from an inconsistent set of theoretical principles. That is, they may be employing inferential restrictions. Many historical examples of this phenomenon have been brought to light by historians and philosophers of science, including the old quantum theory and Newtonian gravity (see Meheus Reference Meheus2002).

In contrast, the formal variant of QFT offers a satisfactory line of response to the reductio of the interaction picture because the response is principled. The goal of the formal variant is to find a consistent reformulation of the principles of the interaction picture. Again, the consistency of quantum and relativistic principles is desired but not presupposed. Ultimately, the success of this project will hinge on whether there exist realistic models of the resultant axioms. (I will return to this issue in Section 5.)

The short argument that the cutoff variant is unsatisfactory is that it does not satisfy the definition of QFT. Setting aside the principled-pragmatic distinction, the cutoff variant cannot, strictly speaking, be considered a relativistic theory because the fields are not Poincaré covariant.

The long argument that the cutoff variant is unsatisfactory appeals to the principled-pragmatic distinction. Whether the cutoff variant of QFT falls into the category of a principled or a pragmatic response to the reductio depends on how the cutoffs are interpreted. If the cutoffs are regarded as a convenient device that is employed for the purpose of facilitating the derivation of predictions, then the cutoff variant offers a pragmatic response to the inconsistency of the interaction picture. However, the cutoff variant also contains the resources for a principled response. As explained above, the cutoff variant rejects common assumptions of the interaction picture and Haag's theorem; consequently, the principles of the cutoff variant are consistent. In general, consistency is achieved by reducing the theory to a finite number of degrees of freedom by defining the fields on a spatial lattice of finite extent. However, to provide a genuine principled response to the inconsistency of the interaction picture, the cutoffs must be regarded as essential elements of the theory and not merely dispensable add-ons to facilitate calculation. On the latter, instrumental view, the content (i.e., core theoretical principles) of the cutoff variant of QFT is identical to that of the infinitely renormalized variant; it is merely the approach to renormalization as a tool for deriving predictions that differs.Footnote ²⁰ The way in which the cutoffs are viewed will have repercussions for their interpretation. If the cutoffs are taken seriously, then they must be interpreted realistically; that is, space is really discrete and of finite extent according to the cutoff variant of QFT.Footnote ²¹ Thus, the cutoffs must take particular fixed values (though these may not be presently known). If the cutoffs are not taken seriously, then they may be interpreted instrumentally; that is, space is really continuous and of infinite extent according to the cutoff variant of QFT. This is compatible with assigning the cutoffs arbitrary values. The upshot is that the cutoff variant of QFT can be regarded as a candidate formulation of relativistic quantum theory only if it makes sense to regard space as a lattice of finite extent.

Can a case be made for interpreting the cutoffs realistically? If QFT were true, would space be discrete and finite in extent? It is telling that—to the best of my knowledge—nobody defends the position that QFT provides evidence that space is discrete and the universe is finite. Of course, proponents of some quantum theories of gravity have claimed that space is discrete. However, even if these claims are borne out, the fact that quantum gravity indicates that space is discrete would not help settle the question of how to interpret the cutoff variant of QFT because gravitational considerations are external to QFT. The point at issue is whether QFT dictates that space is necessarily discrete and finite in extent, that is, whether the discreteness and finitude of space are a foundational principle of QFT.

One reason that a realistic interpretation of the cutoffs is not compelling is illuminated by considering, once again, the cutoffs as a response to the reductio of the interaction picture. As a response to the reductio, the introduction of the assumption that space is discrete and the universe is finite seems ad hoc. It does circumvent Haag's theorem and produces a consistent set of principles for QFT, but it does not have an independent motivation. In contrast, arguments from quantum gravity that space is discrete are supported by deep theoretical considerations about how gravity is to be quantized. Arguments from cosmology that the universe is spatially finite are based on a combination of theory and a variety of experimental results. The desire for a consistent formulation of QFT is not sufficient to justify the introduction of the assumption that space is discrete and the universe is finite because there is another strategy for obtaining a consistent formulation of QFT: the strategy of modifying other principles of the interaction picture, which is adopted by the formal variant. If we had good reason to believe that this strategy will fail, then perhaps we would have good reason to interpret the cutoffs realistically. However, even in that case, it would remain an option to concede that it is not possible to formulate a consistent relativistic quantum theory. (This issue will be taken up in Section 5.) In sum, the cutoff variant of QFT is unsatisfactory because the cutoffs are not interpreted realistically, and when the cutoffs are interpreted instrumentally, the cutoff variant is subject to the same criticisms as the infinitely renormalized variant.

The formal variant is the only one of the three variants to provide an adequate response to the reductio of the interaction picture. Viewed in these terms, the choice among the variants of QFT is not (merely) a matter of picking the variant that is easiest to subject to analysis and interpretation; the choice is a matter of determining the content of QFT. Each of the three variants of QFT adopts a different set of theoretical principles. In this sense, each of the three variants prescribes a different foundation for QFT. Since the content of the theory is what is at stake, it is not surprising that the matter is settled by foundational considerations rather than purely interpretive considerations. The formal variant is not singled out because it supports a particular ontology or even because the other candidates succumb to interpretive difficulties, but because QFT should be a relativistic quantum field theory (if such a thing is possible) and it is the only variant that satisfies this condition. In some respects, the situation in QFT parallels the situation in nonrelativistic quantum mechanics. In nonrelativistic quantum mechanics, alternative, physically distinct formulations of the theory (e.g., Bohmian mechanics) were also sought out in response to foundational problems with the standard formulation of the theory. The significant point of contrast is, of course, that the consistency of relativistic and quantum principles is a different kind of foundational problem than the measurement problem.

As an aside, the foregoing considerations about consistency may also pose a difficulty for the interpretation of infinitely renormalized QFT. A consistent set of theoretical principles may all be true simultaneously; an inconsistent set is certainly not, in any possible world.Footnote ²² If a theory contains an inconsistent set of theoretical principles, some members of the set must be false. A metaphysical interpretation of a theory should be based on the true theoretical principles, but not the false ones. The need to identify the true principles complicates the project of interpreting an inconsistent theory. This must be borne in mind when interpreting the infinitely renormalized variant of QFT because it is not known whether this variant is consistent.

5. But We Possess No Realistic Models of Any Set of Rigorous Axioms for QFT!

The position that I have been advocating—that the formal variant of QFT should be treated as the official formulation of QFT—seems counterintuitive in an important respect: it is not clear that any of the proposed sets of rigorous, consistent principles for QFT holds for any realistic system! To date, no rigorous model for a realistic interaction has been obtained—for any axiomatization of QFT. This is why I have been referring to the Glimm-Jaffe model for the $( \phi ^{4}) _{2}$ interaction. David Wallace worries that “pending the discovery of a realistic interacting [model for an axiomatization of QFT] … we have only limited reason to trust that our results apply to the actual world” (Reference Wallace2006, 34). In particular, the lack of realistic interacting models means that, strictly speaking, formal QFT does not make any empirical predictions about our world! This is a reasonable concern; if the formal variant of QFT is to be viable, it must be addressed. The first step is to observe that, as explained in Section 2, one would expect the empirical predictions of any as-yet-to-be-discovered realistic interacting model to agree with the empirical predictions of the infinitely renormalized and cutoff variants (modulo the use of approximation methods). One might ask, Should not the fact that, at the present time, the alternative variants have enjoyed superior empirical success in the real world weigh in favor of the alternatives to formal QFT?

This concern can be alleviated by regarding the formal variant of QFT as a program that has yet to be completed and recognizing that, even at its current stage, it constitutes an improvement over the alternatives in salient respects. This axiomatic program cannot be considered complete until models of a rigorous set of axioms are found for realistic interactions. The starting point for the program is the interaction picture and the reductio of the interaction picture underwritten by Haag's theorem. As Haag himself recognized in the paper that contains the first presentation of the eponymous theorem (Haag Reference Haag1955), the first step was to reject the assumption of the interaction picture that there is a time at which the representation for the interacting system is unitarily equivalent to the representation for the free system.Footnote ²³ That this was the first step was confirmed by the first success of the program: the construction of the rigorous model for the $( \phi ^{4}) _{2}$ interaction by Glimm and Jaffe in the early 1970s. This model satisfies all the assumptions of the Hall-Wightman version of Haag's theorem with the sole exception of the assumption that there is a time at which the representation is unitarily equivalent to the Fock representation for a free system. However, even before the $( \phi ^{4}) _{2}$ model had been obtained, it was clear that further refinements to the interaction picture would be necessary to obtain a consistent set of principles applicable to realistic interactions. Recall that Haag's theorem pertains only to the divergence in vacuum self-energy, which can be treated by introducing a long-distance cutoff. For realistic interactions, there are other types of divergences (e.g., ultraviolet divergences, which are treated using short-distance cutoffs). It would be very useful if it were possible to prove a sequence of Haag-type no-go theorems: for example, a (Haag)² theorem that establishes that a certain set of the principles embraced by the interaction picture imply that the system under consideration is a free system when the dimension of space-time is greater than two. Such a sequence of theorems might give an indication of which assumptions of the interaction picture would need to be revised or abandoned to treat particular interactions.

By the early 1960s, Wightman (Reference Wightman1959) had proposed one set of axioms for QFT and Haag and Kastler (Reference Haag and Kastler1964) had proposed another set of axioms. These axiom systems were conjectures about how the principles of the interaction picture would need to be refined in order to consistently accommodate realistic interactions. The provisional nature of these proposed axiomatizations was recognized from the outset. In one of the first textbook presentations of the formal approach to QFT, Jost (Reference Jost1965, xi) coined the term “general field theory” because he did not like the connotations of the term “axiomatic.” Haag gives the following explanation of why he also prefers to employ Jost's terminology: “the word ‘axiom' suggests something fixed, unchangeable. This is certainly not intended here. Indeed, some of the assumptions are rather technical and should be replaced by more natural ones as deeper insight is gained. We are concerned with a developing area of physics which is far from closed and should keep an open mind for modifications of the assumptions, additional structural principles as well as information singling out a specific theory within a general frame” (Reference Haag1996, 58).

For the axiomatic program to achieve completion, realistic models of some set of principles for relativistic quantum theory must be found. The construction of realistic models for either (or both) the Wightman or Haag-Kastler axiomsFootnote ²⁴—or any other set of axioms that has been proposed, for that matter—would complete the program, but this is not the only way in which the program could be completed. It is also possible that the sets of principles proposed by Wightman and Haag-Kastler must be further refined before they admit realistic models. For example, Bender (Reference Bender2007) advocates dropping the requirement that the Hamiltonian be Hermitian, and Rivasseau (Reference Rivasseau2007) makes a case for adopting noncommutative space-time.

This approach to the formal variant gains further plausibility when QFT is viewed in its proper historical context. QFT is hardly unusual in requiring refinements over a period of time. There are many cases in which an inconsistent formulation of a theory was replaced by a consistent one in the course of its historical development. There is also historical precedent for an informally formulated theory later being given a formal reformulation. Arguably, the evolution of Newtonian mechanics from Newton's Principia to its modern-day textbook formulation is an example of this.

From this perspective, the fact that no rigorous model for a realistic interaction has been constructed is not an argument for disregarding the formal variant of QFT. In fact, precisely the opposite is the case: the Wightman and Haag-Kastler axiomatizations improve on the interaction picture by refining its axioms. The rigorous models for idealized interactions that have been constructed—interactions that exhibit tamer instances of the same types of divergences that plague realistic interactions—provide evidence that the refinements that have been made are on the right track. In contrast, the infinitely renormalized and cutoff variants of QFT do not institute appropriate refinements of the principles of the interaction picture, as I have argued. The fact that no rigorous model for a realistic interaction has been obtained is not an argument against the formal variants and for the alternatives, but this fact should not be overlooked by philosophers studying the foundations and interpretation of QFT. The provisional nature of the proposed axiomatizations of QFT carries the implication that foundational and interpretive conclusions based on these axiomatizations are also provisional.

What if the formal variant of QFT is a program that it is not possible to complete? That is, what if it turns out not to be possible to find a consistent set of relativistic and quantum principles that are applicable to realistic interacting systems? This is a hypothetical question because mathematical physicists are still working toward constructing models for realistic interactions. The results that have been obtained so far do not give any reason to believe that this is an unattainable goal. However, it is interesting to speculate on what the failure of the program to formulate an applicable yet consistent relativistic quantum theory would mean. One interpretation of the result is that relativistic and quantum principles are inconsistent. This seems to be what Streater and Wightman had in mind when they wrote in the introduction to their textbook PCT, Spin and Statistics, and All That that “the Main Problem of quantum field theory turned out to be to kill it or cure it: either to show that the idealizations involved in the fundamental notions of the theory (relativistic invariance, quantum mechanics, local fields, etc.) are incompatible in some physical sense, or to recast the theory in such a form that it provides a practical language for the description of elementary particle dynamics” (Reference Streater and Wightman2000, 1). Another way of interpreting the failure of the formal program would be as a victory for the cutoff variant: as discussed in Section 4, failure to obtain a consistent set of principles on infinite, continuous space could be cited as a justification for treating space as finite and discrete. A third alternative is that the no-go result for the formal program could be ambiguous.Footnote ²⁵ The requirement is that relativistic quantum theory include recognizably relativistic principles and recognizably quantum principles; however, there is a certain amount of latitude in deciding what counts as a relativistic principle and what counts as a quantum principle. In any case, it is worth pursuing the formal program because either success or failure would be a significant result for the foundations and interpretation of QFT.

6. In Defense of Sophistication

I have been arguing against basing the interpretation of QFT on the cutoff variant of the theory. In a paper entitled “In Defence of Naiveté: The Conceptual Status of Lagrangian Quantum Field Theory,” David Wallace defends the cutoff variant of QFT. By “interpretation” I mean the activity of giving an answer to the following hypothetical question: “If QFT were true, what would reality be like?” In contrast, the interpretive question that Wallace focuses on is “Given that QFT is approximately true, what is reality (approximately) like?” The fact that QFT does not furnish a true description of the actual world makes this a substantial point of disagreement. QFT marries special relativity and quantum theory but does not incorporate general relativity. Following Cao (Reference Cao1997), Wallace frames the debate in terms of which one of two attitudes should be adopted to the foundational status of QFT:Footnote ²⁶

1. (i) “The current situation is genuinely unsatisfactory: we should reject the cutoff theories …Footnote ²⁷ and continue to look for nontrivial theories defined at all lengthscales,” or

2. (ii) “QFT's as a whole are to be regarded as only approximate descriptions of some as-yet-unknown deeper theory [theory X], which gives a mathematically self-contained description of the short-distance physics” (Reference Wallace2006, 45).

Wallace elaborates that “the difference between [(i)] and [(ii)] is that the former rejects current QFT in toto, and looks for mathematically rigorous versions of our current QFTs: QED, $\lambda \phi ^{4}$, the Standard Model, etc. By contrast [(ii)] accepts that these current theories are indeed best understood in terms of Lagrangian QFT, and looks for a deeper-level theory in which Lagrangian QFT as a whole can be grounded” (45). In order to resolve the “attendant foundational problem” with option (ii)—that is, the problem of how we can “give a clean conceptual description of a theory which can be rigorously defined only as the low-energy limit of another theory which we do not yet have”Footnote ²⁸ (46)—Wallace adopts what I have been calling the cutoff variant of QFT. A short-distance cutoff that is much shorter than the length scales at which QFT is applied and tested is imposed.

Recall that in Section 4 I argued that the cutoff variant is unsatisfactory because it offers only a pragmatic response to the reductio of textbook QFT. These arguments are not straightforwardly applicable to Wallace's position because the above arguments are predicated on the assumption that interpreting QFT involves describing what reality would be like if QFT were true. In accordance with attitude (ii), Wallace instead regards QFT as providing an approximate description of the actual world at suitably large distance scales. As he puts it, “success in [algebraic QFT, string theory, or another theory-of-everything candidate] … would, of course, revolutionise physics, but that success would scarcely change the current status of Lagrangian QFT: as an inherently approximate, but nonetheless extraordinarily powerful tool to analyse the deep structure of the world” (Reference Wallace2006, 75). This stance allows Wallace to argue that there is a principled physical justification for introducing the cutoffs without necessarily being forced to defend the view that space is discrete and finite in our world. He lists three alternatives that each provide a “physical justification … for imposing a cutoff in relativistic QFT” (43). Two of these possible physical justifications involve the physics imposing an effective cutoff for QFT without space being discrete. For example, it is possible that the field-theoretic description breaks down at a very high energy scale and is replaced by a different kind of theory (e.g., string theory; 44). Wallace's attempted defense of the cutoff variant is not successful. My first line of objection is that, even if it were granted that QFT should be regarded as an approximate guide to the ontology of relatively large distance scales, it does not follow that the cutoff variant of QFT succeeds in describing large-scale ontology. I take issue with the premise that the content and interpretation of the cutoff variant are approximately equivalent to those of algebraic QFT (the leading strain of the formal variant of QFT). The more important line of objection is the second one: that QFT should not be regarded as merely an approximate guide to the ontology of relatively large distance scales. Wallace's insistence that we respect the fact that QFT is not our final theory actually supports the pursuit of the formal variant of QFT.

An underlying assumption of Wallace's argument is that there is approximate agreement on matters of ontology between the cutoff variant of QFT and algebraic QFT (an instance of what I have been calling the formal variant). This assumption is incorrect. As Wallace represents matters, “we can see that Lagrangian QFT (as I have defended it) is not really in conflict with A[lgebraic] QFT at all. Success in the A[lgebraic] QFT program would leave us with a field theory exactly defined on all scales, and such a theory would be a perfectly valid choice for ‘theory X’: furthermore, even if we found such an exact QFT it would not prevent us from defining low-energy, ‘effective’ QFTs—which would not be well defined without a cutoff” (Reference Wallace2006, 41; see also 35, 75). One of the main conclusions of the foregoing sections is that the variants of QFT differ with respect to both content and matters of interpretation. Changing the subject from precise agreement to approximate agreement does not affect matters. Wallace contends that cutoff QFT approximately satisfies the axioms of QFT.Footnote ²⁹ For example, the axioms of Poincaré covariance and local commutativity are strictly false but approximately satisfied over large distance scales (Wallace Reference Wallace2006, 50). However, the cutoff variant does not have even approximately the same content as algebraic QFT because the cutoff variant has a finite number of degrees of freedom and therefore does not admit unitarily inequivalent representations; in contrast, algebraic QFT has an infinite number of degrees of freedom and therefore admits unitarily inequivalent representations.Footnote ³⁰ Spontaneous symmetry breaking is one case in which these unitarily inequivalent representations are put to use (see Earman Reference Earman2004 for an exposition and preliminary analysis of this case). Cutoff and algebraic QFT admit different interpretations for the same reason. A theory according to which quanta exist is not approximately equivalent to a theory according to which quanta do not exist. The flaw in Wallace's reasoning is that the fact that we do not care about what QFT tells us about short-distance scales does not license chopping short-distance scales out of the theory. Cutting off the theory at some short-distance scale has the effect of changing the content of the theory as a whole, including its description of long-distance scales. The difference in physical content is also reflected in the fact that the variants include different sets of theoretical principles. A consequence of these differences in content is that the theories support different ontologies.

Wallace does address the issue of unitarily inequivalent representations. He summarizes his argument as follows: “From a realist perspective the sting of the representation ambiguity has largely been drawn. Locally, any representation ambiguity is artificial, caused by the presence of unphysical degrees of freedom beyond the high-energy limit of the theory's validity. Globally, there may indeed be representation ambiguities—depending on cosmology, and the topology of the universe—but the inaccessible information which they encode is ‘respectable’, analogous to the classical inaccessibility of the long-distance structure of the universe” (Reference Wallace2006, 58). In the present context, this argument is not compelling because the issue of whether the “degrees of freedom beyond the high-energy limit of the theory's validity” need to be taken into account is precisely the point of contention.

A source of motivation for Wallace's position is the thought that renormalization group methods provide a warrant for subjecting the cutoff variant of QFT to interpretation. As Wallace points out, renormalization group methods establish that the imposition of a short-distance cutoff “has no practical consequences” for the predictions of the theory (Reference Wallace2006, 44). However, this contingent fact about some QFTs is not by itself sufficient warrant for the conclusion that the cutoff variant should be subject to interpretation. Renormalization group methods establish that the predictions of the cutoff variant are in approximate agreement with the predictions of the infinitely renormalized variant, but they do not furnish any evidence about approximate agreement of either the theoretical principles or the interpretations supported by the cutoff and infinitely renormalized variants. A fortiori, renormalization group methods have no bearing on whether or not there is approximate agreement between either the theoretical principles or the interpretations of the cutoff and algebraic variants of QFT. This brings the discussion back to underdetermination.Footnote ³¹

There is a more fundamental difficulty with Wallace's position. Wallace contends that the correct attitude to adopt toward QFT is to accept that our “current theories [i.e., the standard model, QED, etc.] are indeed best understood in terms of Lagrangian QFT, and looks for a deeper-level theory in which Lagrangian QFT as a whole can be grounded” (Reference Wallace2006, 45). The problem with this approach is that it gets things the wrong way around. QFT, as the predecessor theory, is valuable for the hints that it gives about the content of the successor theory, theory X. Theory X is the theory that furnishes an adequate description of short-distance physics and, thus, includes quantum gravity. But, as the name indicates, theory X is an as-yet-undiscovered theory. How are we to determine the content of theory X? One of the starting points for formulating theory X is QFT.Footnote ³² Wallace commits himself to the following brand of scientific realism (at least with respect to Lagrangian QFT): Lagrangian QFT is “an inherently approximate, but nonetheless extraordinarily powerful tool to analyse the deep structure of the world” (75). If (Lagrangian) QFT is to play this role, then theory change from QFT to theory X must involve approximate continuity at the theoretical level and not merely approximate continuity at the empirical level. This makes the theoretical content of QFT particularly relevant for finding theory X. As I have stressed, the content of QFT is precisely what is at issue in the choice among variants of QFT. How the disagreements about the appropriate theoretical principles for QFT get settled is germane to the development of theory X. Resting content with Lagrangian QFT is thus a poor strategy for finding theory X; the better strategy is to continue to work on these questions about the content of QFT.

This is an entirely general point about theoretical change that does not hinge on the details of theory X. Of course, until we actually find theory X and can determine the respects in which there is approximate continuity between theory X and QFT, we have no way of knowing how helpful it would have been to work out the details of QFT. The fact remains that—given our limited knowledge at the present time—continuing to pursue the development of QFT is a good strategy. However, one can hypothesize about how pursuing the axiomatic program might illuminate the content of theory X. Entering into the realm of pure speculation, here are some examples. As Wallace himself points out, the fact that the cutoff variant is approximately (but not exactly) Poincaré covariant is of little help for finding theory X; approximate Poincaré covariance at large-distance scales does not provide any indication about whether theory X is covariant or not (“understood as, say, the absence of a preferred spacetime foliation” [Reference Wallace2006, 51–52]). In contrast, either a formal variant of QFT that contains exact Poincaré covariance as a principle or the knowledge that it is not possible to consistently combine Poincaré covariance with other field-theoretic principles (e.g., knowledge that the axiomatic QFT program fails) would be useful for finding theory X. If the debate were to be settled in favor of the cutoff variant with realistically construed cutoffs, this might lend support to approaches to quantum gravity according to which space is discrete. More fundamentally, if it were to turn out that even special relativity and quantum theory are incompatible (if, say, the axiomatic program proves to be incompletable), then the strategy for combining general relativity and quantum theory could be affected. More generally, it is conceivable that the quest for a rigorous model of a realistic model will uncover a more general mathematical framework that could prove more suitable for formulating quantum gravity. (For hints in this direction, see Bender Reference Bender2007 and Rivasseau Reference Rivasseau2007.) I concede that these examples are pure speculation. I hope that this does not deflect attention from the argument that settling questions about the theoretical content of QFT might aid the search for theory X. Resting content with the cutoff variant of QFT because it is empirically adequate at large-distance scales would be a strategic mistake because it would hinder the search for theory X.

7. Conclusion

This paper has explored the implications of a genuine case of underdetermination in contemporary physics. The infinitely renormalized, cutoff, and formal variants of QFT are empirically equivalent in the sense that they are empirically indistinguishable on the basis of past and future scattering experiments. The three variants differ with respect to physical content; each variant adopts a different set of theoretical principles. The three variants also admit different interpretations. For example, the cutoff variant supports a quanta ontology but the formal variant does not. I argued that the proper moral to draw from this case is not antirealism, but that empirical considerations narrowly construed are not the only grounds on which to choose among rival theories. In this case, consistency is also a relevant criterion because quantum field theory is, by definition, the theory that integrates quantum theory and the special theory of relativity. Consistency is relevant to QFT for theoretical reasons—not for practical reasons (e.g., the derivation of predictions). As a result, it is necessary to either formulate a consistent theory or else show that this criterion cannot be satisfied (i.e., that there is no consistent theory with both quantum and special relativistic principles). The infinitely renormalized variant fails to satisfy this criterion because its theoretical principles are not manifestly consistent. The cutoff variant fails to satisfy the criterion because, while its theoretical principles are consistent, they are not well motivated. The formal variant is the only variant that satisfies the criterion; its set of theoretical principles are both consistent and well motivated. Neither the infinitely renormalized nor cutoff variant furnishes an argument that a consistent formulation of QFT is impossible; such an argument would require making the case that the axiomatic program cannot be completed.

This covers the terms “underdetermination” and “inconsistency” in the title. Idealization also plays an interesting role in this case. The standard criticism of the formal variant of QFT—that it has yet to be established that any set of rigorous axioms applies to the real world because no realistic models have been constructed—can be construed as a complaint that the formal variant of QFT is an idealization. In response to this criticism, I have argued that if it does turn out that none of the sets of axioms that have been proposed admit realistic models, then the proposed axiomatizations could be modified. Thus, if it does turn out that the proposed axiomatizations are idealizations in this sense, the idealization is in principle dispensable; the idealization could be removed by appropriate modification of the axioms (modulo the possibility that there is no set of relativistic plus quantum axioms that admits realistic models).

The cutoff variant is an idealization in another sense: when the theoretical principles of the cutoff variant are interpreted literally, the cutoff variant describes a world in which space is discrete and of finite extent; this is an idealization in the sense that the possible worlds in which QFT is true are presumed to be worlds in which space is continuous and infinite. This idealization is indispensable insofar as it is not possible to remove the cutoffs entirely. (Removing the cutoffs by taking infinite limits would turn the cutoff variant into the infinitely renormalized variant, which would mean adopting a different set of theoretical principles and introducing the attendant set of problems.)Footnote ³³

The significance of idealization in quantum statistical mechanics (QSM) has recently been the subject of debate (see Callendar Reference Callendar2001; Ruetsche Reference Ruetsche2003; Batterman Reference Batterman2005). Since there are many similarities between QFT and QSM, one might expect this debate about idealization to map onto QFT; however, idealization plays opposing roles in the two cases. The root issues in the QSM case are the same as in the QFT case: an infinite number of degrees of freedom and unitarily inequivalent representations. In QSM, the thermodynamic limit is taken, which is an idealization insofar as the system under consideration (e.g., a steaming cup of coffee) is represented as containing an infinite number of particles. The thermodynamic limit, which makes available unitarily inequivalent representations, must be invoked to represent phase transitions (e.g., a transition from a liquid to a gas). Another point of similarity is that renormalization group methods are employed in both QFT and QSM. However, there is a crucial difference between QSM and QFT with an infinite number of degrees of freedom (i.e., either the infinitely renormalized variant or the formal variant): whereas the description of a system as containing an infinite number of particles furnished by QSM is taken to be false, the description of space as continuous and infinite that is furnished by QFT with an infinite number of degrees of freedom is taken to be true. As far as idealization is concerned, the parallel should not be drawn between QSM and QFT with an infinite number of degrees of freedom, but between QSM and QFT with a finite number of degrees of freedom (i.e., the cutoff variant). In the context of QFT, representing space as being discrete and finite is the same sort of idealization as, in the context of QSM, representing a system as containing an infinite number of particles. Both idealizations are also indispensable, in the above sense. However, the fact that there is no alternative to QSM that both is capable of handling phase transitions and does not employ the idealization means that there is an argument for adopting QSM that is not available for QFT with cutoffs.