# Theories Without Models:
Uncontrolled Idealizations in Particle Physics

Antonis Antoniou  University of Bonn<EMAIL_ADDRESS>and  Karim P. Y.
Thébault  University of Bristol<EMAIL_ADDRESS>
(Date: Draft of .)

###### Abstract.

The perturbative treatment of realistic quantum field theories, such as
quantum electrodynamics, requires the use of mathematical idealizations in the
approximation series for scattering amplitudes. Such mathematical
idealisations are necessary to derive empirically relevant models from the
theory. Mathematical idealizations can be either controlled or uncontrolled,
depending on whether current scientific knowledge can explain whether the
effects of the idealization are negligible or not. Drawing upon negative
formal results in asymptotic analysis (failure of Borel summability) and
renormalization group theory (failure of asymptotic safety), we argue that the
mathematical idealizations applied in perturbative quantum electrodynamics
should be understood as uncontrolled. This, in turn, leads to the problematic
conclusion that such theories do not have theoretical models in the standard
understanding of this term. The existence of unquestionable empirically
successful theories without theoretical models has significant implications
both for our understanding of the theory-model relationship in physics and the
concept of empirical adequacy.

###### Contents

  1. 1 Theories Without Models: An Argument
  2. 2 Background
  3. 3 Theoretical Models and Empirical Adequacy
  4. 4 Models in Perturbative QFT
  5. 5 Breaking the Theory-Model Link
  6. 6 Implications

##  1\. Theories Without Models: An Argument

In this paper, we will articulate and explore a chain of reasoning encoded in
the following argument pattern.

  1. P1.

Perturbative quantum field theory is best understood as a framework for
theories where individual theories are picked out within the framework via
particular action principles.

  2. P2.

For a theory to have theoretical models is for a representation of the
quantities predicted by the theory to be derivable either: i) as deductive
consequences of the theory; or ii) as a result of an approximate derivation
based upon controlled mathematical idealizations.

  3. P3.

Contemporary particle physics features perturbative quantum field theories,
such as quantum electrodynamics, in which the derivation of predictions
requires an ineliminable appeal to uncontrolled idealizations.

  4. C1.

Contemporary particle physics includes theories, such as quantum
electrodynamics, without theoretical models.

P1 we take to be uncontroversial and straightforwardly justified by
contemporary scientific practice. P2 is a standard definition drawn from the
philosophical literature and we take it to be also uncontroversial, although
upon to contestation and revision with good reason. P3 is the controversial
premise, and much of the paper will be devoted to defending it in the context
of the well-known formal challenges to the rigorous formulation of
perturbative quantum field theories. Given our arguments in favour of P3 are
accepted, and one wishes to reject the conclusion, either on the grounds of
being unintuitive or due to other negative implications, the obvious
implication via contraposition is then that we should revise our definition of
theoretical models to reject P2.

We are sympathetic to such a response but note that such a revision will have
non-trivial knock-on implications for the philosophy of science. In
particular, changing our definition of a theoretical model to include theory-
model links via uncontrollable idealizations will have implications for non-
arbitrary connections between theories and models and the definition of
empirical adequacy. We will consider these implications in the final Section
6. There we will consider the idea of connecting empirically relevant models
to a theory via reference to further theories or frameworks. That is, of
modifying the definition of empirical adequacy to allow the models on which
the adequacy is based to be linked to the theory via exogenous arguments,
which may include idealizations controlled via other theories or frameworks.
This approach has the advantage of mirroring the attitude of many practicing
scientists and has the interesting broader implication that perhaps
philosophers should look beyond the theory as a unit of analysis in the
context of evaluations of empirical adequacy.

Before we proceed to situate our analysis in the various background
literatures, let us make the aims of this paper completely clear by indicating
what this paper is not about. Here we are not arguing that particle physics
does not have models per se, nor that there is a fundamental or basic
inconsistency or inadequacy in the framework of perturbative quantum field
theory. Indeed, models of various non-theoretical sorts abound in particle
physics and there exist various important formal results that indicate
precisely in which circumstances the idealizations involved in perturbative
quantum field theory can be understood to be controllable. Our aim here is to,
first, demonstrate that in the context of unquestionably empirically
successful examples of individual perturbative quantum field theories applied
in particle physics, such as quantum electrodynamics, the conditions for the
controllability of the idealizations in question can be expected to fail and
to, second, point out that this implies that these theories do not have models
in the standard sense of theoretical model. We take this conclusion to be the
result of the combination of uncontroversial results in perturbative quantum
field theory and standard definitions in the philosophy of science and, as
such, to be all the more worth of interest given the modesty of the individual
argumentative steps involved.

##  2\. Background

###  2.1. Mathematical Idealizations in Particle Physics

Mathematical idealizations are formal simplifications of the structure of a
mathematical model or system of equations that aim to facilitate the provision
of explicit analytical or numerical solutions. The classic simple example of a
mathematical idealization is the small angle approximation in the solution of
the differential equation for pendulum motion and we will return to this
example later. In modern physics, mathematical idealizations are often based
upon truncation series expansion. For example, approximating the exponential
function by the first few terms of the Taylor expansion, i.e. $e^{x}\approx
1+x+\frac{x^{2}}{2}$. Mathematical idealizations should be distinguished from
other forms of idealizations, much discussed in contemporary philosophy, which
typically focus on the processes of constructing model systems or
representations by distorting or omitting features of their target
systems.111In his seminal paper on idealization McMullin (1985) explicitly
distinguishes between ‘mathematical idealization’ and ‘construct
idealization’. The discussions in Pincock (2007) and Batterman (2009) mainly
concern the former type in which the focus is on the mathematical treatment of
the models, while discussions in Weisberg (2007), Morrison (2015) and Portides
(2021) mainly concern the latter. See also Norton (2012) for related
discussion.

A controlled idealization is an idealization for which all the necessary
resources are available in current scientific knowledge to explain why the
effect of the introduced assumptions in a model is, up to required degree,
negligible.222Our deployment of this concept builds upon the idea of a
‘controllable idealization’ due to (Sklar 2000, pp.44-5). See also Batterman
(2005, 2014); Wayne (2011); King (2016); Knuuttila and Morgan (2019). A
controlled mathematical idealization is then a simplification of the formal
structure of a mathematical model or system of equations in which all the
necessary resources are available in current scientific knowledge to explain
why the effects of the relevant formal simplifying assumption are, up to a
required degree, negligible. Typically, in controlling a mathematical
idealization a scientist is able to provide estimates of the deviation between
the theoretical predictions of the idealized model and the exact solutions of
the original mathematically intractable theoretical model. In what follows, we
will distinguish between the situation where we do not at present have the
resources to control an idealization but we have good reason to expect that
this is only a practical limitation and the situation where we have good
reason to believe that we will never be in the situation to control the
idealization in question. The first we will call uncontrolled idealizations
and the second uncontrollable idealizations.

The standard model of particle physics is a marvel of empirical science and
has been variously tested to astounding degrees of accuracy. The framework
within which the standard model is presented is perturbative Quantum Field
Theory (QFT).333There have been various attempts to cast the standard model in
terms of non-perturbative quantum field theory. However, to date, such
attempts have been met with limited success. Here we focus exlcusively on
perturbative approaches with the framework of Lagrangian Quantum Field
Theories and neglect analysis of models in Algebraic Quantum Field Theory
(AQFT) approaches. An issue of particular interest, worthy of consideration in
future work, is the implications for our analysis of work on perturbative
approaches to AQFT (Rejzner 2016). Following Wallace (2006, 2018) and
Koberinski (2021) our analysis is based on an understanding of perturbative
QFT as a general theoretical framework of principles and general mathematical
constraints from which certain well-defined Lagrangian quantum field theories
describing particular interactions are constructed, such as quantum
electrodynamics (QED) and quantum chromodynamics (QCD).444Here we deviate
slightly from the terminology of Koberinski (2021), since he prefers to refer
to the Lagrangian QFTs as dynamical models, although he admits that he uses
the term ‘dynamical model’ in the way most would use the term ‘theory’. These
Lagrangian QFTs necessarily come with an implicit cut-off limit implying that
they are effective field theories describing phenomena within a particular
range of energy scales. The connection of these theories with the empirical
data is achieved via the construction of empirically relevant models of
specific interactions within the theory’s domain (e.g., a meson-meson
scattering for QED), usually by perturbation methods. In what follows, quantum
electrodynamics is thus referred to as the ‘background theory’ from which
empirically relevant models of certain interactions are derived with the use
of approximation techniques.

The perturbative treatment of the class of realistic models in quantum field
theories requires an approximation of the scattering amplitudes in terms of a
divergent power series of the coupling parameter. The divergence occurs (i)
from the fact that the total sum of the series is not known to converge or
known not to converge and/or (ii) from the fact that each individual term of
the series diverges at very high and very low energies leading to the so-
called ultraviolet and infrared divergences respectively. As will be shown,
the necessary mathematical treatment of these challenges in the case of
important examples of QFTs, such as quantum electrodynamics, requires the use
of uncontrolled idealisations which break the formal link needed to identify
the background theory with its theoretical models. In particular, the
justification for the truncation of the series after the first few terms can
only be established if the series is strongly asymptotic, by showing that it
is Borel summable, cf. Miller (2023). Correspondingly, the justification for
the renormalizability of a quantum field theory via renormalization group
methods can only be achieved by showing that a theory is asymptotically safe
by calculating the corresponding beta-function, cf. Huggett (2002). However,
these requirements have not been demonstrated to hold in important examples of
QFTs, and there is evidence that they do not in fact hold in QED at least.
Thus, the introduced mathematical idealizations are uncontrolled, and
plausibly they should be understood as uncontrollable.

The general understanding of perturbative quantum field theory that we adopt
is close in spirit to recent discussions on the role of theories, models,
approximations and renormalization techniques in quantum field theories. We
conclude this subsection by giving a brief overview of this recent related
literature. In an analysis related to but distinct from our own, Fraser (2020)
argues that perturbative methods in quantum field theory do not produce
models, rather they only produce approximations of physical quantities like
cross-sections. In more recent work, Fraser’s view on the role of
approximations in quantum field theory has been further elaborated by de Olano
et al. (2022) by showing via certain historical examples that approximations
are instrumental in determining the empirical content of perturbative quantum
field theories and should be recognised as a distinct and equally important
category of theoretical output compared to idealized systems. de Olano et al.
(2022) further stress that the integral role of approximations in assigning
empirical and physical content to a model cannot be easily accounted for by
extant philosophical approaches to scientific modelling. Koberinski (2021)
also provides a historical study of the mathematical developments of Yang-
Mills theories to highlight the fact that quantum field theories are rarely
axiomatic systems with a neat set of deductive consequences, but are rather
modular and rely heavily on key conceptual and mathematical tools that can be
treated independently. Similarly, Miller (2023) argues that the state-space
semantics developed by Beth (1960) and Fraassen (1970) fails to capture
perturbatively evaluated observables in cases of divergent series, and
proposes a slight modification that allows the accommodation of these results.
These results complement and support our own approach. As such, we take
ourselves to be operating in the mainstream of the philosophy of quantum field
theory, despite the counter-intuitiveness of our conclusion.

###  2.2. Models and Theories

Under what might be called the ‘canonical view’ on theoretical models, to be a
model of a theory is to stand in the relation to the theory that is equivalent
to that between a logical model and a formal theory expressed as a set of
axioms or formal sentences. Such a view of theoretical models is common to all
major approaches to the structure of scientific theories. That is, whether or
not one holds the semantic view of theories under which theories are
identified with the theoretical models, it is still standard to consider the
models of a theory as an important aspect of a theory’s structure and to
define such theoretical models via the canonical view.555See, for example,
Savage et al. (1990); French and Ladyman (1999); Lutz (2012); Winther (2021).
Most vividly, the models of general relativity are unambiguously specified by
parings of stress-energy tensors and spacetime geometries that solve the
Einstein Field Equations. A highly influential articulation of the canonical
view is due to van Fraassen (1980), who indicates that theoretical models
should be understood as deductive products of the background theory in the
sense that they are derived from a set of fundamental equations: ‘[t]he sense
in which a theory offers or presents us with a family of models – the
theoretical models – is just the sense in which a set of equations presents us
with the set of its own solutions […] When the equations formulate a
scientific theory, their solutions are the models of that theory’ (ibid.
p.311).

Significantly, just like mathematical equations, a theory may have models that
we have not yet discovered or fully articulated, but nonetheless exist.
Furthermore, again just like mathematical equations, a theory may be
associated with models which have not been demonstrated to be deductive
consequences of the theory but rather to be under a set of simplifying
assumptions. Using the terminology of the last section, one can understand as
theoretical models those models whose deductive relation to the theory is
based upon a controlled mathematical idealization. We will articulate this
idea further in the following section. For the time being the important point
is that in the context of examples of QFTs, such as quantum electrodynamics,
where the theory-model association is based upon mathematical idealizations
that are uncontrolled, there is no longer a clear sense in which the models in
question are models of the theory.

We thus end up with a deeply unintuitive situation where we have theories
without models. On close inspection the situation turns out to be not just
unintuitive but also concerning since theoretical models can be expected to
play a number of important roles within scientific theories. A feature of
particular significance is that on most accounts it is the theoretical models
that allow theories to make contact with empirical evidence. This feature is
not only pivotal to van Fraassen’s constructive empiricist view of science but
can be expected to be central to any account of how scientific theories are
applied in practice. In particular, the empirical adequacy of a theory is
standardly understood to assessed by the predictive success of the theoretical
models it provides. Thus we arrive at the significant worry that the existence
of theories without models not only undermines the pristine picture of
theories and theoretical models, but also indicates that the empirical
adequacy of theories such as quantum electrodynamics cannot be cogently
expressed. We will return to this worry in Section 6.

The autonomy of models from theories as a criticism to the semantic view of
theories has of course been discussed in the past in other contexts.666For
instance, Portides (2005) criticises the semantic view by showing how the
liquid drop model of the nucleus was introduced in the 1930s to explain –
amongst other things – the Weizsäcker semi-empirical formula of the binding
energy of the nucleus. Portides convinsingly argued that certain crucial
assumptions of the model lack the necessary theoretical justifications which
would allow one to consider the liquid drop model as a theoretical model
(ibid., p.1294). In a similar spirit, Cartwright et al. (1995) and Morrison
(1999) argued for the autonomy of models from theories using the examples of
the Londons’s model of superconductivity and Prandtl’s model of ideal fluids
respectively. However, such arguments typically appeal to examples of
phenomenological models in physics. The main characteristic of
phenomenological models is that their construction is primarily guided by
empirical observation and experimental data, maintaining minimal dependence on
the background theoretical framework. As a result, such models often violate
basic theoretical principles and fundamental laws of the background physical
theory, and hence, phenomenological models exhibit a type of non-theoretical
models in virtue of their independence and autonomy from the theory. Our worry
is rather different. That is, our claim is that there is a very different type
of non-theoretical models in physics – for which the models of perturbative
quantum field theory are a prime example – in which the relationship between
theories and models breaks down not because these models are primarily guided
by phenomenological considerations; but rather, because the mathematical
treatment of these models results to the introduction of uncontrollable
idealizations due to the lack of a rigorous mathematical or physical
justification.

The class of problematic models that we identify is similar to what Redhead
(1980) calls floating models. For Redhead, these are models that are
disconnected from a background theory due to a computational gap arising from
the scientist’s inability to justify the validity of the mathematical
approximations used for their construction, but at the same time, unlike
phenomenological models, fail to provide successful empirical predictions.
They are thus, in a sense, ‘floating’ in that they are detached both from
theory and the experimental data. Their value is to be found in the fact that
they often serve as preliminary steps for probing various essential features
of the theory and exploring the possible ways of refining it in order to yield
further empirically successful models. However, unlike Redhead’s floating
models, the models of perturbative quantum field theory exhibit remarkable
agreement with experimental data despite the fact that they are not connected
to the background theory via a secure approximate derivation based upon a
controlled mathematical idealizations. This is, we contend, a deeply
challenging situation that has not been adequately addressed in the vast
literature on theories and models in science.

##  3\. Theoretical Models and Empirical Adequacy

In this section we will specialize our analysis of theoretical models to the
particular case of van Fraassen’s constructive empiricism and consider the
relationship to empirical adequacy and mathematical idealizations in that
context. As noted above, we take our analysis to generalise to other
approaches, and thus the reader should see our work here as illustrating the
general features of the canonical view on theoretical models and the
relationship to empirical adequacy.

According to van Frassen, in articulating a theory one begins with a set of
fundamental principles and laws and, without taking into consideration any
kind of empirical data, constructs a set of models which can be deductively
derived from these principles. These models are then compared to the
‘appearances’, i.e., the structures which can be described in experimental and
measurement reports. A physical theory is empirically adequate ‘if it has some
model such that all appearances are isomorphic to empirical substructures of
that model’ (van Fraassen 1980). Van Fraassen’s emphasis on the importance of
theoretical models of a theory stems from his strong views in favour of the
semantic view, according to which the structure of a theory is nothing but its
class of mathematical models. For van Fraassen (1980, 1989), the presentation
of a scientific theory consists of a description of a class of state-space
types, where state spaces are understood as a generic collection of
mathematical objects (such as vectors, functions, and numbers) denoting the
physical state of a system. Theoretical models are understood in this context
as mathematical structures that are deductively derivable from the basic
principles of the theory, denoting sequences of states that form a trajectory
in the state space over time. The connection of these models to the physical
world is described by (van Fraassen 2008, pp. 168-170) in terms of an
isomorphic relationship between the theoretical models and the surface models
of an experiment, i.e. the idealized models of the raw data obtained from
experiments.777See Lutz (2014, 2021) for interesting work on generalizing van
Fraassen’s notion of empirical adequacy.  Together with the observable
phenomena, theoretical models occupy the central stage of constructive
empiricism, since these are ‘the two poles of scientific understanding, for
the empiricist [..] The former are the target of scientific representation and
the latter its vehicle’ (van Fraassen 2008, p. 238).

A possible way to understand theoretical models, according to van Fraassen, is
to think of them as specifying:

  1. (i)

a family $N$ of observables (representing physical quantities) each with a
range of possible values

  2. (ii)

a set S of possible states (i.e. the model’s state space)

  3. (iii)

a stochastic response function $P_{s}^{n}$ for each $n$ in $N$ and $s$ in $S$,
which is a probability measure on the range of $n$.

The members of the set $N$ of observables are functions from the model’s state
space to the real line, assigning values to various physical quantities that
are typically represented by the parameters and variables of the model. The
state space $S$ comprises the set of all possible configurations of the
system’s variables, and can be thought of geometrically as an N-dimensional
space whose dimensions are defined by the number $N$ of observables in the
model. A given state $s$ is thus represented by a point in the state space
whose coordinates stand for a possible combination of the values of the
physical quantities of the system represented in the model. Finally, the
stochastic response function $P_{s}^{n}$ is the model’s specification that a
(direct or indirect) measurement of a given observable $n$ will yield a
specific value $n_{i}$, when the state is $s_{i}$.

The overall idea is that a theoretical model fits (or is ‘isomorphically
embeddable to’) a surface model only if the theoretical model has some state
$s$ such that the function $P_{s}^{n}$ contains the surface state $P$ of the
surface model, relative to the given identification of the measurement setups
as measurement of the physical magnitudes $n$ (ibid. p.169). A physical
phenomenon is thus represented both ‘from below’ via the surface model and
‘from above’ via the theoretical model. The fitting of these two models
establishes the link of the theory with the natural world and eventually its
empirical adequacy.

A characteristic example of a theoretical model in physics comes from the
classical harmonic oscillator which satisfies the theory of classical
mechanics. The dynamic behaviour of the system described in this model is
completely determined by the equations of the background theory and is in full
accordance with the theory’s fundamental principles. However, this purely
theoretical model does not allow the direct comparison of the model with the
corresponding surface models of data derived from experiments. Hence, the
additional construction of empirically relevant models – based on these
theoretical models – is often necessary, to compare the quantities predicted
by the models with the outcomes of experimental measurements. Theoretical
models thus provide the mathematical basis on which further empirically
relevant models are built with respect to a corresponding target system. The
aim is to obtain theoretical predictions for experimental observables via
these new models, in order to test the empirical adequacy of the models and
consequently of the background theory.

As van Fraassen notes (ibid. p.311), the process of arriving at a working
model from a background theory can take several forms, often involving the use
of mathematical idealizations. These idealizations typically occur in cases
where the theory provides theoretical models for which no analytic solutions
can be found and thus, further modifications are required in order to make the
models mathematically tractable. For instance, the (theoretical) model of the
ideal pendulum in classical mechanics provides the pendulum’s equation of
motion

(1) $\derivative[2]{\theta}{t}+\frac{g}{l}\sin\theta=0$

where $\theta$ is the angle between the string and the vertical, $g$ is the
local gravitational acceleration and $l$ is the length of the string. Equation
(1) specifies, in principle, the aforementioned stochastic response function
according to which a measurement of $\theta$ will yield a specific value
$\theta_{1}$ given a state $s_{1}$, and is a second order non-linear
differential equation for which no exact analytic solution can be found.
Hence, the introduction of a mathematical idealization is necessary in order
to arrive at a mathematically tractable form of the empirically relevant model
which will eventually enable the comparison to the relevant surface model.

This is achieved by introducing the assumption that the pendulum only swings
in small angles for which we know that

(2) $\sin\theta\approx\theta$

The small angle approximation yields a new linear equation of motion

(3) $\derivative[2]{\theta}{t}+\frac{g}{l}\theta=0$

which has a well known exact solution. Insofar as there are good mathematical
reasons to justify that (2) holds for small angles, the introduced
idealization is controllable and the linearized version of the ideal pendulum
is still a theoretical model, in that it is derivable from the theory modulo
some controlled modifications.

In what follows, we draw on the modelling techniques of perturbative quantum
field theory to argue that in addition to phenomenological models whose
construction aims in accommodating empirical results, there is another
important class of non-theoretical models in physics whose construction comes
from the use of non-controllable idealizations aiming to make them
mathematically tractable and empirically relevant. A careful analysis of the
modelling methodology of quantum electrodynamics within perturbative quantum
field theory indicates that the empirically relevant cut-off models produced
by regularization and renormalization techniques are detached from their
corresponding theoretical models and the background theory, due to the fact
that the required justification for the truncation of the series and the
employment of the renormalization group method has not been established in
practice. This fact breaks down the derivational relationship between the
empirically relevant models and their background theory, leading to problems
both for our understanding of the theory-model relationship and the empirical
adquacy of the theory– according to van Fraassen’s view and more widely.

##  4\. Models in Perturbative QFT

Perturbative Quantum Field Theory is the standard theoretical framework for
constructing models for scattering processes of subatomic particles in
particle physics. Realistic models of interactions are constructed by
performing scattering theory calculation using specific quantum field theory
Lagrangians or Hamiltonians, such as those for QED and QCD. These calculations
allow the construction of empirically relevant models that are capable of
providing theoretical predictions for physical quantities that can then be
compared directly to experimental observables, such as cross sections and
decay widths. A first difficulty in this process arises from the fact that the
construction of four-dimensional interacting models via scattering theory
leads to intractable Hamiltonians since the introduction of interacting terms
typically leads to non-linear equations of motion for which no exact solutions
can be found analytically.

In order to make the interacting models mathematically tractable, physicists
often appeal to perturbation theory, in which the total Hamiltonian $H$ of the
system in question is split into a free and an interacting part:

(4) $H=H_{0}+gV$

where $H_{0}$ is the free Hamiltonian whose eigenstates can be calculated
analytically and $gV$ represents the Hamiltonian of the interacting model with
an interaction potential $V$ parameterised by a coupling parameter $g$. Given
that the coupling strength of the theory is weak – as is the case in QED and
in high-energy QCD – this mathematical idealization is controllable, since it
can be justified by safely assuming that the solutions of the total
Hamiltonian are close enough to the well-known solutions given by the free
Hamiltonian $H_{0}$.

One of the most important quantities to be obtained by applying perturbation
theory is the S-matrix, the operator that maps the initial state of a physical
system undergoing a scattering process to the final state. The importance of
the S-matrix for particle phenomenology experiments boils down to the close
relationship of its elements – also referred to as scattering amplitudes
$\mathcal{M}$ – to the scattering cross sections that are (indirectly)
measured in these experiments. The S-matrix thus contains some of the
theoretical predictions of the empirically relevant model of the theory upon
which the empirical adequacy of the model can be tested.

However, when perturbative methods are applied to realistic quantum field
theories like QED, the final result is an expression of the scattering
amplitudes as a power series of the (bare) coupling parameter $g_{0}$ which
has the following generic form

(5) $\mathcal{M}=\sum_{n}^{\infty}{g_{0}}^{n}\int_{-\infty}^{\infty}dkA_{n}$

where $A_{n}$ stands for a multiple integral over momentum $k$ at each
$n^{th}$ order term of the series. The main idea is that for those
interactions where the coupling constant is relatively small, the first factor
${g_{0}}^{n}$ vanishes as the series proceeds to higher orders, and thus the
contribution of higher order terms decreases significantly. All that is needed
to calculate the coefficients of the series for each order (and consequently
the S-matrix elements) is, therefore, to evaluate a set of multiple integrals
over momentum space.

This is where two significant problems arise however. The first problem
concerns the fact that the total sum of the series is sometimes either not
known to converge or known not to converge, and thus a legitimate question
arises as to whether the series is indeed a good approximation to the quantity
in question. Moreover, the number and the complexity of each of these
integrals increases rapidly for higher orders and thus, in practice, only the
sum of the first few terms is evaluated and compared to experimental results,
often with great success. The second complication stems from the fact that
each individual term of the series often appears to diverge either at very
high energies (as $k\rightarrow\infty$) or at very low energies (as
$k\rightarrow 0$) leading to the so-called ultraviolet and infrared
divergences respectively. This is the notorious ‘problem of infinities’ in
quantum field theory and, as we shall see, the solution to this anomaly comes
from various regularization and renormalization techniques.

The crucial question is whether the mathematical methods employed at this
stage can be rigorously justified, making the introduced idealizations
controllable, as in the simple case of the ideal pendulum. If so, then one is
warranted to maintain that the final products, i.e. the empirically relevant
models that are eventually put to test by comparison with experimental
results, are theoretical models by virtue of being derivable from the
background theory via controllable idealizations. Here is where we encounter a
problem however, since the mathematical treatment of these two complications
indicates that the introduced idealizations are uncontrolled and possibly
uncontrollable due to the lack of a rigorous mathematical proof for (i) the
Borel summability of the series to establish strong asymptoticity and (ii) the
calculation of the corresponding beta-functions in renormalization group
methods to establish asymptotic safety. As a result, the strong relationship
between the final models and the background theory required by the semantic
view breaks down. Let us consider each of these problems in more detail.

##  5\. Breaking the Theory-Model Link

###  5.1. Divergent Series

The first complication concerns the large order behaviour of the perturbative
series as a whole, independently of the offending integrals in the individual
terms. In addition to the fact that the integrals in each term of the series
diverge, there is often no evidence that the total sum of the series
converges, or, to make things worse, there is strong evidence that the sum of
the series at all orders diverges. Given the infinite number of terms and the
increasing intractability of the integrals as one proceeds in higher-order
terms, what happens in practice is that only the first few terms of the series
are usually calculated and compared to experimental results, often by using
different techniques for calculating each term of the series. Nevertheless, in
several cases the sum of the first few series turns out to be in very close
agreement with the experimental results, which leads us to the question of how
the empirical success of the truncated series should be explained.888The
theoretical prediction for the anomalous magnetic moment of the electron is
probably the most famous example of this practice. The first term of the
series was first calculated by Schwinger (1948) and found to be in close
agreement with the then available experimental results. The current state of
the art only allows the calculation of the first five terms which requires –
amongst other things – an evaluation of 12672 Feynman diagrams in the tenth-
order perturbation theory (Kinoshita 2014).

Miller (2023) discusses at length the case of divergent series in perturbative
quantum field theories and notes that a possible justification for the
truncation of the series after the first few terms and the explanation of the
empirical success of the approximating series can be given in terms of strong
asymptoticity.999As Miller notes, this explanation was also given by Dyson
(1952) himself in his arguments for the divergence of the perturbative series.
Unlike convergent series, when a divergent asymptotic series is used for
approximating a function, the sum of the first few terms is very close to the
exact value of the function, and as one includes more and more terms to the
sum, the value of the series increasingly diverges until it becomes infinitely
different. Hence, if the perturbative series for the matrix elements of a
meson-meson scattering process is indeed asymptotic to the unknown exact
solution, it should be no surprise that the sum of the first few terms is
often found to be in agreement with experimental results.

The problem, however, is that while a given function can only have one
asymptotic expansion, the converse is not true. That is, an asymptotic
expansion can correspond to multiple functions, and thus it is possible that
two or more different functions have the same asymptotic expansion. In fact,
it is possible for an infinite number of functions with an entirely different
set of solutions to share the same asymptotic function. Hence, even if a
perturbative expansion in QED is indeed asymptotic to an exact solution, it
does not uniquely specify what that function is. The only possible way to
uniquely determine a function via an asymptotic series is to show that the
series satisfies a strong asymptotic condition, which, roughly speaking,
requires the differences between the exact value of the function and its
series representation to be appropriately small for every order of
perturbation theory.

The condition of strong asymptoticity that shows that a perturbative expansion
uniquely satisfies an exact solution is only established if and only if a
series is Borel summable. Borel summation is a mathematical method that is
particularly useful for summing divergent asymptotic series. Suppose a formal
power series $\sum_{k=0}^{\infty}\alpha_{k}\lambda^{k}$ of a function
$f(\lambda)$ and define the Borel transform $B$ to be its equivalent
exponential series $B(\lambda t)\equiv\sum_{k=0}^{\infty}\frac{dk}{k!}t^{k}$.
If the Borel transform can be used to produce a unique reconstruction of
$f(\lambda)$ then the series is Borel summable.101010For a more rigorous
discussion on Borel summation in the context of perturbative QFT see (Duncan
2012, pp. 400-3).

The important point for our purposes is as follows: If one knows the
asymptotic expansion of a function, the function can be uniquely reconstructed
by Borell summation, thus showing that the series corresponds to a unique
function determined by the background theory. Unfortunately, however, Borel
summability has not only not been demonstrated but is in fact not expected to
hold in phenomenologically interesting models of quantum field theory. Indeed,
Duncan (2012) notes that ‘the property of Borel summability is an extremely
fragile one, and one which we can hardly ever expect to be present in
interesting relativistic field theories’ (p.403, emphasis added). For
perturbative QED in particular, Borel summability is not expected to hold in
general.111111Here we are following the discussion of Miller (2023). For
further discussion in the physics literature see ’t Hooft (1979) Khuri (1981);
Fischer (1997). Thus the explanation for the success of the series after the
truncation of the first terms based on strong asymptoticity is only a
conjecture. The relevant mathematical idealization is therefore uncontrolled
and the derivational relationship between the empirically relevant models in
perturbative QFT and the background theory is, as it stands, obstructed. We
will return to this issue and consider the idea of exogenous control of the
idealization via non-perturbative ‘instanton’ effects in Section 6.

###  5.2. Cutting Off Our Ignorance

The second problem that needs to be addressed in the construction of
empirically relevant models in perturbative QFT is the presence of ultraviolet
and infrared divergences in the individual terms of the power series in (5)
which makes the calculation of the coefficients impossible. The elimination of
these infinities is achieved by various methods of regularization in which a
divergent integral $A_{n}$ is redefined as a function of a new parameter $\xi$
– the regulator – which must satisfy the two following pragmatic constraints:
(i) finite values of the regulator must render the integral $A_{n}(\xi)$
finite and (ii) if the regulator is removed by taking its limit to infinity
the retrieved result is the original divergent integral $A_{n}$.

A standard regularization method is the Pauli-Villars regularization in which
a cut-off limit $\Lambda$ is introduced for the ultraviolet and infrared
domains of high and low energies respectively, beyond which the value of the
integral is taken to be zero.121212Another popular method of regularization is
dimensional regularization where, roughly speaking, the calculations are
initially carried out in a d-dimensional spacetime, and the cut-off
dependencies appear as we take the well-defined limit $d\rightarrow 4$. The
Pauli-Villars method discussed here has the advantage of being physically more
transparent as opposed to dimensional regularization which is considerably
more abstract in nature. Some less popular methods of regularization are the
lattice regularization, the zeta function regularization and the causal
regularization. In the case of ultraviolet divergences, this is equivalent to
the introduction of a mathematical idealization according to which

(6) $\sum_{n}\int_{\Lambda}^{\infty}A_{n}\equiv 0$

The introduction of the cut-off limit $\Lambda$ should be understood here as a
physical construct introduced by fiat, representing our ignorance about the
validity of our methods beyond this limit. It is not a mathematically or
physically justified constraint similar to $(\ref{sin})$ for instance. Rather,
the only available justification for the introduction of the cut-off limit is
instrumental, in that it aims to make the calculations of the series’
coefficients possible. In other words, we simply ignore contributions to the
integrands for values of momentum greater than $\Lambda$ in order to remove
the infinities and make the integrals physically relevant. A similar technique
is used for the removal of the infrared divergences in the low momentum domain
that typically occur in theories with massless particles, such as the photons
of QED. In these cases, an infrared cut-off limit is introduced for a small
but non-zero value of momentum below which the contribution of the integrands
to the sum is neglected.131313For a philosophical discussion on the treatment
of infrared divergences see Miller (2021). These methods eventually remove the
undesirable infinities from the first few terms and thus the series provides
an estimate of the quantity in question – e.g. the S-Matrix element – up to
the order for which the infinities have been removed.141414For textbook style
expositions of regularisation and renormalization methods in quantum field
theory see Peskin and Schroeder (1995), Zee (2010) and Duncan (2012). For
expositions of these methods aiming particularly at philosophers see
Butterfield and Bouatta (2015), Wallace (2018) and Williams (2018). For a
philosophically rich historical discussion see Fraser (2021).

An immediate implication of regularization is that the approximate value of
the scattering amplitudes $\mathcal{M}$ now depends on the arbitrarily chosen
value of $\Lambda$. However, given that there are no particular reasons for
choosing a specific value for $\Lambda$ over a different one, say
$\Lambda^{2}$, this should be worrying; different values of the arbitrarily
chosen regulator will simply give different results for the scattering
amplitudes, which is of course unacceptable. The scattering amplitudes
provided by these models express the empirical content of our theory and thus
they should not be a function of the theorists’ arbitrarily chosen value for
$\Lambda$.151515It should be noted here that the choice of $\Lambda$ is not
always completely arbitrary. For instance, when we want to abstract away from
the interactions of a heavier, higher-energy particle, $\Lambda$ is chosen to
be well below the mass of that particle and above the mass of the particles we
are interested in studying. However, given that the range between the two
masses is significantly large, the choice of the exact value for $\Lambda$ is
still, more or less arbitrary. Butterfield and Bouatta (2015) also point out
that in some cases the background theory hints towards a range of values for
the cut-off limit that are plausible to take (p.448). An example of such a
suggestion for the cut-off limit comes from the Lamb shift in QED, which
suggests the electron’s Compton wavelength as a natural lower limit for
distance $d$. Nonetheless, even in these cases no rigorous mathematical or
physical justification can be given and thus the final choice of $\Lambda$
remains arbitrary.

The solution to this problem came from renormalization, cf. Fraser (2021). In
this method the unwanted $\Lambda$-dependence of the integrals, and
consequently of the scattering amplitudes, is eliminated by extracting the
value of the coupling parameter $g$ from scattering experiments at a
particular energy scale $\mu$. Once the value of the coupling parameter is
obtained, the original ‘bare coupling’ $g_{0}$ is replaced by the new
renormalized value $g_{R}(\mu)$ (sometimes called the ‘physical coupling
parameter’) which is now a function of the energy scale $\mu$ at which the
experiment was performed.

As a concrete example, consider the perturbation series for the scattering
amplitude of a meson-meson scattering process up to the second order
correction. Once the ultraviolet cut-off is introduced, the amplitude
$\mathcal{M}$ becomes a finite ‘cut-off dependent’ quantity of the form

(7)
$\mathcal{M}=-ig_{0}+iC{g_{0}}^{2}[\log(\frac{\Lambda^{2}}{s})+\log(\frac{\Lambda^{2}}{t})+\log(\frac{\Lambda^{2}}{u})]+\order{{g_{0}}^{3}}$

where $C$ is a numerical constant, and the kinetic variables $s,t$ and $u$ are
functions of the square of the energy at which the particles are scattered,
and are related to rather mundane quantities such as the centre of mass energy
and the scattering angle. Once the value of the coupling parameter is
experimentally obtained, the bare coupling parameter $g_{0}$ in (7) is
replaced by the renormalized coupling parameter $g_{R}$. By carrying out some
simple algebraic calculations one is left with a new ‘cut-off independent’
expression

(8)
$\mathcal{M}=-ig_{R}+iCg_{R}^{2}[\log(\frac{s_{0}}{s})+\log(\frac{t_{0}}{t})+\log(\frac{u_{0}}{u})]+\order{g^{3}}$

where the values of $s_{0},t_{0}$ and $u_{0}$ are defined by the particular
energy at which the coupling parameter $g_{R}$ was measured. At this stage,
the dependence on the cut-off limit $\Lambda$ is dropped out, however, as one
may notice the scattering amplitude now depends on the ratio between the
energy scale $\mu$ at which the value of $g_{R}(\mu)$ is measured (captured by
$s_{0},t_{0}$ and $u_{0}$) and the energy scale of the future scattering
experiments by which the empirically relevant model will be tested (captured
by $s,t$ and $u$).

What is so special about the chosen energy scale $\mu$ however? And what if
the renormalized parameter $g_{R}(\mu)$ was measured at a different scale
$\mu^{\prime}$? One might think here that this is not much of an improvement
since the dependence on the arbitrarily chosen value of $\Lambda$ has simply
been shifted to a dependence on the energy scales $\mu$ at which we are able
to conduct experiments and extract the value of the renormalized coupling
parameter. The standard way to deal with this situation in the context of
condensed matter physics is to think of Quantum Field Theory as a theoretical
framework for providing Effective Field Theories that only describe physical
phenomena at a particular range of energy scales (up to $\mu$) and not as a
fundamental theory of physics. This idea leads us naturally to the framework
of renormalization group (RG) theory (Wilson 1971; Fisher 1998).

The renormalization group method provides the mathematical apparatus for a
systematic investigation of the changes in the couplings of a theory with
respect to the changes in the energy scales. Leaving the technical details
aside, the main idea of this modern approach to renormalization is captured by
the renormalization group flow equation

(9) $\mu\derivative{\mu}g_{i}(\mu)=\beta_{i}(g_{i})$

which determines the so-called beta function $\beta_{i}(g_{i})$ as the
differential change of the running coupling parameters $g_{i}(\mu)$ of a QFT
with respect to a small change in energy scale. If the theory happens to have
several coupling constants $g_{i},i=1,...,N$, then the beta function is a
function of all of the couplings in the theory $\beta_{i}(g_{1},...g_{N})$.
One can think of the renormalization group equation as defining an
N-dimensional space of theories in which $(g_{1},...,g_{N})$ are the
coordinates of a particle that flows in the space as the energy scale $\mu$
increases. In the high-energy regime where we have no experimental access to
measure the values for the running coupling parameters $g_{i}(\mu)$, there are
four possible behaviours of the theory’s couplings depending on the beta
function:

  1. (I)

Safety. At a finite flow parameter value the renormalization group flow hits a
point (also known as an attractor) at which the beta function becomes zero for
all running couplings, $g_{i}(\mu)$. This is the so-called fixed point $g*$
which basically implies that the theory becomes scale-invariant above a
certain energy scale since all couplings converge to a fixed constant value.
If a theory demonstrates this behaviour, it is said to be asymptotically safe.

  2. (II)

Freedom. The second possibility is a special case of asymptotic safety in
which the beta functions are negative for all the couplings in the theory and
thus the values of the running couplings decrease as the energies become
higher and higher until they eventually hit a fixed point which is equal to
zero ($g*=0$). Accordingly, for lower energies (and large distances) the
coupling strengths increase rapidly until they become divergent at some low,
but finite, scale. This behaviour is thought to be responsible for the
phenomenon of quark confinement in QCD, and theories of this kind are said to
be asymptotically free.

  3. (III)

Triviality The third possibility is the case in which the beta function is
positive for at least one of the couplings in the theory and there is no non-
trivial UV fixed point. This means that the running couplings $g_{i}(\mu)$
increase indefinitely as the energy scale increases. Theories without a non-
trivial UV fixed point are considered trivial since their continuum limit is
only well-defined if the values of all renormalized couplings, $g_{R}$, are
zero. The physical interpretation of triviality is that the quantum
corrections completely suppress the interactions in the absence of a finite
cutoff: the physical charges are entirely ‘screened off’ if the ultraviolet
cutoff is sent to infinity (Gies and Jaeckel 2004; Crowther and De Haro 2022).
A particular problem associated with lack of a non-trivial continuum limit is
where the renormalization group flow not only does not hit a fixed point but
rather the beta function is found to blow up to infinity at a very large but
finite energy level ($\mu_{Landau}$). Such a theory is said to encounter a
Landau pole (Landau et al. 1954).

  4. (IV)

Non-Integrability. A fourth and less explored possibility, is that the RG flow
exhibits a cyclic or even chaotic behaviour at higher scales. This possibility
was also noted by Wilson himself and although the existence of such cyclic or
chaotic RG flows has not been yet confirmed, several models have been
developed as candidates for limit cycle behaviour (Curtright et al. 2012;
Morozov and Niemi 2003; Leclair et al. 2003; Wilson 1971).

Huggett (2002) presents a detailed discussion explaining how the
renormalization group provides a map that helps us understand the infinities
and the ‘maneuvering’ during the renormalization process, making the derived
models consequences of the background theory. The basic idea is that one uses
the renormalization group and a renormalization scale to generate a family of
renormalized theories from a family of bare theories and then confirm that the
family of bare theories describes well-defined physics in the continuum limit.
The physics of every bare theory is captured by the physics of the
corresponding renormalized theory, which is rendered finite by a fixed cut-off
limit, and hence the physics of the limit of the bare theories is captured by
the limit of the renormalized theories which is finite. The crucial step in
the argument is to justify that we have well-defined continuum physics in the
limit, and this is achieved, as (Huggett 2002, p.274) explains, by the
topology of the parameter space. However, this explanation only holds for
quantum field theories that are asymptotically safe such as QCD, where the
renormalization group flow can be shown to hit a fix point. As Huggett puts it
at the very end of his article: ‘…assuming that a theory is asymptotically
safe, the problem is not terminal but can be fixed using renormalization to
tune it to a suitable point. If, however, a theory is not asymptotically safe,
then […] renormalization as described here will not be possible’ (ibid.,
p.275-6).161616Here it is worth mentioning the work of Franklin (2020) on the
effectiveness of Effective Field Theories. Commenting on the paper of
Butterfield (2014), Franklin notes that the appeal to the fixed point
structure of the RG method in order to justify renormalization is ill suited.
His claim is that RG is best seen as a mathematical framework that codifies
rather than explains the renormalizability of theories in QFT, in that it
allows us to mathematically establish whether or not a theory is
renormalizable, but nonetheless, it does not provide a sufficient explanation
of the physical reasons behind renormalization. Our focus here is on
mathematical relationships rather than explanation and in Franklin’s terms,
our point could be re-expressed as a statement that without a non-trivial UV
fixed point the RG method cannot not successfully achieve its codifying role.

Unfortunately, this rather attractive theoretical diagnostic picture is very
difficult to fully implement in practice. For realistic QFTs there is no fully
reliable way to explicitly calculate the beta functions to all orders. Take
the example of QED. To render the introduced idealizations controllable, one
needs to show that the theory is asymptotically safe by calculating the
corresponding beta functions in realistic QED models, however, no rigorous
proof for the asymptotic safety of QED has been achieved. Rather, what we have
available are further mathematical idealization techniques for approximating
the beta functions in some cases, which further suggest a possible behaviour
of the theory in higher energies. Interestingly, the situation is then that we
are attempting to justify the assumptions introduced by regularization and
renormalization, by introducing further assumptions, creating a form of
regress in which idealizations are justified by further
idealizations.171717This is not to say that this is a vicious regress,
however. Whether the justification of an approximation by means of a further
approximation is epistemically warranted is an interesting open question and
deserves to be studied on its own merit in future work.

In any case, even ignoring the regress worry, the evidence gained from
approximate methods for calculating the beta functions is not in fact
promising for the case of QED. In particular, calculations at the one-loop
level indicate the existence of a Landau pole (Hollowood 2013, p.37-41) and
thus provide evidence against the asymptotic safety of QED . One might hope at
this point that the Landau pole is an artifact of the perturbation calculation
of the beta function and there is some evidence in this regard (Goeckeler et
al. 1998). However, no indication of a non-trivial UV fixed point for QED has
been found so far. Furthermore, evidence from non-perturbative lattice
simulations (Goeckeler et al. 1998; Kim et al. 2001) and the exact
renormalization group approach (Gies and Jaeckel 2004) supports the triviality
of the theory with complete charge screening if the UV cutoff is sent to
infinity.

The standard response to the presumed lack of asymptotic safety in quantum
field theories found in the physics literature is that such theories should
not be understood as fundamental theories and a consistent UV completion can
be obtained by appeal to more fundamental theories. However, it is not
expected that electroweak theory can provide a UV completion of QED and,
indeed, the triviality problem is expected to also occur in the Yukawa-Higgs
sector.181818See for example Hambye and Riesselmann (1997) A full list of
references is given in Eichhorn and Held (2018) reference [4]. Rather, the
hope for a UV completion of QED relies upon appeal to physics beyond the
standard model, such as in asymptotically safe quantum gravity scenarios
(Christiansen and Eichhorn 2017; Eichhorn 2019). Whether or not such a
response to the QED triviality problem ultimately proves successful is,
however, tangential to the issue at hand: we are concerned with the
justification for the idealizations involved in the theories and models
considered in their own right, not exogenous justifications of the
idealizations coming from other theories, cf. Crowther and De Haro (2022).
Moreover, since this putative justification would need to come from theories
we do not in fact have, the supposed resolution has the status of a promissory
note. For the case of QED considered as a theory in its own right, the
mathematical idealization in question is evidently uncontrolled (since no
fixed point has been identified), and potentially also uncontrollable (since
there is evidence that none in fact exists and the theory is trivial).

Let us return to our principal question: whether the empirically relevant
models that are eventually put to test by comparison with experimental data,
can still be thought of as theoretical models of the background theory, based
on the initial non-perturbative models of QFT. Given that one requires a
secure deductive or approximate derivation relation between the background
theory and its models, one needs to show that each of the aforementioned
introduced assumptions is a controlled idealization. For theories that have
been shown to be asymptotically safe to some standard of rigour, there is a
strong case that the relevant idealizations are controllable. However, in
cases such as QED, where asymptotic safety has not been established or indeed
there are indications of triviality, the idealizations involved are
uncontrolled. As such, the empirically relevant models of these theories are
not theoretical models in the canonical sense.

##  6\. Implications

The analysis provided thus far has important implications for the
philosophical understanding of theoretical models and the empirical adequacy
of scientific theories. Let us recap the argument provided in Section 1. The
first premise, P1, was that quantum field theory is a more general framework
from which specific theories such as quantum electrodynamics and quantum
chromodynamics are derived via particular action principles. The second
premise, P2, was that a theory has theoretical models if its predicted
empirical quantities are derivable either as deductive consequences of the
theory or as a result of an approximate derivation involving controllable
mathematical idealizations. The third premise, P3, was that the derivation of
predictions from empirically successful theories in contemporary particle
physics, such as quantum electrodynamics, necessarily involves appeal to
uncontrollable idealizations. These three premises together lead us to the
deeply unintuitive conclusion, C1, that theories in contemporary particle
physics, such as quantum electrodynamics, do not have theoretical models.

The natural response for those seeking to reject the unintuitive conclusion is
to challenge the controversial premise P3. However, as we saw in the previous
section, P3 is, as it stands, remarkably resilient, even in the face of the
sophisticated framework of renormalization group theory. Thus, it might seem
that the only choice is to ‘bite the bullet’ and accept the conclusion of
theories without models, despite its unintuitiveness. However, as we will
argue now, this option is undesirable since C1 has further clearly
unacceptable consequences for empirical adequacy. Consider the following
extension of our argument:

  1. C1.

Contemporary particle physics includes theories, such as quantum
electrodynamics, without theoretical models.

  2. P4.

A theory is empirically adequate only if it possesses theoretical models that
agree with experimental outcomes.191919cf. (van Fraassen 1991, p.12): ‘To be
empirically adequate is to have some model which can accommodate all the
phenomena’.

  3. C2.

The theory of quantum electrodynamics is not empirically adequate.

Clearly, C2 is not just unintuitive but absurd and thus unacceptable. Quantum
electrodynamics is variously understood as the most empirically successful
theory humanity has yet constructed, so any argument that ends up with the
conclusion that it is not empirical adequate must have gone wrong somewhere!
Given that our robust defence of P3 is accepted and that P1 is also accepted
as uncontroversial, the two remaining options to avoid absurdity are then to
put pressure on P2 and P4. Let us consider each of these options in turn.

If theoretical models are not required to be connected to their background
theories via controlled idealizations, then one clearly avoids the absurd
conclusion since one is able to understand the empirically relevant models of
quantum electrodynamics as models of the theory. The cost to the coherence of
our concepts is, however, catastrophic. This is because the expansion of the
definition of a theoretical model to include links via uncontrollable
mathematical idealizations, simpliciter, allows arbitrary mathematical
connections between scientific theories and models. To put it differently,
saying that the models of a theory include models which are the products of
both controllable and uncontrollable idealizations allows any possible
mathematical connection of any possible model with a background theory. Such a
view would lead to a narrowly instrumentalist understanding of the empirical
adequacy of theories which is solely defined in terms of the empirical success
of the derived models, regardless of whether such models are constructed based
upon rigorous and well justified scientific and mathematical processes or not.
Consequently, under such view, any scientific theory in which various ad hoc
mathematical modifications are introduced with the sole purpose of achieving
agreement with data can be considered as empirically adequate insofar as the
derived predictions agree with measurement outcomes. We do not, therefore,
take the route via weakening P2 to be worth pursuing.

The final option available to us is then to reconsider P4 and provide a more
generalised definition of empirical adequacy that avoids the absurd conclusion
C2, even if it does not allow us to avoid the unintuitive conclusion C1. The
problem however, is that at least within the context of constructive
empiricism it is not at all clear how one might reconstruct empirical
adequacy. Recall that within the framework of constructive empiricism a theory
is empirically adequate if the theoretical models of the theory provide
predictions that agree with the contents of measurement outcomes. As van
Fraassen (1980) puts it: ‘[t]o present a theory is to specify a family of
structures, its models; and secondly, to specify certain parts of those models
(the empirical substructures) as candidates for the direct representation of
observable phenomena’ (p.64). There appears to us no natural way to generalise
empirically adequacy to avoid P4 within the constructive empiricist view of
theoretical structure. Of course, whilst constructive empiricism is probably
the philosophical view with the most clear and well developed definition of
empirical adequacy, there is nothing forbidding us to search for a wider
definition.

One option would be to try and find means by which to connect the empirically
relevant models to the theory via reference to further theories or frameworks.
That is, to modify the definition of empirical adequacy to allow the models on
which the adequacy is based to be linked to the theory via exogenous
arguments, which may include idealizations controlled via other theories or
frameworks. This approach has the advantage of mirroring the attitude of many
practicing scientists and has the interesting broader implication that perhaps
philosophers should look beyond the theory as a unit of analysis in the
context of evaluations of empirical adequacy.

In the context of the failure of Borel summability, the exogenous control
approach is illustrated by the idea of appealing to non-perturbative
‘instanton’ corrections as a means to control the idealisation within the
perturbative theory (Marino 2014). The key idea is to use properties of the
non-perturbative solutions to understand the failure of Borel summability. In
particular, we can differentiate two types of situation when there is no Borel
summability: i) we are considering perturbative series around unstable minima;
and ii) there are extra saddle points in the path integral. The second
situation is understood to correspond to non-perturbative effects of the
instanton type. In that context, one can consider ‘lateral Borel resummations’
(i.e. integrals along paths in the complex plane) and the ground state energy
can be reconstructed from the Borel-resummed perturbative series corrected by
Borel-resummed instanton solutions (Marino 2014, p. 37). The idea is, then,
that realistic perturbative quantum field theories which are not Borel
summable, admit a non-perturbative analysis with instanton corrections (or
generalisations thereof). Formally, the relevant properties have only been
shown to hold in toy models and thus the idealization in question is certainly
not controlled as yet. However, there may be scope for experimental
demonstration of the existence of instantons in the standard model (Amoroso et
al. 2021) and this would go a long way towards provision of the kind of
exogenous control that we take the idealizations of perturbative quantum field
theory to require in order to anchor their empirical adequacy.

There are also plausible possibilities for exogenous arguments that could
provide control of the idealisations in the context of renormalization. These
also come from a non-perturbative analysis but are exogenous to an even
greater degree since they require us to include additional effects due to
gravity. In particular, in the context of failure of asymptotic safety in
quantum electrodynamics, scientists in the field of quantum gravity propose
control of the relevant idealization by appeal to the existence of a fixed
point in a unified theory of quantum gravity. This wider definition would be
aligned to the general attitude that UV divergences of QFT point to quantum
gravity as discussed by Crowther and De Haro (2022) and to the particular
motivations for the asymptotic safety programme (Eichhorn 2019).202020See also
Crowther and Linnemann (2019).

A related idea would be to try and offer a more relaxed definition of
empirical adequacy based on a distinction between appropriate and
inappropriate uncontrolled idealizations. Such an approach would require a
robust framework for distinguishing between uncontrollable idealizations that
are appropriate for the derivation of empirical predictions, such as the ones
appearing in quantum electrodynamics, and other arbitrary and ad hoc
idealizations to make models mathematically tractable. Although intuitively
some idealizations seem plainly wrong, e.g. the introduction of an arbitrary
number of free parameters to fit the data, the presented analysis of the
status of idealizations in quantum electrodynamics indicates that there is a
grey zone in which it is not clear whether a certain (uncontrolled)
mathematical idealization is appropriate or not.

Whichever option is pursued, we take ourselves to have demonstrated that the
problems for the articulation of theoretical models and empirical adequacy in
the context of perturbative quantum field theory are severe ones and warrant
serious attention from philosophers and, perhaps, also scientists themselves.

## Acknowledgements

We are greatly appreciative to Richard Dawid, Sean Gryb and Tim Koslowski for
extremely helpful comments on a draft manuscript. Earlier versions of this
work have also greatly benefited from valuable feedback from James Fraser,
Michael Miller and Alex Franklin to which we express our gratitude.

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