# How Haag-tied is QFT, really?

David<EMAIL_ADDRESS>
Marian<EMAIL_ADDRESS>
Chris<EMAIL_ADDRESS>§§§This work was produced through equal
contribution from each author.

###### Abstract

Haag’s theorem cries out for explanation and critical assessment: it sounds
the alarm that something is (perhaps) not right in one of the standard way of
constructing interacting fields to be used in generating predictions for
scattering experiments. Viewpoints as to the precise nature of the problem,
the appropriate solution, and subsequently-called-for developments in areas of
physics, mathematics, and philosophy differ widely. In this paper, we develop
and deploy a conceptual framework for critically assessing these disparate
responses to Haag’s theorem. Doing so reveals the driving force of more
general questions as to the nature and purpose of foundational work in
physics.

##  1 Introduction

> The edifice of science is not raised like a dwelling, in which the
> foundations are first firmly laid and only then one proceeds to construct
> and to enlarge the rooms. Science prefers to secure as soon as possible
> comfortable spaces to wander around and only subsequently, when signs appear
> here and there that the loose foundations are not able to sustain the
> expansion of the rooms, it sets about supporting and fortifying them. This
> is not a weakness, but rather the right and healthy path of development.

[Hilbert 1905, 102; translation by Corry 2004, 127]

Proven over six decades ago, Haag’s theorem appears to present a problem for
particle physics. The theorem seems to block a key technique—namely, the
interaction picture and its attendant calculational methods—that has been
widely used to generate successful predictions. It is clear that the theorem
points to some sort of problem, driven by the empirical success of the
calculations employing the interaction picture on the one hand and the logical
force of the theorem on the other. Thus, while particle physics has secured
for itself a “comfortable space” around which to wander, Haag’s theorem
appears as a sign that the foundation are too loose “to sustain the expansion
of the rooms.”

This paper aims to provide a framework for possible answers to a single, if
double-faced, question: What does Haag’s theorem tell us about quantum field
theory, present and future? Several divergent answers have been given already.
Indeed, these will shape the paper’s framework substantially. Nevertheless,
the shape the framework should take is less straightforward than it may at
first seem. Even before getting to the nitty-gritty analysis of Haag’s
theorem, a framework must grapple with the problem of viewpoint, as the
following exercise makes clear:

Regardless of your actual field, take whatever career stage you are at—early
graduate student, doctoral candidate, early-, middle- or late-career
researcher—and imagine yourself instead as a particle physicist. You may
imagine yourself as an experimentalist or a theoretician, expert in QED or
QCD—whatever comes to mind. Regardless, several things are true of you. First,
you are committed to the development of particle physics (no matter what this
means in practice). Second, you are steeped in, and reliant on, the
interaction picture for your research in and teaching. And third, you have
just learned of Haag’s theorem and the trouble that it spells for the
interaction picture. How might you respond?

Set the exercise up again, except now you are a mathematician committed to
contributing to the development of QFT (no matter what this means in
practice). Second, you are steeped in the implications of Haag’s theorem, and
you are intimately familiar with the axiomatic or algebraic approach to
quantum field theory. And third, you believe that a full, conceptually
coherent physically realistic replacement for the interaction picture is
(presently) unavailable.

Set it up one last time, except now you are a philosopher of science. You may
consider yourself a realist or an instrumentalist, interested in metaphysics
or methodology—whatever comes to mind. Regardless, several things are true of
you. First, you are committed to understanding the foundations of QFT (no
matter what this means in practice). Second, you are familiar both with the
major advances in conventional (Lagrangian) QFT that use the interaction
picture as well as those based on algebraic QFT (AQFT).555See [Fraser, 2011]
and [Wallace, 2011] for the classic debate over these two formulations of QFT.
But third, it is unclear to you if, or how, these advances can form a
consistent whole.

It is not a given that your physicist, mathematician, and philosopher selves
will share a single outlook on RQFT, even before considering Haag’s theorem.
Nor is it clear that they should. Inevitably, this will affect your reaction
to our guiding question. Thus, a major contribution of our framework will be
to highlight the influence these extra-Haagian outlooks have on understanding
Haag’s theorem and on assessing its implications for philosophy and for
theoretical physics. As we show in section 4, many (but not all) of the
disagreements about Haag’s theorem derive ultimately from different extra-
Haagian outlooks, such that the disagreement is far less about Haag’s theorem
itself than it is about how to do (foundations of) physics.

This work also aims to contribute to a larger discussion in philosophy of
physics. As a quintessential and live example of work at the foundations of
physics, the discussions of Haag’s theorem draw our attention to important
methodological questions: What role does (should) foundational work play in
progress in physics? How is foundational work coordinated with non-
foundational work, or how should it be? And, what does (should) foundational
work even look like? These are undoubtedly heady questions, and we feign no
complete answers. Nevertheless, our framework will reveal some of the answers
that are being given by the authors under survey, and in so doing these
answers open themselves to investigation. As we conclude the paper, we suggest
several investigations we expect will sober up future discussions of these
heady methodological questions.

The remainder of the paper is structured as follows. In section 2 we provide a
synopsis of a standard proof of Haag’s theorem and discuss the sense in which
it raises an alarm that something is not right with the interaction picture.
In section 3 we motivate the need for a framework for understanding the
literature on Haag’s theorem. The framework itself is given in section 4. The
framework employs and extends Hilbert’s construction analogy: the framework
understands each author as something like a contractor giving their assessment
of the problem in the foundations of physics heralded by Haag’s theorem, their
recommendation for repair work on the foundations, and their expectations of
the needed long-term maintenance or future renovations to QFT’s area of the
edifice of science. This section applies the framework to seven leading
contemporary viewpoints on Haag’s theorem; the key results of this application
are given in table 1. Section 5 demonstrates the framework’s judicious balance
of conceptual structure and flexibility in order to bring clarity to the space
of responses to Haag’s theorem; it further argues that, at the end of the day,
the most important lesson for philosophers to take from Haag’s theorem is that
we need to put our own energies into clearly answering meta-level questions
regarding the nature and purpose of foundational work in physics. Concluding
remarks are given in section 6.

##  2 Haag’s Theorem

###  2.1 History of Haag’s Theorem

Haag’s theorem is the culmination of two approaches in early quantum field
theory. On the one hand, there had arisen a practical approach to calculating
the results of scattering experiments in relativistic quantum field theory.
This approach relied on the so-called interaction picture to model
interactions [Schwinger, 1948b]. The approach was wildly successful, in
particular in its use to calculate the anomalous magnetic moment of the
electron [Schwinger, 1948a]. On the other hand, spurred by Wigner’s precise
characterization of special relativity’s implications for quantum theory
[Wigner, 1939], there had arisen a more mathematical approach to understanding
the structure of relativistic quantum field theories. Naturally, relativistic
QFT’s mathematical structure was made transparent on this approach,
characterized by a number of basic assumptions. However, this was not true of
the interaction picture. Haag, in his lecture series given at CERN in
1953–4,666See [Haag, 2010] for his recollections of this aspect of the
history. Note that Haag’s placing of the lecture series in 1953–4 conflicts
with [Lupher, 2005], who places it in 1952–3. began by recounting the basic
assumptions necessary for relativistic QFT (irreducible Hilbert space
representation of the Poincaré group, etc.). Then, Haag used the mathematical
tools at his disposal to characterize the interaction picture (Fock space,
unitary intertwiner). The problem (Haag’s theorem) was that the interaction
picture, thus characterized, could not be used to represent non-trivial
interactions.

Haag’s theorem exists in several versions, notably [Haag, 1955] and [Hall and
Wightman, 1957]. Opinions differ widely on its precise implications. However,
loosely, the theorem states that any suitable representation for an
interacting quantum field must be unitarily equivalent to a Fock
representation for a free field. This means that any quantum field that is
unitarily equivalent to a free field must also be a free field. Obviously, a
field with a nontrivial interaction cannot be unitarily equivalent to a free
field. Thus, Haag’s theorem seems to show that if the interaction picture is
mathematically consistent, it can only describe trivial interactions, in which
the interacting field is in fact free.

###  2.2 The Interaction Picture

Haag’s theorem is often construed as a no-go theorem for the use of what’s
called the interaction picture. This picture has been widely used for the
calculation of many physical quantities that have matched experimental results
to a high degree of accuracy, for example the celebrated computation of the
anomalous magnetic dipole moment of the electron by [Schwinger, 1948a], and is
a mainstay of undergraduate and graduate textbooks and has facilitated the
calculation of many physical quantities that have matched experimental results
to a high degree of accuracy. 777 As one crude measure of how salient the
interaction picture is for actual physics, a search for the exact phrase
“interaction picture” in Physics Reviews conducted on Nov 4, 2021 returned
4,377 hits. The interaction picture forms the basis of calculations using
time-dependent perturbation theory. Crucially, the Gell-Mann and Low theorem
makes use of the interaction picture. The original paper [Gell-Mann and Low,
1951] presenting this formula has (as of Nov 4, 2021) 1544 citations recorded
on google scholar. This is likely a rather low estimate of how widely the
formula is in fact used since the formula is now at the level of common
knowledge among physicists, and thus in many cases it might be used without
citation. For instance, the Gell-Mann and Low theorem was used to rigorously
derive the Bethe–Salpeter equation, which describes the bound states of a
system of two fermions in a relativistic formalism [Salpeter and Bethe, 1951].
This equation was used in subsequent quantum electrodynamics calculations of
the fine structure of Hydrogen-like atoms [Salpeter, 1952], Helium [Douglas
and Kroll, 1974] and heavy atoms [Mohr et al., 1998] and contemporary
precision calculations of the electron magnetic moment [Aoyama et al., 2015].
Likewise, the equations have been utilized in the calculation of nucleon-
nucleon potentials [Partovi and Lomon, 1969, Partovi and Lomon, 1972, Lomon,
1976, Partovi, 1978] and the derivation of scaling laws for interactions with
large momentum transfer, confirmed by scattering experiments [Brodsky and
Farrar, 1975], among many other examples. As its name suggests, the
interaction picture is one way to model interacting fields in conventional
quantum field theory. Let us suppose we have a field, $\phi$, with conjugate
momentum $\pi$, generally taken (either explicitly or implicitly) to obey the
equal time canonical commutation relations and the Wightman axioms.

The interaction picture is intermediate between the Schrödinger and Heisenberg
pictures. In the Schrödinger picture of quantum mechanics, states evolve in
time under the full Hamiltonian, whilst operators are stationary. In the
Heisenberg picture of quantum mechanics, the operators evolve under the full
Hamiltonian, whilst states are stationary. To form the interaction picture, we
split the full Hamiltonian into a free and a (time-dependent) interaction
part, $H=H_{F}+H_{I}$. The evolution of operators is governed by the free
Hamiltonian, $H_{F}$, so the fields are free. The evolution of states is
governed by the interaction part, $H_{I}$.

We stipulate that the operator fields coincide with those of the Heisenberg
picture at some time, $t_{0}$. Let $V(t_{2},t_{1})$ represent the unitary
evolution of the interaction picture states from time $t_{1}$ to $t_{2}$,
generated by the interacting part of the Hamiltonian, $H_{I}$,

(1)
$V(t_{2},t_{1})=e^{-iH_{I}(t_{2}-t_{1})}=e^{+iH_{F}(t_{2}-t_{1})}e^{-iH(t_{2}-t_{1})}.$

We call this operator the intertwiner, or Dyson operator. Then, at all times,
$t$, the Heisenberg (subscript $F$) and interaction (subscript $I$) operators
are related by the intertwiner as follows,

(2) $\displaystyle\phi_{I}(x,t)$
$\displaystyle=V^{-1}(t,t_{0})\phi_{H}V(t,t_{0}),$ (3)
$\displaystyle\pi_{I}(x,t)$ $\displaystyle=V^{-1}(t,t_{0})\pi_{H}V(t,t_{0}).$

As we will see, it is central to Haag’s theorem that the relation of the
interaction field, $\phi_{I}$, to the free field, $\phi_{F}$ is characterized
by a unitary map.

Generally, we seek to calculate physical amplitudes, taken in the limit in
which the fields are free at times $t\rightarrow\pm\infty$. This is meant to
capture the intuition that in an interaction, particles begin infinitely far
apart (and hence not interacting) and then separate again infinitely far apart
after an interaction. The interaction picture may be useful if we can treat
the effects of $H_{I}$ as a small, time-dependent perturbation on the
evolution under $H_{F}$. In perturbation theory we perform approximate
calculations by expanding the desired physical quantities in powers of the
small interaction, $H_{I}$.

###  2.3 Proof of Haag’s Theorem for Spin-free, Neutral, Scalar Fields

The proof of Haag’s theorem assumes the Wightman Axioms. For convenience we
number the axioms according to the convention of [Seidewitz, 2017].888This
proof largely follows the proof of [Hall and Wightman, 1957], using some of
the notation and conventions of [Seidewitz, 2017] and [Earman and Fraser,
2006] in order to make the role of each assumption transparent.

####  2.3.1 Wightman Axioms

######  Axiom 0.

States. We have a physical Hilbert Space, $\mathcal{H}$, for which the states,
$\ket{\phi}$, are rays, such that,

  1. 1.

The states transform according to a continuous unitary representation of the
Poincare group, $U(\mathbf{\Delta x},\Lambda)$, under Poincare
transformations, $\\{\mathbf{\Delta x},\Lambda\\}$.

  2. 2.

There is a unique, invariant vaccum state, $\ket{0}$, in $\mathcal{H}$,
invariant under $U$: $U(\mathbf{\Delta x},\Lambda)\ket{0}=\ket{0}$.

  3. 3.

Let $U(\mathbf{\Delta x},\Lambda)=e^{iP^{\mu}\mathbf{\Delta x}_{\mu}}$. Then,
$P^{\mu}P_{\mu}=-m^{2}$. We interpret $P^{\mu}$ as an energy-momentum operator
and $m$ as a mass. The eigenvalues of $P^{\mu}$ lie in the future lightcone.

######  Axiom 1.

Domain and continuity of fields. The field $\phi(x)$ and its adjoint
$\phi^{\dagger}(x)$ are defined on a domain $D$ of states dense in
$\mathcal{H}$ containing the vacuum state $\ket{0}$. The $U(\mathbf{\Delta
x},\Lambda)$, $\phi(x)$ and $\phi^{\dagger}(x)$ all transform vectors in $D$
to vectors in $D$.

######  Axiom 2.

Field transformation law. The fields transform under Poincare transformations
as,

(4) $U(\mathbf{\Delta x},\Lambda)\phi(x)U^{-1}(\mathbf{\Delta
x},\Lambda)=\phi(\Lambda x+\mathbf{\Delta x}).$

######  Axiom 3.

Local commutativity. If $x$ and $x^{\prime}$ are two spacetime positions,

(5)
$[\phi(x),\phi(x^{\prime})]=[\phi^{\dagger}(x),\phi^{\dagger}(x^{\prime})]=0$

Furthermore, if $x$ and $x^{\prime}$ are space-like separated,

(6) $[\phi(x),\phi^{\dagger}(x^{\prime})]=0.$

######  Axiom 4.

Cyclicity of the vacuum. The vacuum state $\ket{0}$ is cyclic for the fields,
$\phi(x)$. That is, polynomials in the fields and their adjoints, when applied
to the vacuum state, yield a set $D_{0}$ dense in $\mathcal{H}$.

####  2.3.2 Proof of Haag’s theorem

Haag’s theorem follows from the results of two other theorems. For clarity of
exposition, here we only prove the theorem for neutral, scalar fields; the
generalization for other types of fields follows in a straightforward manner.
From here on, it is convenient to decompose four-vectors as
$x=(t,\mathbf{x})$, where the right hand side are the temporal and spatial
components, respectively.

######  Theorem 1.

Equality of equal-time vacuum expectation values. Let $\phi_{1}$ and
$\phi_{2}$ be two field operators, with associated conjugate momentum
operators, $\pi_{1}$ and $\pi_{2}$, defined in respective Hilbert spaces
$\mathcal{H}_{1}$ and $\mathcal{H}_{2}$, satisfying the Wightman axioms listed
above, and satisfying the equal time commutation relations,

(7) $\displaystyle[\phi_{i}(t,\mathbf{x}),\pi_{i}(t,\mathbf{x}^{\prime})]$
$\displaystyle=i\delta(\mathbf{x}-\mathbf{x}^{\prime}),$ (8)
$\displaystyle[\phi_{i}(t,\mathbf{x}),\phi_{i}(t,\mathbf{x}^{\prime})]$
$\displaystyle=[\pi_{i}(t,\mathbf{x}),\pi_{i}(t,\mathbf{x}^{\prime})]=0.$

Suppose that there exists a unitary operator $G$ such that, at some specific
time $t_{0}$,

(9) $\displaystyle\phi_{2}(t_{0},\mathbf{x})$
$\displaystyle=G\phi_{1}(t_{0},\mathbf{x})G^{-1},$ (10)
$\displaystyle\pi_{2}(t_{0},\mathbf{x})$
$\displaystyle=G\pi_{1}(t_{0},\mathbf{x})G^{-1}.$

We call G an intertwiner for the fields $\phi_{1}$ and $\phi_{2}$. Then the
equal-time vacuum expectation values of the fields coincide. (Note that so
far, this holds only at the particular time, $t_{0}$.)

###### Proof.

Let $U_{i}(\mathbf{\Delta x},\mathbf{R})$ be a continuous, unitary
representation of the inhomogeneous Euclidean group of translations,
$\mathbf{\Delta x}$, and three-dimensional rotations $\mathbf{R}$, defined on
each $\mathcal{H}_{i}$, $i=1,2$. Let us further suppose that transformations
$U_{i}(\mathbf{\Delta x},\mathbf{R})$ induce Euclidean transformations of the
field

(11) $\displaystyle U_{i}(\mathbf{\Delta
x},\mathbf{R})\phi_{i}(t_{0},\mathbf{x})U_{i}^{-1}(\mathbf{\Delta
x},\mathbf{R})$ $\displaystyle=\phi(t_{0},\mathbf{R}\mathbf{x}+\mathbf{\Delta
x}),$ (12) $\displaystyle U_{i}(\mathbf{\Delta
x},\mathbf{R})\pi_{i}(t_{0},\mathbf{x})U_{i}^{-1}(\mathbf{\Delta
x},\mathbf{R})$ $\displaystyle=\pi(t_{0},\mathbf{R}\mathbf{x}+\mathbf{\Delta
x}),$

as in axiom 0.1. From our supposition (equations 9 and 10), it follows that

(13) $U_{2}(\mathbf{\Delta x},\mathbf{R})=GU_{1}(\mathbf{\Delta
x},\mathbf{R})G^{-1}.$

But since the representations possess unique invariant vacuum states
$\ket{0}_{i}$ such that $U_{i}(\mathbf{\Delta
x},\Lambda)\ket{0}_{i}=\ket{0}_{i}$, as in axiom 0.2.,

(14) $c\ket{0}_{2}=G\ket{0}_{1},$

where $c$ is a complex number of absolute value 1, $|c|=1$. In other words, up
to a phase factor, $G\ket{0}_{1}$ is the vacuum state for field $\phi_{2}$ at
time $t_{0}$.

It further follows that the equal-time vacuum expectation values (Wightman
functions, also known as correlation functions) of the two fields are the
same,

(15)
$\prescript{}{1}{\bra{0}}\phi_{1}(t_{0},\mathbf{x}_{1}),\ldots\phi_{1}(t_{0},\mathbf{x}_{n})\ket{0}_{1}=\prescript{}{2}{\bra{0}}\phi_{2}(t_{0},\mathbf{x}_{1}),\ldots\phi_{2}(t_{0},\mathbf{x}_{n})\ket{0}_{2},$

for $\mathbf{x}_{1},\mathbf{x}_{2}$ up to at most $\mathbf{x}_{4}$, all at the
same fixed time, $t_{0}$.999If at least one of $\phi_{1}$ or $\phi_{2}$ is a
free field, then this can be proven to hold for all Wightman functions, i.e.
for $\mathbf{x}_{1},\mathbf{x}_{2}$ up to any $\mathbf{x}_{n}$. ∎

######  Theorem 2.

Jost-Schroer theorem. For any free scalar field $\phi$, the two-point vacuum
expectation values are given by

(16) $\bra{0}\phi(x)\phi^{\dagger}(x_{0})\ket{0}=\Delta^{+}(x-x_{0}),$

where $\Delta^{+}$ is the advanced Feynman propagator101010For example, see
[Duncan, 2012, pages 209-210 and 256],

(17) $\Delta^{+}(x-x_{0})=(2\pi)^{-3}\int
d^{3}p\frac{e^{i[-\omega_{\mathbf{p}}(x^{0}-x^{0}_{0})+p\cdot(\mathbf{x}-\mathbf{x}_{0})]}}{2\omega_{\mathbf{p}}},$

and $\omega_{p}=\sqrt{\mathbf{p}^{2}+m^{2}}$.

If, for any arbitrary field, the vacuum expectation values are given by
equation 16, then that field is a free field.

This result was first proved by Jost [Jost, 1961] for fields of positive mass,
and extended to fields of zero mass by Pohlmeyer [Pohlmeyer, 1969].

######  Theorem 3.

Haag’s theorem for scalar fields. Let $\phi_{1}$ be a free, scalar field,
which therefore satisfies equation 16. Let $\phi_{2}$ be a second, locally
Lorentz-covariant scalar field. Let us assume that $\phi_{2}$ is unitarily
related to $\phi_{1}$ at time $t_{0}$, as in equations 9 and 10. Further let
us assume that the conjugate momenta fields (written as adjoints
$\phi_{1}^{\dagger}$ and $\phi_{2}^{\dagger}$) satisfy the hypotheses of
Theorem 1. Then $\phi_{2}(x)$ is also a free field.

###### Proof.

Theorem 1 tells us that the equal time vacuum expectation values of the two
fields, $\phi_{1}$ and $\phi_{2}$ must coincide at $t_{0}$, i.e.,

(18)
$\prescript{}{1}{\bra{0}}\phi_{1}(t_{0},\mathbf{x}_{1})\phi_{1}(t_{0},\mathbf{x}_{n})\ket{0}_{1}=\prescript{}{2}{\bra{0}}\phi_{2}(t_{0},\mathbf{x}_{1})\phi_{2}(t_{0},\mathbf{x}_{n})\ket{0}_{2}.$

Since the field $\phi_{1}$ is free, it follows from Theorem 2 (equation 16)
and equation 20 that the two-point vacuum expectation values for the two
fields coincide at $t_{0}$, i.e.,

(19)
$\prescript{}{2}{\bra{0}}\phi_{2}(t_{0},\mathbf{x}_{1})\phi_{2}^{\dagger}(t_{0},\mathbf{x}_{n})\ket{0}_{2}=\prescript{}{1}{\bra{0}}\phi_{1}(t_{0},\mathbf{x}_{1})\phi_{1}^{\dagger}(t_{0},\mathbf{x}_{n})\ket{0}_{1}.$

So far, this holds only at $t_{0}$. However, any two spacelike separated
position vectors, $(t_{1},x_{1})$ and $(t_{2},x_{2})$, can be brought into the
equal time plane $t_{1}=t_{2}$ by a Lorentz transformation. Thus, the Lorentz-
covariance of $\phi_{2}$ allows us to extend the satisfaction of equation 19
to any two spacelike positions, and then, by analytic continuation, to any two
positions:

(20)
$\prescript{}{2}{\bra{0}}\phi_{2}(x_{1})\phi_{2}^{\dagger}(x_{n})\ket{0}_{2}=\Delta^{+}(x-x_{0}).$

Therefore, by Theorem 2, $\phi_{2}(x)$ must be a free field.

∎

###  2.4 Implications of Haag’s Theorem

By the criteria used by particle physicists, the interaction picture has
undoubtedly produced numerous notable successes (see note 7). However, the
interaction picture is generally taken to depend upon all of the assumptions
needed for Haag’s theorem, including Poincaré invariance and the existence of
a unitary operator relating the two fields, that is, that the fields are
unitarily equivalent. The apparent clash between the interaction picture and
Haag’s theorem arises as follows. If the fields, $\phi_{F}$ and $\phi_{I}$,
obey the Wightman axioms, and we have a unitary operator intertwining the two
fields at even a single time (as in equations 9 and 10), and $\phi_{F}$ is
free, then according to Haag’s theorem $\phi_{I}$ must also be free. So the
interaction must be trivial.

So Haag’s Theorem appears to be a no-go theorem for calculations that use the
interaction picture. At a glance, it would be mathematically inconsistent to
use the interaction picture for calculations involving any of the non-trivial
interactions that we care about in particle physics. However, a closer look
reveals a whole labyrinth of philosophical, mathematical, and physical issues
at stake in understanding the full significance of Haag’s theorem. We take
that closer look in the next section.

##  3 What the Haag Is Going on?

There are a few points on which all parties generally agree. First, judged by
its own particular standards of success, particle physics –including the
interaction picture– is highly successful.

Second, there is a consensus that Haag’s theorem poses a bona fide problem for
the standard presentation of the interaction picture, pre-renormalization. The
theorem itself is mathematically correct: as Klaczynski puts it, “it is a
mathematical theorem in the truest sense of the word; it brings with it the
‘hardness of the logical must’.” [Klaczynski, 2016]. Further, it is not
disputed that the assumptions of Haag’s theorem hold in the case of the
standard textbook presentations of the interaction picture.111111For example,
standard presentations assume, either implicitly or explicitly, a Poincaré
invariant theory, and that there are distinct free and interacting fields,
related by a unitary operator. No one, to our knowledge, argues that the
problem posed by Haag’s theorem is illusory.121212It may very well be that
many practicing physicists hold such a view, and that, precisely because they
see the problem posed by Haag’s theorem as at most rooted in an unrealistic
idealization of non-interacting states at temporal infinity, they therefore
choose not to address Haag’s theorem in their textbooks, lecture notes, or
research articles. We can do little more than speculate that such an unspoken
consensus fully explains the dearth of references to Haag’s theorem in
standard accounts of QFT—simple ignorance of the theorem may be just as strong
of a causal factor. David Tong’s lecture notes on QFT, for instance, do not
explicitly mention Haag’s theorem; but they do say that the assumption of non-
interacting states in the interaction picture is wrong and should be replaced
by the interactions-are-always-on interpretation of the LSZ reduction formula
[Tong, 2012] p. 54-55, 79-80. Thus, while it is possible that this dismissive
response to Haag’s theorem is widespread, too little of it exists in print to
be extensively covered in the remainder of this paper.  It is, rather, the
severity and appropriate remedy of the problem that is subject to debate.

However, the severity of the problem raised by Haag’s theorem clearly stops
short of spelling the demise of that research program predicated on the
application of quantum theories of fields to scattering experiments, commonly
called particle physics.131313Though see the discussion of Kastner (section
4.6) and Seidewitz (section 4.7) below for genuine proposals of new physical
theories, each drawing some motivation from Haag’s theorem.  Even the
practitioners of AQFT, while operating far afield from the details of the
Standard Model’s phenomenology, still conceive of their work as contributing
to the scientific enterprise whose primary goal is to develop the quantum
theory of fields as the appropriate theoretical apparatus for understanding
scattering experiments. So, then, what is going on with Haag’s theorem such
that these two stances —the interaction picture has been used successfully,
and Haag’s theorem poses a substantive problem for the interaction picture—can
be held together? This central tension is widely recognized as a point of
agreement. As Teller succinctly puts it,

> Everyone must agree that as a piece of mathematics Haag’s theorem is a valid
> result that at least appears to call into question the mathematical
> foundation of interacting quantum field theory, and agree that at the same
> time the theory has proved astonishingly successful in application to
> experimental results. [Teller, 1995, 115]

And yet, despite this initial agreement, extant responses to Haag’s theorem
form a confusing lot. The literature on Haag’s theorem reflects a number of
different assessments of the import of the theorem for both (mathematical)
physics and philosophy. Moreover, the extent to which these different
assessments make meaningful contact with each other is often unclear. In this
section, we briefly illustrate the nature of the confusion in this literature.
First, the confusion is not about the status of Haag’s theorem as a
mathematical result. All parties agree that the original theorem, and its
several generalizations, have valid proofs. The confusion enters when trying
to trace out the ramifications of Haag’s theorem for the foundations of QFT.
[Earman and Fraser, 2006] put it well, saying that “the theorem provides an
entry point into a labyrinth of issues that must be confronted in any
satisfactory account of the foundations of QFT” (p. 334). Once we have entered
into this labyrinth via Haag’s theorem, we encounter a host of conceptual and
interpretive issues, enmeshed in technical issues of mathematics and physics,
making even the range of options for a way forward through the labyrinth
unclear, much less which one may be the best.

A reader interested in Haag’s theorem and its implications for the use of the
interaction picture in physics may first look for insight from Earman’s and
Fraser’s seminal paper, “Haag’s theorem and its implications for the
foundations of quantum field theory.” They conclude, “On any reading Haag’s
theorem undermines the interaction picture and the attendant approach to
scattering theory” [Earman and Fraser, 2006, 333]. So, the reader naturally
thinks, the interaction picture is no good. And yet for Duncan, “the proper
response to Haag’s theorem is simply a frank admission that the same
regularizations needed to make proper mathematical sense of the dynamics of an
interacting field theory at each stage of a perturbative calculation will do
double duty in restoring the applicability of the interaction picture at
intermediate stages of the calculation” [Duncan, 2012, 370]—the interaction
picture survives! Miller concurs, adding that the success of calculations
delivered from regularized and renormalized theories is explained by the
conjecture that “perturbative expansions are asymptotic to exact solutions of
a theory that generates them” [Miller, 2018, 818]. So, the reader concludes
triumphantly, the interaction picture works!

Still more paths begin to emerge, however. According to Klaczynski, these
renormalized theories evade Haag’s theorem precisely by denying that the
interaction picture exists [Klaczynski, 2016]. Maiezza and Vasquez agree,
arguing that “due to Haag’s theorem, it is impossible to define QFT starting
from the interaction picture with free fields” [Maiezza and Vasquez, 2020, 10]
(italics in original); indeed, they seem to argue that the interaction picture
fails precisely because of the failure of the conjecture Miller relies on to
save it. Yet confusingly, Maiezza and Vasquez also disagree with Klaczynski on
what saves perturbative calculations from Haag’s theorem.

The paths so far, while many, nevertheless seem to turn on what to say about
the mathematical coherence of the interaction picture. Thus, a labyrinth
though it may be, the reader thinks, I can at least see its basic structure.
The reader has judged too soon, however, for it is not only the interaction
picture per se that is at stake, but the metaphysics: “either the assumptions
of Haag’s theorem do not hold, in which case there is no particle notion
applicable to a scattering experiment at intermediate times, or they do, in
which case the particle notion applicable at intermediate times is
incommensurable with the ingoing/outgoing particle notions, if the interaction
is non-trivial” [Ruetsche, 2011, 252] (italics in original). So, the reader
thinks, the path out of the Haagian labyrinth requires the banishment of
particles and the embracing of fields! Not so, says Kastner: particles exist,
and fields must be banished [Kastner, 2015].

The reader is thus confronted with paths out of the Haagian labyrinth
diverging on both metaphysical and mathematical grounds and, worse still, she
can’t tell from her place in the labyrinth whether or where these paths
coincide. As if this weren’t bad enough, a fog sets in. Are we even trapped at
all?, the reader asks, [Seidewitz, 2017, 356] in hand, for Haag’s theorem
arises in traditional QFT only because time is not “treated comparably to the
three space coordinates, rather than as an evolution parameter.” Thus, the
Haagian labyrinth could have been entirely avoided had time been treated in a
relativistically sensible manner at the outset.

How should the reader react to this? Is there a Haagian labyrinth? And if so,
which path will lead us out? The framework given in the next section gives a
fixed structure for organizing and assessing this confusing network of
responses to Haag’s theorem, thereby creating several distinct mappings of the
Haagian labyrinth. More general lessons from studying these maps are given in
section 5.

##  4 The Framework: Assessment, Repair, and Renovation and Maintenance

Before presenting the framework, we should carefully consider its goal. The
purpose of this framework, as we said in the introduction, is to fruitfully
structure and organize answers to the question “What does Haag’s theorem tell
us about quantum field theory, present and future?” Two desiderata for such a
framework are fairly obvious. First, it should sensibly organize the answers
that have already been given to this question. As such, we will apply the
framework to prominent answers in the literature. Second, the framework should
leave significant room for future developments. Given the interdisciplinary
nature of the study of Haag’s theorem, and given the recent increase of
interest from physicists ([Klaczynski, 2016], [Maiezza and Vasquez, 2020], and
[Seidewitz, 2017]), we expect there is much more to be said on this topic. A
good framework for organization, therefore, must have space to accommodate
these expected future developments.

There are additional desiderata, which are less obvious but of no lesser
importance. These arise principally from the interdisciplinary nature of the
research on Haag’s theorem. Third, the framework should organize various
extant and possible responses to Haag’s theorem without itself providing a
judgement as to which response is best. Thus, while a successful framework
should assist the scholarly community in evaluating the success or failure of
a given response to Haag’s theorem with respect to or along a given criterial
axis, it should in no way bias one axis as more or less important than any
other axis. In particular, the criteria should not mark out one outlook as
more important or relevant to a given criteria simply in virtue of that
outlook being more interested or invested in that criteria. For example,
“which assumption(s) of Haag’s theorem must we give up?” is not unbiased
insofar as it is of preeminent interest to axiomatic investigators; likewise,
“what notion of particle, if any, is circumscribed by QFT?” is not agnostic
insofar as it is of preeminent interest to the metaphysically-inclined
philosopher. The importance of this desideratum is born of this framework’s
goal to organize, but not finalize, answers as to the significance of Haag’s
theorem. Nevertheless, for proper subsets of responses that coincide on at
least one axis, sub-frameworks biased along that axis can provide further
helpful organization (see section 5).

Fourth and finally, the framework should force the various extra-Haagian
outlooks to be more explicitly acknowledged. In particular, it should
illuminate what is at stake in discussion of Haag’s theorem separately for
mathematicians, physicists, and philosophers. This last desideratum is meant
to draw attention away from products of miscommunication and toward more
honest and fruitful dialogue.

Our framework consists of the following three criterial axes along which
responses to Haag’s theorem can be assessed, categorized, and critically
compared. These axes have been chosen to expand upon Hilbert’s leading
construction analogy, namely, as the assessment and diagnosis, immediate
repair, and longer-term renovation or maintenance of a building (the “edifice
of science”) that is continually under construction. Following the analogy,
the authors surveyed here may be thought of as subcontractors brought in to
assess and diagnose the building’s status in light of Haag’s theorem. Thus, in
our capacity as organizers of the varied responses to Haag’s theorem, we the
authors may be thought of as something like the general contractor whose job
is to arrange and make comparable the subcontractors’ assessments and provide
expert advice to the building’s owner as to how to proceed with the
construction. We accomplish this first aim in section 4 and the second aim in
section 5.

  * •

Assessment: What precisely is the problem posed by Haag’s theorem, if any? For
what objective(s) is this a problem?

  * •

Repair: How should this problem be remediated?

  * •

Maintenance or Renovation: Where should resources (time, attention, grant
funding, conference and journal platforms, etc.) for the next (relevant) phase
of research be allocated?

We now turn to applying this framework to organize seven major responses to
Haag’s theorem from several different perspectives. The results of applying
our framework are summarized in table 1. Each result is discussed in detail in
the following subsections. For clarity, each subsection begins with a
discussion of the extra-Haagian outlook at play; however, note that the
content of these discussions has been determined through application of our
framework.

Name |  Assessment |  Repair |  Renovation or Maintenance
---|---|---|---
Earman & Fraser |  interaction picture (IP) rests on a set of inconsistent assumptions |  Abandon the interaction picture |  Interpret and assess AQFT and constructive QFT
Fundamental Interpreters |  Standard particle notion inconsistent with axioms of QFT |  Abandon standard particle interpretation |  Develop univocal interpretation of axioms of QFT
Duncan & Miller |  pre-renormalization IP is inconsistent |  Renormalization (breaks Poincaré invariance) |  Continue studying renormalization and EFTs, don’t worry
Klaczynski |  IP is inconsistent |  Renormalization leads us to abandon the IP |  replace the IP’s unitary intertwiner
Maiezza & Vasquez |  The full perturbative series has renormalon divergences |  Resummation methods (but these are ambiguous) |  New resummation methods and physical insights are needed
Kastner |  Physics is non-local |  Abandon notion of an independent field |  Develop direct-action theories
Seidewitz |  Use of time as the evolution parameter is at fault |  Revise axioms to introduce new evolution parameter |  Extend parameterized QFT to gauge theories and non-Abelian interactions
Table 1: Framework application summary

###  4.1 Earman and Fraser

##### Extra-Haagian outlooks.

For Earman and Fraser, the most pressing question is, what is QFT? As they see
it, answering this question requires an honest and complete marriage of SR
with QM. As such, QFT cannot settle for the weakening of any fundamental
postulates of either theory—dishonest or partial marriages of SR with QM are,
by definition, not QFT.

This requirement of a complete and honest marriage is a significant departure
from orthodoxy. For one, it immediately rules out the two most dominant
approaches to constructing QFTs. On the one hand, renormalized canonical QFT
is ruled out, according to Fraser, because the theory is so mathematically
ill-defined that we cannot check if it has honestly married SR with QM. On the
other hand, the introduction of cutoffs in canonical QFTs, while
mathematically acceptable, means that the marriage is incomplete—SR concerns
infinite, continuous space, not finite lattices. But for another, such a
requirement is born of an attitude toward theory development that is
unorthodox in contemporary science (if not philosophy of physics). Rather than
demand full unification up front, the average theoretical particle physicist
contents themselves with combining SR and QM to the best of their current
ability. At least for now, our current best practice gives up the strict
Poincaré invariance demanded by SR. Arguably, the ‘best of their ability’ has
improved over the decades such that, the average theoretical particle
physicist could claim, we will inevitably find such a complete and honest
marriage if one is forthcoming. In the meantime, we have inarguably successful
partial (resp. dishonest) marriages of SR with QM with which we can keep
experimental and theoretical particle physics going. While it is likewise a
live option for Fraser (and presumably Earman) that an honest and complete
marriage of QFT is not possible, this is a question of logic, not practical or
empirical feasibility. This leaves only AQFT as a means for characterizing
QFT.141414Fraser seems optimistic that, with more time, the as-of-yet
unfinished project of constructing a fully rigorous (according to AQFT
standards) interacting QFT in four spacetime dimensions could be completed.
But she is clear that this has not happened yet, and that in principle it
could be shown to be impossible. Fraser writes, “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 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.” [Fraser, 2009] (p. 563).

##### Assessment.

As discussed above, the goal for Earman and Fraser is to determine what QFT
is, i.e., to completely and honestly unify SR and QM. A successful
unification, if it is possible, will at least be able to model interacting
systems. Thus, according to Earman and Fraser, Haag’s theorem is a
demonstration of the logical impossibility of modeling interactions using the
interaction picture in our would-be QFT (assumed to obey the Wightman axioms).
All that the interaction picture is capable of modeling is, provably, free
fields—a spectacular failure. As such, Haag’s theorem is perhaps the first of
many signs that SR and QM are, in the end, irreconcilable.

##### Repair.

The repair work needed is in the task of modeling interactions in QFT. This
might proceed either by abandoning the interaction picture altogether (e.g.
with Haag-Ruelle scattering), or else by substantially revising it through
giving up one or more of its assumptions. While assessing this second possible
response, Earman and Fraser find the robustness of Haag’s theorem, and its
generalizations, to be significant: “Subsequent no-go results do not show that
field theorists do not have to worry about Haag’s theorem because some of its
assumptions do not hold in all cases of interest; rather, what the subsequent
results show [is] that even more assumptions have to be abandoned in order to
obtain well-defined Hilbert space descriptions of interacting fields” (318).
Thus, Earman’s and Fraser’s preferred remedy for Haag’s theorem is to abandon
the interaction picture altogether in favor of alternative ways of modeling
interactions in QFT.

Note that this accords with Earman and Fraser’s stated goal of unifying SR and
QM. Regularization and renormalization techniques are not options available to
them as they change the game or execute the unification dishonestly,
respectively. Thus, the only option is to give up the interaction picture:
Haag’s theorem does not pose a problem for QFT per se, but it “does pose
problems for some of the techniques used in textbook physics for extracting
physical prediction from the theory” [Earman and Fraser, 2006](p. 306). QFT,
therefore, is not to be identified with textbook physics. The textbook physics
is one attempt at doing QFT, and Haag’s theorem exposes serious problems in
that particular attempt. The immediate technical moral of Haag’s theorem is
that (better attempts at) QFT must embrace the use of unitarily inequivalent
representations of the CCR.151515“[A] single, universal Hilbert space
representation does not suffice for describing both free and interacting
fields; instead, unitarily inequivalent representation of the CCR must be
employed” (p. 333). The philosophical insight from this technical result is
that it designates the role of unitarily inequivalent representations as a
distinctive feature of QFT in contrast to QM.161616“Haag’s theorem was
instrumental in convincing physicists that inequivalent representation of the
CCR are not mere mathematical playthings but are essential in the description
of quantum fields” (p. 319).

##### Renovation.

As a consequence of this logical diagnosis and logical remedy, Earman and
Fraser advocate for future philosophical work and resources to be deployed in
philosophical projects about AQFT and its attendant non-interaction-picture
approaches to interactions. There are at least two such broad philosophical
projects that benefit from the ways in which AQFT-practicing mathematicians
have “digested the lesson of Haag’s theorem” [Earman and Fraser, 2006] (p.
334). First there is the project of interpreting the mathematical structures
used in AQFT and in constructive QFT. Second, and relatedly, there is the
subtle matter of assessing the implications of theorems of AQFT for the
philosophy of those areas of fundamental physics that make use of QFT.

###  4.2 The Particle Problem

There is a segment of the philosophical literature addressing Haag’s theorem
that is primarily motivated by the question, as put by Laura Ruetsche, ‘Is
particle physics particle physics?’ [Ruetsche, 2011, 190]. In this context,
Haag’s theorem is most often taken to be a no-go theorem for a particle
interpretation of QFT (most notably [Halvorson and Clifton, 2002] and [Fraser,
2008]). Others disagree with this assessment (e.g. [Wallace, 2011] and [Bain,
2011]).

##### Extra-Haagian outlooks.

In this literature, the primary extra-Haagian divide is between those adhering
to what [Ruetsche, 2011] calls the ideal of pristine interpretation and those
that do not. As she noted when coining the phrase, this ideal is rarely
explicitly stated. Nevertheless, sympathy for the ideal of pristine
interpretation appears widespread, particularly in the philosophy of
QFT.171717See [Arageorgis, 1995, 119–125] for an early interpretation-focused
discussion of Haag’s theorem. We do not discuss this work here because his
status as pristine or not is unclear. [Halvorson and Clifton, 2002], [Fraser,
2008], and [Bain, 2011]181818Bain’s precise view on the ideal of pristine
interpretation is less clear, but we include him here because of the
fundamentality of his notion of particle and its perspicuity in the LSZ
formalism. See, e.g., [Bain, 2013] for work suggesting this classification is
incorrect., for example, approach the particle problem from a methodological
stance of pristine interpretation. What binds pristine interpreters in the
philosophy of QFT together is a desire for a unimodal interpretation of the
theory of quantum fields. These authors demand that an interpretation specify
what metaphysical entities are fundamental, in the sense that they are present
across all applications (models) of the theory of quantum
fields.191919Ruetsche characterizes the pristine interpreter as one who
believes that “a theory’s laws on their own—or even in concert with “a few
general principles” in the form of duly abstract mathematical, metaphysical,
methodological, etc., considerations—delimit what worlds are possible
according to that theory”; in particular, “those worlds should [not] be
characterized in different ways for different extranomic (factual or material
or explanatory or maybe even practical) circumstances” [Ruetsche, 2011, 4].
For simplicity, here we prefer to talk of interpretations that are fundamental
or not rather than of physical possibility being determined by a theory’s laws
on their own or not. A fundamental interpretation is one whose generalizations
are determined solely by the theory’s laws (with the same caveat given by
Ruetsche). Implicit in this is the requirement that all representations be
specifiable uniformly and precisely enough that a single, mathematically-
precise interpretive schema is available throughout.

Despite widespread sympathy for pristine interpretation in the philosophy of
QFT literature, there remain dissenters. In [Wallace, 2011, 124] and [Wallace,
2001], for example, scale-variant and emergent concepts, respectively, are
given pride-of-place in interpretation. Likewise, Ruetsche herself argues at
length that “[w]hat set of possible worlds we associate with a theory of
QM$\infty$ can depend on what we’d like to do with that theory” [Ruetsche,
2011, 352]. Both authors champion non-pristine, non-fundamental
interpretations of QFT, and they thereby approach the particle problem and
Haag’s theorem with a very different outlook. While this difference in outlook
is often implicit, Fraser (in response to [Wallace, 2001]) is transparently
skeptical of even the cogency of an ontology with non-fundamental entities or
which is not built on an exact similarity relation with the mathematical model
at hand [Fraser, 2008, 857–8]. This difference in outlook clearly informs
reactions to the particle problem.

More generally, the pristine interpreters of QFT share an understanding of how
no-go theorems should be addressed. On the one hand, they care almost
exclusively about the fundamental-interpretive implications of Haag’s
theorem—its implications for scattering theory and calculations therein deemed
irrelevant. Thus, this group fits uneasily alongside the other authors here
discussed—with the possible exception of Earman and Fraser and
Seidewitz—insofar as use or application of QFT plays no role in the discussion
nor, indeed, motivation for discussion of no-go theorems. They furthermore
expect a response to no-go results like Haag’s theorem to apply for all
theories of quantum fields. Whereas Duncan and Miller satisfy themselves with
identifying violations of the assumptions of Haag’s theorem in particular
settings, thus potentially making available in those settings a classic (non-
fundamental) particle interpretation, here a satisfactory response must apply
across all settings. This feeds directly into the resource allocations
proposed by pristine interpreters: since a response must work across all
settings, axiomatic approaches to inquiry are naturally preferred. For this
reason, satisfying responses are typically sought from AQFT; the effective
field theory approach, in particular, cannot provide a satisfying response to
the pristine interpreter because it makes no pretense of applying one set of
concepts across all settings.

##### Assessment.

For pristine interpreters, Haag’s theorem is a diagnostic no-go result
concerning interpretation: QFT cannot be interpreted as fundamentally about
particles. For instance, in their [Halvorson and Clifton, 2002, 24], Halvorson
and Clifton blame Haag’s theorem, among other no-go results, when they
conclude that the theory of quantum fields “does not permit an ontology of
localizable particles; and so, strictly speaking, our talk about localizable
particles is a fiction.” Similarly, as Ruetsche summarizes the thinking of
pristine interpreters, it is due to Haag’s theorem that “within the confines
of the interaction picture, no fundamental particle interpretation can frame
explanations of particle physics phenomenology and its compliance with
calculations mediated by Feynman diagrams” [Ruetsche, 2011, 253]. Fraser, too,
comes to this conclusion on the basis of Haag’s theorem [Fraser, 2008]. Thus,
for these authors, Haag’s theorem sits alongside results like the
Reeh–Schlieder theorem [Reeh and Schlieder, 1961], the Unruh effect [Crispino
et al., 2008] [Earman, 2011], or the Hegerfeldt [Hegerfeldt, 1998a]
[Hegerfeldt, 1998b] and Malament [Malament, 1996] theorems (and extensions
thereof [Halvorson and Clifton, 2002]) as evidence against the existence of a
fundamental particle interpretation for RQFT. Indeed, canon seems to
dictate—even for non-fundamental interpreters —that these results be discussed
together when addressing the particle problem (e.g., [Bain, 2000, 380]
[Halvorson and Clifton, 2002, 20] [Fraser, 2008, 842] [Ruetsche, 2011, Chs.
9–11] [Wallace, 2001, §2.4]

The most obvious path to a fundamental particle interpretation—relying on what
we call the standard particle notion—is blocked by Haag’s theorem in the
following way. This path relies on a representation of the Weyl relations
having a (global) number operator, which can be understood physically as
counting the number of particles. Since number operators only exist for
representations unitarily equivalent to a free-field Fock representation
(theorem 3.3 of [Chaiken, 1968]), and since Haag’s theorem implies that the
latter cannot represent an interacting field, a particle interpretation for
interacting fields cannot rely on the existence of (global) number
operators.202020See references in [Fraser, 2008, 847] for further detail. See
also [Heathcote, 1989] pp. 91-97 for a discussion of the consequences of
Haag’s theorem for Fock space representations and axiomatic approaches to
modeling interactions (and see [Earman and Fraser, 2006] section 7 for their
critical assessment thereof). But the situation is even worse, it would seem:
violating any one of the assumptions of Haag’s theorem would seem to undermine
any notion of particle that could do the work we demand of a fundamental
interpretation [Ruetsche, 2011, 253]. Thus, Haag’s theorem seems to rule out
fundamental particle interpretations of RQFT.

Crucially, the existence of a global number operator is necessary for the
standard fundamental particle interpretation. That is, the existence of a
global number operator is necessary for a particle interpretation to meet the
fundamental interpreter’s demand for a single, mathematically-precise
interpretive schema applicable everywhere. This condition is meant to ensure
both that would-be particles are like particles—i.e., are sufficiently
localizable and countable—as well as that they are so in the same way
throughout an interaction. Put so crudely, it is this latter aspect of the
criterion that most directly rules out emergence- and coalescence-type
particle notions like that found in [Wallace, 2001].212121See also [Feintzeig
et al., 2021] for recent work on accounting for emergent particle phenomena
using the classical limit of QFT.

##### Repair.

Pristine interpreters have proposed two kinds of responses to the problems
posed by Haag’s theorem (and other no-go theorems) for a fundamental particle
interpretation. Because of the strictures of the ideal of pristine
interpretation, each is, in a sense, revisionary.

On the one hand, some bite the bullet and accept that there are no fundamental
particles. Doing so entails providing an alternative fundamental ontology and,
consequently, alternative explanations of erstwhile “particle” physics
phenomena. This is the more typical repair proposal. For instance, Halvorson
and Clifton conclude that “relativistic quantum field theory does permit talk
about particles—albeit, if we understand this talk as really being about the
properties of, and interactions among, quantized fields” [Halvorson and
Clifton, 2002, 24]. Similarly, albeit with less patience for the talk of
particles, Fraser concludes that special relativity and the non-linearity of
the field equation for an interacting system—key ingredients for Haag’s
theorem—conspire against a quanta interpretation, i.e., “there is no quanta
interpretation and there are no quanta” [Fraser, 2008, 858]. However, it
should be noted that the most obvious way of constructing a field ontology
appears to run into the same problems as the particle ontology [Baker,
2009].222222However this argument relies on the restriction that the field
wave functionals are square-integrable. If we relax this restriction, then the
space of wave functionals will be larger than the space of particle wave
functions (see [Jackiw, 1990] and [Sebens, 2022], but see [Wallace, 2006] for
a defense of the square-integrability restriction). In each case, a revision
of our ontology—or at least, ontological speech—is proposed.

On the other hand, some propose revision of the notion of particle required
for a fundamental particle interpretation. Doing so entails revising the
relevant definitions at play in the no-go on particles, e.g., the approach to
scattering theory (Haag–Ruelle) [Reed and Simon, 1980], the condition for
localizability and state equivalence (LSZ formalism) [Bain, 2000], or the
definition of particle itself232323After arguing that the Received View’s
notion of particle, which is inconsistent with relativistic quantum field
theories, relies implicitly on the existence of an absolute temporal metric,
[Bain, 2011] concludes by pointing to several potential modifications of the
definition of particle., in order to “save” the talk of fundamental particles,
especially the explanatory work done by this talk. This repair proposal is
less typical. Though each proposal may save (some of) the talk of fundamental
particles, each nevertheless proposes revision of QFT or the standard
definition of particle.

For non-pristine interpreters, Haag’s theorem alone does not demand any
repair. Instead of proposing repairs to either our ontology or QFT, it
suffices for [Wallace, 2001] to show that a concept of particle can be
recovered from a field-theoretic description. Given Wallace’s background
presumption that approximate concepts are legitimate in physics, it does not
matter that the particle concept he deploys is vaguely defined and applies
only to bosonic, massive QFTs; likewise, the satisfaction with approximate
concepts removes the sting of Haag’s theorem for a fundamental particle
notion. For Ruetsche, Haag’s theorem is only one chapter (namely, Ch. 9) in
the story of particles [Ruetsche, 2011, Chs. 9–11], and that it rules out the
standard particle interpretation in general is unremarkable in the larger
story. Indeed, she concludes, “[s]ometimes particle physics is, adulteratedly,
particle physics, and that’s a good thing” [Ruetsche, 2011, 260].

##### Renovation.

The long-term desire of the pristine interpreter group—including each of the
above repairs—is to develop a pristine interpretation of QFT. Given this
group’s expectations for how one should be developed, a mathematically
precise, general, and consistent formulation of QFT is a goal. Thus, expected
renovations for this group are largely mathematical. We should note some
reason for pessimism about the attainment of this goal, however. While the
pristine interpreters have worked from the Wightman axioms for over half a
century now, sufficiently realistic models of these axioms have not been
forthcoming. In particular, these models must be able to represent
interactions. This has proven especially challenging, and there is no
guarantee that such models are even possible. One noteworthy recent
development on what kinds of field theories can be rigorously constructed in
four space-time dimensions is [Aizenman and Duminil-Copin, 2021]. Likewise,
revisions of the relevant definitions (e.g., Bain’s re-definition of
‘particle’) have led to difficulties [Fraser, 2008]. Long term renovations in
this area will likely require deep conceptual work rather than mathematical
advancement. Finally, the non-pristine interpreter does not see Haag’s theorem
itself as reason for any renovation.

###  4.3 Duncan and Miller

Tony Duncan [Duncan, 2012] and Michael Miller [Miller, 2018] argue that at
least one of the assumptions needed to prove Haag’s theorem is violated at
some point in the actual process of calculating scattering theory results in
perturbative QFT. Thus, in Duncan’s words, we can “stop worrying” about Haag’s
theorem. These calculational processes include the methods of regularization
and renormalization.

A similar view on Haag’s theorem is given in [Fraser, 2017]. Here, James
Fraser is primarily concerned with the much broader question of the status of
perturbative QFT and diagnosing the ‘real problem’ with this area of physics.
He argues that perturbative QFT is “a method for producing approximations
without addressing the project of constructing interacting QFT models” (4). As
a consequence of this view, the threat of inconsistency posed by Haag’s
theorem is defused for much the same reasons as given by Duncan and Miller.
Fraser concludes section 4, “The perturbative method simply does not assert
the set of claims shown to be inconsistent by Haag’s theorem” (18). We set
aside [Fraser, 2017] for the remainder of this section since Haag’s theorem is
not the primary target of that article; but readers interested in the broader
question of how to assess perturbative QFT are encouraged to look there for an
important contribution on that topic.

##### Extra-Haagian outlooks.

For Duncan and Miller, it is a given that the interaction picture has been
consistently applied in calculating various specific theoretical predictions.
For Duncan, QFT as it is used in particle physics—including its use of the
interaction picture—is “the most powerful, beautiful, and effective
theoretical edifice ever constructed in the physical sciences” [Duncan, 2012,
iv]. His goal, therefore, cannot coincide exactly with Earman and Fraser’s of
strictly unifying SR with QM. Rather, Duncan’s broader goal for the book is to
provide a “deep and satisfying comprehension” of QFT by addressing the
important conceptual issues for which the traditional, “utilitarian” texts
fail to provide careful explanations (pp. iii–iv).

Likewise, Miller is focused on understanding the QFT noted for its empirical
successes [Miller, 2018, 802], hence not the QFT of Earman and Fraser.
Ultimately, Miller’s aim is to address “a general tension that exists in much
of the literature engaged in the philosophical appraisal of the foundations of
quantum field theory” (p. 803). This tension is essentially that between
Earman and Fraser and the fundamental interpreters’ approach to QFT on the one
hand and the empirically-tractable QFT on the other: while the former is
mathematically rigorous and hence (relatively) easy to interpret using
standard philosophical tools, it has yet to produce a realistic model, so it
is unclear how it could inform claims about the actual world; conversely,
while the latter has generated wildly successful empirical predictions, it has
done so through changes to the mathematical formalism whose interpretive
significance is far from obvious (p. 803).

For both authors, then, the preferred approach seems to be to bring the
philosopher’s penchant for logical and conceptual clarity to bear on QFT as it
is actually used. Given that both are sure (up to fallibility) that QFT is
being applied consistently, questions about its foundations will naturally not
concern whether they are consistent or correct but rather how they are so.

##### Assessment.

Called in to assess the status of our QFT building in light of Haag’s theorem,
what Duncan and Miller are assessing is not whether QFT is consistently using
the interaction picture, but how. Both Duncan and Miller recognize the logical
nature of the problem posed by Haag’s theorem—i.e., as the conflict between
the interaction picture and free-field representation assumptions—and they
take consistency to be a requirement for a viable theory. Indeed, they
recognize that a response to Haag’s theorem must come from negating at least
one of its assumptions. However, unlike Earman and Fraser, their goal is to
“explain why theoretical predictions for realistic experimental observables
give empirically adequate results” [Miller, 2018, 803]. These predictions in
fact use the interaction picture, so they cannot just jettison the interaction
picture. Rather, they aim to identify where within the practical calculational
techniques such violations of the assumptions must already take place for
reasons entirely independent of Haag’s theorem.

[Duncan, 2012, pages 359-370] and [Miller, 2018] each argue that we circumvent
Haag’s theorem in the messy calculational details of how the interaction
picture is used in practice. This takes place in a two-step process. In the
first step, regularization evades Haag’s theorem at the price of also breaking
the Poincaré invariance (and generally the unitarity) of the S-matrix, which
we generally consider to be desirable. In the second step, renormalization
allows us to remove the regularization, thus restoring the Poincaré invariance
and unitarity of the theory. According to [Duncan, 2012, page 370], “the
proper response to Haag’s theorem is simply a frank admission that the same
regularizations needed to make proper mathematical sense of the dynamics of an
interacting field theory at each stage of a perturbative calculation will do
double duty in restoring the applicability of the interaction picture at
intermediate stages of the calculation.” Thus, in their assessment, Haag’s
theorem poses no problem to our QFT building’s integrity.

##### Repair.

The interaction picture in practice makes use of renormalization and
regularization techniques that, in pulling “double duty”, both produce finite
perturbative results and defuse the assumptions of Haag’s theorem. The
renormalization and regularization ‘repairs’ also evade Haag’s theorem as a
positive side effect. No additional repair is needed.

##### Maintenance.

Maintenance is straightforward: maintain current practices of proper use of
renormalization and regularization242424However, some methods of
regularization do not break Poincaré invariance. This raises the question as
to how this story should apply in such cases. [Miller, 2018, pages 814-815]
suggests that the basic structure of the argument still stands. Any such
regularization technique must provide a means for controlling infrared
divergences. In so doing, one expects that one or more of the assumptions of
Haag’s theorem must have been violated through the regularization process.
techniques. For the conceptually and logically curious, following in Duncan
and Miller’s footsteps, this maintenance will likely also involve identifying
precisely which assumptions have been violated, and where. While it remains to
be shown that the perturbative power-series converge and thus correspond to
exact non-perturbative objects post-regularization and -renormalization,
[Miller, 2018, page 815] contends that this problem is unrelated to Haag’s
theorem.

###  4.4 Klaczynski

##### Extra-Haagian outlooks.

Like the authors above, Klaczynski recognizes a bifurcation in QFT research.
On the one hand there is canonical QFT. Canonical QFT has been spectacularly
successful not only for making precise predictions but, as Klaczynski
emphasizes, “predict[ing] the existence of hitherto unknown particles
[Klaczynski, 2016, 2]. Nevertheless, canonical QFT is mathematically ill-
defined: “canonical QFT presents itself as a stupendous and intricate jigsaw
puzzle. While some massive chunks are for themselves coherent, we shall see
that some connecting pieces are still only tenuously locked, though simply
taken for granted by many practising physicists, both of phenomenological and
of theoretical creed” [Klaczynski, 2016, 2]. One major contributor to this
ill-definedness is owed to the use of renormalization.

On the other hand is constructive QFT, which use operator theory and
stochastic analysis to attempt to construct models of quantum fields in a
mathematically well-defined manner. A number of important results have been
obtained by this research program, including many triviality results that can
be seen as calling into question basic features expected of any rigorous QFT.
Nevertheless, the construction of a renormalizable theory in 4
dimensions—i.e., realistic—has neither been achieved nor seems achievable
using current methods [Klaczynski, 2016, 3]. That is, no rigorous and
realistic model exists. Klaczynski’s aim is to reconcile canonical and
constructive QFT by elucidating the coherence brought about by
renormalization.

##### Assessment.

Given the above, Klaczynski approaches Haag’s theorem intent on understanding
what it says about canonical QFT. At first blush, the result appears to be
negative. For instance, the Gell-Mann-Low formula, relating the ground states
of the interacting and non-interacting fields, is built on the assumption that
the time-evolution operator in the interaction picture, which relates the two
fields, is unitary; this is exactly what Haag’s theorem rules out (recall note
7 on the significance and wide-spread use of the Gell-Mann-Low formula.).
However, on closer inspection, the contradiction is resolved, if nevertheless
unfavorably: like Fraser and Duncan, Klaczynski blames Haag’s theorem for the
divergence of the perturbative expansion of the Gell-Mann-Low formula. This
leads Klaczynski to conclude that the interaction picture is ill-defined and
trivial [Klaczynski, 2016, 59].

While he points to similar symptoms as Duncan and Miller, Klaczynski’s
conclusion is more severe. In his final assessment, the interaction picture
itself, relying as it does on a unitary intertwiner, is flawed—even
renormalization does not save the interaction picture. While standard
regularization methods may break Poincaré invariance, this is physically
unacceptable; moreover, Poincaré invariance broken by regularization is
restored when we take the adiabatic limit. This means that Haag’s theorem
again applies, albeit now to the renormalized theory [Klaczynski, 2016, 62–3].

##### Repair.

Like Duncan and Miller, Klaczynski thinks that renormalization procedure still
repairs the problem, but it works for a different reason. When we renormalize
our theories, we replace the bare quantities with their renormalised
counterparts. This process of renormalization does not merely cancel the
divergences, but also fundamentally changes the theory, by bringing about a
coupling-dependent mass shift. As a result, the renormalized free and
interacting fields now have different masses. Two quantum fields of different
mass are overwhelmingly likely to be unitarily inequivalent252525The
mathematical ill-definedness of the renormalized terms makes it hard to prove
that renormalized free and interacting fields will be unitarily inequivalent
in general. However, theorem X.46 of [Reed and Simon, 1980] is used to argue
that it is “plausible beyond doubt” [Klaczynski, 2016, page 68].. As such, the
renormalized interacting and free theories are almost certainly unitarily
inequivalent, and so the conditions for Haag’s theorem do not apply in our
renormalized theories.

Nevertheless, the un-renormalized and renormalized theories are still
related—just not by a unitary transformation, as physicists still believe.
Thus, we are working with something like the interaction picture insofar as
the two theories are still related by an intertwiner of sorts, but unlike the
interaction picture insofar as this intertwiner is manifestly non-unitary. So
whilst renormalization is the correct treatment, physicists have not fully
grasped how it allows us to evade Haag’s theorem. By using renormalization to
fix the divergences, physicists have “muddled our way through to successfully
applying perturbation theory” [Klaczynski, personal communication, Sept. 8,
2021].

##### Renovation.

Given that the interaction-picture presentation of canonical QFT still
dominates, renovation is necessary. This has two parts. First, physicists have
misunderstood, or at least mis-described, the tools they are using when
performing scattering calculations with renormalized fields. While Haag’s
theorem says this tool cannot be the interaction picture, that does not mean
renormalization is unconstrained or incoherent. Rather, renormalization
“follows rules which have a neat underlying algebraic structure [the Hopf
algebra] and are not those of a random whack-a-mole game” [Klaczynski, 2016,
page 4].

Second, this program of identifying the structure of renormalization must
continue. First and foremost, there are some mathematical lacunae in this
process, as well as procedures that are not defined wholly rigorously; these
should be addressed. However, at the end of the day, Klaczynski believes that
renormalized quantum field theory “provides us with peepholes through which we
are allowed to glimpse at least some parts of that ‘true’ theory” [Klaczynski,
2016, page 4]. As such, the ultimate goal here would be to glimpse what we can
of the ‘true’ theory through the peepholes this work affords.

###  4.5 Maiezza and Vasquez

##### Extra-Haagian outlooks.

Like Klaczynski, Maiezza and Vasquez are interested in determining the precise
mathematical structure of canonical QFT. However, they seek a “consistent and
generic (non-perturbative) formulation of QFT” [Maiezza and Vasquez, 2020, 2],
in contrast to the perturbative formulation of canonical QFT. Given the
centrality of the interaction picture to canonical QFT, the natural starting
point for finding such a consistent and generic formulation of QFT would be to
use the interaction picture, starting with free fields. But, Maiezza and
Vasquez ask, is it possible to build such a non-perturbative formulation from
the interaction picture and perturbative renormalization? Or, rather, is
something new needed?

##### Assessment.

In Maiezza and Vasquez’s assessment, Haag’s theorem reveals a central problem
when we try to improve upon the standard methods of perturbatively
renormalized canonical QFT. We see this problem clearly when we consider the
entire perturbative expansion series. The problem stems from vacuum
polarization: because of vacuum polarization, the full power-series expansion
diverges262626This was first noted in [Dyson, 1952] concerning QED.. Maiezza
and Vasquez prove that renormalon singularities arise in the total
perturbative series, and trace these back to Haag’s theorem. These renormalon
singularities “are the concrete manifestation” of the impossibility of
generating an unambiguous finite result in such cases [Maiezza and Vasquez,
2020, page 10]. 272727Renormalons are singularities that arise as a function
of the complex transform parameter when a formally divergent series is summed
using Borel summation, see [Beneke, 1999] for a review. Attempts to generate a
finite result for the whole series through analytic continuation methods (such
as using a Borel-Laplace transformation) necessarily rely on a choice of
arbitrary constants. Maiezza and Vasquez describe these dependencies on an
arbitrary choice as “renormalization ambiguities.” Thus, because of these
ambiguities, ultimately arising from Haag’s theorem, the interaction picture
with perturbative renormalization cannot lead to their desired non-
perturbative QFT.

##### Repair.

According to Maiezza and Vasquez, there is no repair we can make at this time
to address this problem. There remains a concerning flaw in perturbative
renormalization, namely, the renormalized series’ dependence on an arbitrary
choice of constant. As Maiezza and Vasquez put it, perturbative
renormalization “cannot be a self-consistent cure, because perturbative
renormalization needs to be completed, or in practice resummed” [Maiezza and
Vasquez, 2020, page 10].

In particular, they argue that the repair suggested by Klaczynski—essentially,
to replace the interaction picture’s unitary intertwiner with a non-unitary
one—cannot work. The Dyson operator relating the free and interacting fields
is unitary by construction, in such a way that simply undoing the time-ordered
product present in its definition (as suggested by [Klaczynski, 2016]) does
not therefore make it non-unitary [Maiezza and Vasquez, 2020, 5]. Something
more is needed to repair the interaction picture, in particular to make it
suitable for calculations using non-perturbative QFT.

That said, Maiezza and Vasquez are quick to note that the practical uses made
of the interaction picture with perturbative renormalization need no repair:
they evade Haag’s theorem. Haag’s theorem is, after all, a non-perturbative
result, in that it applies to the entire power-series expansion, not just the
perturbative expansion at some finite order. Because the power-series
expansion is asymptotic, truncating at finite orders can give meaningful
approximations, while also missing out on the non-perturbative effects that
lead to the complete series’ divergence. Thus, while regularization and
renormalization do not ‘fix’ the problem posed by the divergences tracked by
Haag’s theorem, this is essentially because Haag’s theorem does not apply in
cases where the series is truncated. Perturbative renormalization fixes the
divergences of individual terms within the series, but it does not address the
non-perturbative divergences of Haag’s theorem.

##### Renovation.

They suggest two forms of renovation. First, in the short run, research
efforts should aim at improving our understanding of the non-perturbative
singularities that arise, such as renormalons. Further research should be
directed into mathematical techniques to better understand neglected non-
perturbative effects (for example [Maiezza and Vasquez, 2021b, Maiezza and
Vasquez, 2021a]). These include resurgence, a method for reading off certain
non-perturbative effects from the perturbative expansions (for a review, see
[Dunne and Ünsal, 2016]).

Second, further work is needed work _within_ canonical QFT to account for non-
perturbative effects. Ultimately, new physical insights are needed to complete
quantum field theory. These must move beyond simply reformulating the
perturbative quantum field theory framework [Maiezza and Vasquez, personal
communication, Sept. 8, 2021].

###  4.6 Kastner

##### Extra-Haagian outlooks.

[Kastner, 2015]’s outlook differs from the previous authors. Insofar as she
seeks the correct formulation of our theory of quantum phenomena, she sits
with the physicists above. However, insofar as she thinks this correct
formulation requires a fundamental reinterpretation of quantum phenomena, she
sits with the fundamental interpreters, who are overwhelmingly philosophers.
Crucially, she has already given up on mediating interactions locally in favor
of nonlocal, direct interactions between field sources. This is a profound
departure from QFT.

##### Assessment.

Naturally, given her prior commitment to abandoning QFT, Kastner’s assessment
of QFT in the light of Haag’s theorem is grim: “QFT is not the correct model;
a different, yet empirically equivalent, model is needed” (57). This goes
beyond others’ assessments locating the problem specifically with the
interaction picture. Rather, Kastner locates the problem ‘further back’
conceptually. For her, QFT is wrong from outset: the foundational assumption
of QFT—namely, that we use fields as mediators between particles (crudely, as
“a ‘bucket brigade’ that is invoked in order to restrict causal influence to a
local, continuous conveyance from spacetime point to spacetime point” (59))—is
the problem. The revival of direct-action theories (DATs) is the natural
solution for that problem. Thus, for DAT advocates such as Kastner, Haag’s
theorem “simply tells us what we already know: the interaction picture of
quantized fields does not really exist” [Kastner, 2015, 58]. Haag’s theorem is
therefore an additional motivation for abandoning QFT in favor of DATs.

##### Repair.

The treatment for such a deep problem is a major change in the modeling
procedure for this area of physics. If we accept that Haag’s theorem shows us
that it simply does not work to model interactions by field operators creating
and destroying Fock space states, then, Kastner urges, we are to instead model
interactions through a direct, nonlocal interaction between sources of the
field [Kastner, 2015, 57]. On this direct-action approach, one does away with
fields as mediators of interactions, replacing them with non-local direct
actions between charge sources. The key ingredient for avoiding Haag’s
theorem, according to Kastner, is that real and virtual particle propagation
be treated distinctly. In particular, this means that virtual processes should
not be assigned statehood, as they are in traditional QFT.

Kastner believes that this is a viable repair because she is convinced that
DATs—which differ radically from QFTs—are empirically equivalent to QFTs. This
places significant weight on (purported) demonstrations of the equivalence of
direct-action theories and QFT (e.g., [Narlikar, 1968]). If the demonstrations
are sound, and they extend such that all currently-used QFTs are
approximations to some DAT, then they would go some way towards explaining how
calculations in renormalized quantum field theories are able to generate
successful predictions: QFT’s “mathematical inconsistencies can be rendered
inconsequential since they can be understood as arising from its ‘makeshift,’
nonfundamental character” [63].282828However, this is a non-trivial ‘if’,
especially given the recent results suggesting that ‘empirical equivalence’ is
a less straightforward concept than philosophers often presume. See
[Weatherall, 2019] and references therein for more detail.

Kastner further invokes the abandonment of the idea of the ether due to the
Michelson-Morley experiment to justify abandoning QFT, saying that abandoning
interactions through local quantum fields in response to Haag’s theorem would
have an analogous “interpretive elegance” (58). She also quotes Wheeler (who
co-developed with Feynman a direct-action theory of electromagnetism) and
Wesley, who liken Haag’s theorem to the EPR experiment insofar as it presents
a serious challenge to the assumption that the laws of nature are entirely
local. We repair all these problems, including those raised by Haag’s theorem,
by abandoning locality in favor of DATs.

##### Renovation.

The prescribed renovation, on Kastner’s view, is to develop a research program
for DATs. This research program includes the rejection of alternative
responses to Haag’s theorem such as Haag-Ruelle scattering and constructive
QFT; Kastner views these workarounds as “ad hoc, approximate, or partial
measures” (63). Future work in the research program would need to assess the
question of the calculational, pragmatic viability of DATs as well as the
crucial need to check thoroughly for empirical equivalence with all of the
accepted theoretical results of ordinary QFT. Kastner cites work from Narlikar
and Davies developing a DAT empirically equivalent to QED; more work is needed
to determine if other sectors of the Standard Model of particle physics can be
recovered in the DAT approach. Furhter rennovation efforts should also explore
the possibility that a DAT approach would lead to novel, empirically testable
predictions, as much of theoretical particle physics is currently disappointed
with the lack of direction from particle accelerator experiments for new lines
of research.

Given the wide extent of Kastner’s proposed work—not least, replacing the
entire Standard Model with DATs—it would perhaps be more appropriate to
describe the proposal as an entirely new foundation than as a renovation of
the one that exists.

###  4.7 Seidewitz

##### Extra-Haagian outlook.

Seidewitz’s outlook is interesting in that it substantially coincides with
several of the others already discussed, while at the same time departing from
them. First, he seems to think of theory interpretation in a way similar to
Earman and Fraser and the fundamental interpreters. Like the latter, he
appears to take mathematical rigor as a precondition for providing a
satisfactory interpretation [Seidewitz, 2017, 369]. However, unlike Earman and
Fraser, Seidewitz is not satisfied taking the axioms of AQFT as gospel;
indeed, his theory substantially revises these axioms in light of his
conviction that the former axioms are an incorrect marriage of special
relativity with quantum theory. This departs from Earman and Fraser insofar as
Seidewitz is not merely marrying special relativity and quantum theory as they
were traditionally understood. Rather, Seidewitz shares with the physicists a
willingness to revise our theories in an effort to find the correct
unification of them. Seidewitz also departs from the fundamental interpreters
insofar as a univocal characterization of particle-ness is not forthcoming.
This is because particles now come in two varieties—virtual and real—and a new
definition of ‘particle’ will have to account for the significant mathematical
distinction between the two.292929In particular, we should expect (an analog
of) the (global) number operator characterization to hold only for real
particles, given that the space-like evolution possible for virtual particles
prevents their characterization as Fock space states to be created or
destroyed (where this is done by operators used to define number operators).
While a univocal characterization of each variety of particle may still be
forthcoming, nevertheless this is a departure from the expectations of
fundamental interpreters.

Second, Seidewitz’s outlook coincides with Kastner’s. On the one hand,
processes can evolve in a space-like way. That is, physics is non-local.
Specifically, virtual processes need not evolve in a local fashion. On the
other hand, particles come in two varieties—virtual and real. Indeed, the two
agree on how these varieties are constituted with respect to locality.
However, Seidewitz disagrees with Kastner that the non-locality of physics is
a reason to abandon field theories. Rather, he suggests an alternative
formulation of quantum field theory.

##### Assessment.

Thus, [Seidewitz, 2017]’s assessment of Haag’s theorem is that it is the
symptom of a larger problem, namely the inconsistent treatment of time in
traditional QFT. Seidewitz reads Haag’s theorem as a corollary of two previous
results, where these previous results are the more significant. First, let
$\hat{\psi}_{1}(x,t)$ and $\hat{\psi}_{2}(x,t)$ be field operators defined in
Hilbert spaces $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$, respectively, and
suppose there exists a unitary operator $\hat{G}$ such that, at a specific
time $t$, $\hat{\psi}_{2}(x,t)=\hat{G}\hat{\psi}_{1}(x,t)\hat{G^{-1}}$. Then
(Theorem 1) the equal-time vacuum expectation values of the fields at time $t$
are the same. Second (Theorem 2), a given field’s two-point expectation values
satisfy a certain equation (Eq. (3) in Seidewitz) iff that field is free.
Letting $\hat{\psi}_{1}$ be a free field and $\hat{\psi}_{2}$ a field related
to the former as in Theorem 1, Haag’s theorem merely observes that
$\hat{\psi}_{2}$ also satisfies Theorem 2’s equation for spacetime points at
time $t$ (by Theorem 1) and, by the Lorentz-covariance of $\hat{\psi}_{2}$ and
analytic continuation, extends $\hat{\psi}_{2}$’s satisfaction of the equation
to any two positions.

As Seidewitz sees it, Haag’s theorem “essentially relies on a conflict between
the presumption that the fields are Lorentz-covariant and the special
identification of time in the assumptions of Theorem 1” [Seidewitz, 2017,
360]. This special identification is apparent in that the time-evolution is
given by the (frame-dependent) Hamiltonian. This requires particles to evolve
on time-like trajectories. Essentially, the (frame-dependent) Hamiltonian
operator is playing double-duty as the generator of time translation, in
addition to the generator of state evolution, with $t$ as the evolution
parameter for each. However, because of this, the translation group of $t$—the
unitary operators $\hat{G}(t)$—coincides with the group of Lorentz
transformations, which guarantees that one can extend the coincidence
$\hat{\psi}_{2}(x,t)=\hat{G}\hat{\psi}_{1}(x,t)\hat{G^{-1}}$ at a specific
time $t$ to all times $t$.

##### Repair.

The immediate repair is to break the identification of the Hamiltonian with
the generator of time translation and the energy observable. Seidewitz
proposes we do so by dropping (i) the requirement that there is a unique
Poincaré-invariant vacuum state and (ii) the spectral condition, i.e., the
requirement that states are on shell. Essentially, this involves dropping the
assumption that states transform according to an irreducible representation of
the Poincaré group. This frees us up to define vacuum states as relative to
the choice of a Hamiltonian operator, where the Hamiltonian is no longer
equated with the time-translation generator. Instead, a Hamiltonian is
required only to be Hermitian, commute with all spacetime translations, and
have a unique null eigenstate. Then, one can consider the one-dimensional
group generated by each Hamiltonian, where $\lambda$ (instead of $t$) is the
(now frame-independent!) evolution parameter. (Physically, $\lambda$ can be
thought of as the parameterization of a particle’s path in spacetime—now
unburdened from the constraint that the path be timelike.) Haag’s theorem is
evaded by limiting the equality of vacuum expectation values (Theorem 1) to
equal values of $\lambda$, which is neither surprising—each Hamiltonian has a
unique Poincaré invariant vacuum state—nor problematic—one can no longer use
Lorentz transformations to extend the equality any farther. In particular, one
cannot demonstrate that the fields coincide (Haag’s theorem). Thus,
interacting fields need not coincide with free fields, and yet a version of
the interaction picture is reinstated since interacting fields can be related
to free fields by a unitary intertwiner [Seidewitz, 2017, 369].

##### Renovation.

Like Kastner, the renovation Seidewitz proposes is far-reaching. Most
importantly, it will involve expanding the reach of parameterized QFT. As it
stands, the theory does not cover gauge theories or non-Abelian interactions,
nor does it resolve all of the problems with standard QFT. As such,
significant work is required before this approach is of practical use.
Nevertheless, insofar as it is conceptually closer to constructive QFT, one
might reasonably suspect that the renovations required will be less thorough-
going than those proposed by Kastner.

###  4.8 Framework Results Summary

Our framework accomplishes two main objectives. First, it maps out an
otherwise wild jungle of scholarship on Haag’s theorem. This organization is
especially helpful as groundwork for making the interdisciplinary exchanges of
ideas on Haag’s theorem more efficient: it is not always easy for physicists,
philosophers, and mathematicians to communicate. This is in part due to
appropriately different aims, as well as differences in how these separate
communities have developed their own forms of discourse, vocabulary, and
standards of rigour.

Second, our framework goes beyond the organization of viewpoints on Haag’s
theorem by pulling each viewpoint’s underlying disciplinary and methodological
values to the foreground (see again table 2). How one diagnoses and treats
Haag’s theorem is profoundly influenced by what one expects of QFT, and of the
interaction picture in particular, as well as one’s expectations of
theoretical physics in general. One’s expectations for the mathematician’s and
the philosopher’s appropriate relationship to theoretical physics also has an
influence. Each response discussed here is shaped by the authors’ background
commitments, research purposes, and expectations of theoretical physics—what
we have called Extra-Haagian outlooks. A breakdown summary of these Extra-
Haagian outlooks is given in table 2.

Each viewpoint surveyed and assessed above has a distinctive purpose. Duncan
and Miller are both concerned with executing a very specific explanatory task
concerning the successful implementation of the interaction picture despite
Haag’s theorem, in the current practice of physics. Klaczynski is nearby doing
a similar explanatory task of physical practice, but does not see the result
as having saved the interaction picture. In contrast, Kastner and Seidewitz
are interested in developing genuinely new physical theories. Maiezza and
Vasquez are somewhere in between these previous two, seeking to explain how
Haag’s theorem eventually leads to renormalon divergences. They use this
explanation to suggest further avenues for mathematical research that can make
progress on the foundations of this area of physics. In this sense, their
research aims at developing valuable technical groundwork for new physical
theories, while stopping short of offering a new theory as Kastner and
Seidewitz do. Finally, Earman and Fraser and the fundamental interpreters are
doing the distinctly philosophical work of sorting out the downstream
interpretive implications of Haag’s theorem in conversation with the wider
goals and conceptual foundations of QFT. Bringing these separate research aims
to the surface helps to address the confusions described in section 3.

Name |  |  Extra-Haagian Outlooks
---|---|---
Earman & Fraser |  |  QFT is, by definition, the unification of SR and QM. Therefore, giving up any postulate from either theory in order to defuse Haag’s theorem is not an acceptable response.
Fundamental Interpreters |  |  The larger project is one of fundamental theory interpretation. Hence, a satisfying interpretation must apply across all models of QFT. AQFT’s axiomatic structure makes it the preferred formulation for interpretive work.
Duncan & Miller |  |  The question of focus is: How can it be that actual calculations in theoretical physics, based on the interaction picture, have been successful despite Haag’s theorem? This is a specific question for explaining the success of current practices, to be contrasted with more foundational questions in QFT.
Klaczynski |  |  The question is how can we reconcile the success of scattering calculations with the consequences Haag’s theorem? If we are committed to keeping the axioms of QFT, then all that is left to revise is the interaction picture itself.
Maiezza & Vasquez |  |  We need to understand the implications of Haag’s theorem for physically realistic quantum field theories. A perturbative procedure cannot give a complete answer to the problems posed by Haag’s theorem. We must also attempt to trace the implications of Haag’s theorem for non-perturbative methods.
Kastner |  |  Kastner has a background commitment to the existence of non-local interactions. More generally, Kastner has a background expectation that much of mainstream formulations and interpretations of QFT are misguided. In this light, Haag’s theorem becomes one more piece of evidence that the foundations of this area of physics need major revisions.
Seidewitz |  |  QFT incorrectly treats time as an evolution parameter. This can be fixed by adding an evolution parameter, resulting in a new and improved field theory: parameterized QFT. The theory is mathematically rigorous, hence is suitable for interpretation. In particular, the theory is non-local.
Table 2: Extra-Haagian outlooks summary.

Finally, in bringing these background values and guiding aims to the
forefront, our framework exposes, for general methodological consideration,
several desiderata for any satisfactory theory of scattering experiments.
First, we must be able to generate a consistent system for modeling
interactions, that is, we must escape the inconsistency threat to the
interaction picture demonstrated by Haag’s theorem. Beyond that, we require
the ability to consider interactions with either massive or massless particles
(an issue raised in [Earman and Fraser, 2006], [Klaczynski, 2016], and
[Kastner, 2015]), as well as a modeling procedure that is can be implemented
in a realistic number of spacetime dimensions—the longstanding obstacle for
constructive program within AQFT. Some in the philosophy community also
require of QFT that it be formulated in a way amenable to traditional
interpretive methods (e.g [Fraser, 2011] and [Halvorson and Clifton, 2002]).
Finally, amongst those who take Haag’s theorem as additional fuel on the fire
of theory change, more far afield concerns over the desire to solve seemingly
unrelated problems become relevant, such as doing away with gauge
arbitrariness and the potential of a new formulation of QFT to lead to
unification with gravity. [Kastner, 2015] and [Seidewitz, 2017] seem to have
these considerations in the forefront of their thinking, and [Maiezza and
Vasquez, 2020] perhaps have them in the background. And all of this is, of
course, in addition to the basic requirement of empirical adequacy.

##  5 Cartography of the QFT Foundations Labyrinth

From what has been said so far, one may be drawn towards ordering the
different viewpoints surveyed here, perhaps along a scale of radical-ness of
treatment or concern for mathematical rigor or some other axis. We want to
emphasize that it is a feature of our framework that it is sufficiently
flexible to allow for multiple different orderings on this space of
viewpoints. Our framework gives structure and clarity to the literature on
Haag’s theorem, but it intentionally does not fix any one line of debate as
the most important issue, nor does it settle any one of the many lines of
debate to be drawn. The framework thus makes it possible to draw these lines
of debate clearly, as if it were producing competing maps of the labyrinth of
conceptual issues at the foundations of QFT. In this section we give three
such maps.

In table 3, we map viewpoints in terms of their response to the bare logical
problem posed by the contradiction between Haag’s theorem and the interaction
picture: Which assumptions should be abandoned in order to escape the
contradiction? On this mapping, some authors locate the problem in the QFT
axioms while others point to key assumptions in the interaction picture.
Within these two branches, viewpoints further diverge. Note, too, that some
respondents’ viewpoints are left out, in virtue of not responding to Haag’s
theorem in this way. In particular, neither Maiezza and Vasquez nor the
fundamental interpreters have tended to respond to Haag’s theorem by
considering the rejection of (some of) its assumptions.

While the bare logical problem raised by Haag’s theorem is widely agreed upon,
it is not the only illuminating way to survey the terrain of the literature on
Haag’s theorem. In figure 1 we give an alternative mapping according to the
driving motivation in each response: some authors are primarily motivated by
questions about conceptual foundations/interpretation, whereas others are
motivated by a desire to explain and assess physics practice. Authors further
diverge according to which issue at the foundations or within physics practice
is most pressing. Extra-Haagian outlooks play a key role in this mapping.

Figure 2 shows a third option for mapping out the different positions in terms
of which disciplinary speciality is best suited for the kind of repair and
maintenance work needed. Loosely, we can conceive of three general areas where
energy could be directed in response to Haag’s theorem. First, energy could be
put into developing new mathematical techniques. This could include new
mathematical models, renormalization methods, and methods such as
regularization and resurgence to better deal with the asymptotic series.
Second, energy could be put into theoretical physics, such as studying cutoff
invariance, beta functions, or more generally seeking out new physical
insights. Third is philosophical work readdressing how we interpret quantum
field theory (or whatever theory should replace it). However, none of these
areas are clear cut and all overlap somewhat.

These three mappings are likely not exhaustive. They are meant to illustrate
the value of our framework in simultaneously giving a fixed structure for
understanding each response (the structure of assessment/repair/maintenance &
renovation), while also leaving enough conceptual room to accommodate these
various further cartographic options.

The mapping in figure 1 is particularly helpful in that it reveals where there
are points of genuine disagreement. Moving upwards through the diagram, the
first disagreement is about whether, and how, regularization and perturbative
renormalization allow us to circumvent Haag’s theorem. Here, Duncan and
Miller, Klaczynski, and Maiezza and Vasquez each come to contradicting
conclusions. According to Duncan and Miller, the regularization process allows
us to circumvent Haag’s theorem by temporarily breaking the Poincare
invariance of the theory. However, Klaczynski argues that Haag’s theorem
cannot be circumvented in this way. Instead, he argues that the fault lies
with the interaction picture: once we have fully renormalized our theory,
there is no longer a unitary intertwiner. Maiezza and Vasquez assert that the
intertwiner is unitary by definition, thus Haag’s theorem cannot be fully
evaded by a perturbative procedure like renormalization. The problems of
Haag’s theorem still appear as divergences when we consider the entire
perturbative series of our renormalized theory. These three views cannot all
be correct.

A second clear disagreement emerges with regards to the status of conventional
quantum field theory. Both Kastner and Seidewitz propose alternatives to
conventional QFT. Seidewitz proposes modifying the axioms of QFT, in which we
drop the requirement that states transform according to an irreducible
representation of the Poincaré group. In Kastner’s proposal, we replace
quantum field theory with a direct action theory, in which interactions take
place non-locally between charge-sources. Even if both approaches can
recapture our empirical data, they are fundamentally different theories, built
on different assumptions. They cannot both be right.

Having now thoroughly and multiply mapped out the Haagian labyrinth, the
question remains: how do we get out? The foregoing suggests that this is not
so simple. Each proposed path is sensible enough on its own. However, as the
last two paragraphs emphasize, the paths cannot all overlap—a non-trivial
choice must still be made. To make this choice, we require further
information. We have highlighted the extra-Haagian outlooks throughout this
section because they together provide a glimpse of where that further
information should come from.

The central lesson of our framework is this: charting a way out of the Haagian
labyrinth will require us to answer meta-level questions of what foundational
work is, and about what it does for physics. We highlight the following three
as central:

  1. 1.

What role does (should) foundational work play in progress in physics?

  2. 2.

How does (should) foundational work coordinate with non-foundational work?

  3. 3.

And, what does (should) foundational work even look like?

All those who are lost in the Haagian labyrinth would agree that they are
doing work at the foundations of physics. Doing this work inevitably involves
taking a stand on these meta-level questions. Nevertheless, these questions
are rarely addressed head-on. However, doing so is necessary in order to
adjudicate between different escape routes from the Haagian labyrinth.

We can phrase this in terms of our general contractor metaphor, too. You have
received all of your subcontractors’ assessments, proposed repairs, and any
renovation plans they recommend for your QFT building in response to Haag’s
theorem. Each has even toured the building with you to evidence their
assessment and lobby for their proposal. Nevertheless, you remain unsure of
how to proceed: you want to ensure the building’s integrity as its rooms
continue to expand, but precisely how the building will evolve is beyond you.
Our advice as to how to make an informed decision is to reconsider the
fundamental design principles that inform the relationship of the building and
its rooms to its foundation. Indeed, you may also investigate that
relationship in similar buildings, past and present.

Name |  |  Problem Source |  |  We should give up on…
---|---|---|---|---
Duncan & Miller |  |  QFT Axioms |  |  exact Poincaré invariance.
Kastner |  |  QFT Axioms |  |  independent fields.
Seidewitz |  |  QFT Axioms |  |  time as evolution parameter.
Klaczynski |  |  Interaction Picture |  |  a unitary intertwiner.
Earman & Fraser |  |  Interaction Picture |  |  the entire IP. 303030There are several different ways of doing this. For instance, Haag–Ruelle scattering gives up the idea of building out from free-field representation. Alternatively, LSZ formalism. [Earman and Fraser, 2006], [Duncan, 2012, pages 268-281]
Table 3: Diagnoses of those who take the import of Haag’s theorem to be the
identification of a contradiction in the foundations of physics, specifically
between the axioms of QFT and the assumptions of the interaction picture.

For our own part, we plan to carry out the following investigations. First,
how should we think about the relationship between (foundational) no-go
theorems and non-foundational physics? We believe there is a more constructive
way of construing no-go theorems that would more closely and more fruitfully
connect foundational and non-foundational work. Second, what distinguishes
fruitful uses of axiomatization from unfruitful uses in physics? Here we plan
to investigate the origins of, and expectations for, the Wightman axioms and
Haag’s theorem. Finally, how precisely are the assumptions of Haag’s theorem
undermined using the various regularization, renormalization, and resummation
methods? Duncan and Miller offer good reason to believe that at least one
assumption is undermined at some point during many calculations, but they
neither verify this for all methods nor characterize precisely where in a
given calculation the assumptions are undermined. Both have implications for
the significance of Haag’s theorem.

In most general sense, what “game” are we playing? Investigate the union of SR
and NRQM Explain/assess/improve theoretical physics in practice What is the
logical structure of the foundations of QFT? What is the fundamental ontology
of ‘particle’ physics? Earman & Fraserfundamental interpreters Does Haag’s
theorem call for radical revisions to QFT? Yes No SeidewitzKastner Do current
perturbative renormalization techniques circumvent Haag’s theorem? Yes:
Regularization breaks Poincaré invariance Existing practice with IP is
salvaged Duncan & Miller Yes: Renormalization entails a non-unitarity
intertwiner We need to replace the IP Klaczynski No: Need to consider the
whole perturbative expansion Will require new physical insight Maiezza &
Vasquez Figure 1: A mapping of the responses to Haag’s theorem according to
the driving motivations.

What kind of work do the repairs, maintenance and/or renovations call for?
MathematicsPhysicsPhilosophyRenormalizationModelsEffective Field
TheoryPhysical InsightCutoff InvarianceNew Theoretical CommitmentsFundamental
InterpretationNew Interpretive MethodsSeidewitzEarman & FraserMaiezza &
VasquezKlaczynskiDuncanDuncan & MillerMaiezz & VasquezEarman &
FraserWallaceRuetscheBainKastner Figure 2: What kind of work do the repairs,
maintenance and/or renovations call for?

##  6 Conclusion

Haag’s theorem cries out for explanation and critical assessment: it sounds
the alarm that something is (perhaps) not right in one of the standard way of
constructing interacting fields to be used in generating predictions for
scattering experiments. Viewpoints as to the precise nature of the problem
(assessment), the appropriate solution (repair), and subsequently called-for
developments in areas of physics, mathematics, and philosophy (maintenance or
renovation) differ widely. Moreover, the extant literature presenting these
differing views constitutes a complex mix of arguments at cross-purposes,
generating substantive confusion as to the precise issues to be addressed. In
this paper, we have worked to address this confusion by cataloging and
comparing a number of these viewpoints, and we have developed and applied a
framework for understanding these distinct viewpoints. The application of our
framework reveals each authors’ background disciplinary and methodological
commitments and expectations of QFT—what we have termed ‘extra-Haagian’
outlooks. While regimenting and structuring the literature, our framework does
not fix any one line of debate regarding viable or preferable responses to
Haag’s theorem. Rather, within our framework we have mapped out three distinct
ways of drawing the possible lines of debate, and we hope that these maps will
prove useful for future work on many issues within the labyrinth of the
foundations of QFT.

Finally, in light of these maps of the Haaging labyrinth, charting a way out
requires that we turn our attention to meta-level questions regarding the
nature of foundational work in physics. What role does (should) foundation
work play in the progress in physics? How does (should) foundational work
coordinate with non-foundational work? What does (should) foundational work
even look like? Each practitioner, be they physicist, mathematician, or
philosopher, must have at least some implicit answer to these questions in
order to find a path out. It is perhaps more seriously incumbent upon the
philosopher to develop and debate explicit answers to these questions.

## Bibliography

  * [Aizenman and Duminil-Copin, 2021] Aizenman, M. and Duminil-Copin, H. (2021).  Marginal triviality of the scaling limits of critical 4D Ising and $\lambda\phi_{4}^{4}$ models.  Annals of Mathematics, 194(1):163 – 235.
  * [Aoyama et al., 2015] Aoyama, T., Hayakawa, M., Kinoshita, T., and Nio, M. (2015).  Tenth-order electron anomalous magnetic moment: Contribution of diagrams without closed lepton loops.  Physical Review D, 91(3).
  * [Arageorgis, 1995] Arageorgis, A. (1995).  Fields, particles, and curvature: Foundations and philosophical aspects of quantum field theory in curved spacetime.  PhD thesis, University of Pittsburgh.
  * [Bain, 2000] Bain, J. (2000).  Against particle/field duality: Asymptotic particle states and interpolating fields in interacting qft (or: Who’s afraid of haag’s theorem?).  Erkenntnis, 53(3):375–406.
  * [Bain, 2011] Bain, J. (2011).  Quantum field theories in classical spacetimes and particles.  Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 42(2):98–106.
  * [Bain, 2013] Bain, J. (2013).  Emergence in effective field theories.  European journal for philosophy of science, 3(3):257–273.
  * [Baker, 2009] Baker, D. J. (2009).  Against field interpretations of quantum field theory.  The British Journal for the Philosophy of Science, 60(3):585–609.
  * [Beneke, 1999] Beneke, M. (1999).  Renormalons.  Phys. Rept., 317:1–142.
  * [Brodsky and Farrar, 1975] Brodsky, S. J. and Farrar, G. R. (1975).  Scaling laws for large-momentum-transfer processes.  Phys. Rev. D, 11:1309–1330.
  * [Chaiken, 1968] Chaiken, J. M. (1968).  Number operators for representations of the canonical commutation relations.  Communications in Mathematical Physics, 8(2):164–184.
  * [Crispino et al., 2008] Crispino, L. C., Higuchi, A., and Matsas, G. E. (2008).  The unruh effect and its applications.  Reviews of Modern Physics, 80(3):787.
  * [Douglas and Kroll, 1974] Douglas, M. and Kroll, N. M. (1974).  Quantum electrodynamical corrections to the fine structure of helium.  Annals of Physics, 82(1):89–155.
  * [Duncan, 2012] Duncan, A. (2012).  The conceptual framework of quantum field theory.  Oxford University Press.
  * [Dunne and Ünsal, 2016] Dunne, G. V. and Ünsal, M. (2016).  New Nonperturbative Methods in Quantum Field Theory: From Large-N Orbifold Equivalence to Bions and Resurgence.  Ann. Rev. Nucl. Part. Sci., 66:245–272.
  * [Dyson, 1952] Dyson, F. J. (1952).  Divergence of perturbation theory in quantum electrodynamics.  Phys. Rev., 85:631–632.
  * [Earman, 2011] Earman, J. (2011).  The unruh effect for philosophers.  Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, 42(2):81–97.
  * [Earman and Fraser, 2006] Earman, J. and Fraser, D. (2006).  Haag’s theorem and its implications for the foundations of quantum field theory.  Erkenntnis, 64(3):305.
  * [Feintzeig et al., 2021] Feintzeig, B. H., Librande, J., and Soiffer, R. (2021).  Localizable particles in the classical limit of quantum field theory.  Foundations of Physics, 51(2):1–31.
  * [Fraser, 2008] Fraser, D. (2008).  The fate of ‘particles’ in quantum field theories with interactions.  Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Modern Phys., 39(4):841–859.
  * [Fraser, 2009] Fraser, D. (2009).  Quantum field theory: Underdetermination, inconsistency, and idealization.  Philosophy of Science, 76(4):536–567.
  * [Fraser, 2011] Fraser, D. (2011).  How to take particle physics seriously: A further defence of axiomatic quantum field theory.  Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, 42(2):126–135.
  * [Fraser, 2017] Fraser, J. (2017).  The real problem with perturbative quantum field theory.  The British Journal for the Philosophy of Science, 71.
  * [Gell-Mann and Low, 1951] Gell-Mann, M. and Low, F. (1951).  Bound states in quantum field theory.  Phys. Rev., 84:350–354.
  * [Haag, 1955] Haag, R. (1955).  On quantum field theories.  Kgl. Danske Videnskab. Selakab, Mat.-Fys. Medd., 29.
  * [Haag, 2010] Haag, R. (2010).  Some people and some problems met in half a century of commitment to mathematical physics.  The European Physical Journal H, 35(3):263–307.
  * [Hall and Wightman, 1957] Hall, D. and Wightman, A. S. (1957).  A theorem on invariant analytic functions with applications to relativistic quantum field theory.  Det Kongelige Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser, 31(5):1–41.
  * [Halvorson and Clifton, 2002] Halvorson, H. and Clifton, R. (2002).  No place for particles in relativistic quantum theories?  Philosophy of science, 69(1):1–28.
  * [Heathcote, 1989] Heathcote, A. (1989).  A theory of causality: Causality= interaction (as defined by a suitable quantum field theory).  Erkenntnis, 31(1):77–108.
  * [Hegerfeldt, 1998a] Hegerfeldt, G. C. (1998a).  Causality, particle localization and positivity of the energy.  In Bohm, A., Doebner, H.-D., and Kielanowski, P., editors, Irreversibility and causality semigroups and rigged Hilbert spaces, pages 238–245. Springer.
  * [Hegerfeldt, 1998b] Hegerfeldt, G. C. (1998b).  Instantaneous spreading and einstein causality in quantum theory.  Annalen der Physik, 7(7-8):716–725.
  * [Jackiw, 1990] Jackiw, R. (1990).  Analysis on Infinite-Dimensional Manifolds—Schrödinger Representation for Quantized Fields, pages 78–143.  World Scientific.
  * [Jost, 1961] Jost, R. (1961).  Properties of wightman functions.  In Caianiello, E., editor, Lectures on the Many-body Problems, pages 127–145. Academic Press.
  * [Kastner, 2015] Kastner, R. E. (2015).  Haag’s theorem as a reason to reconsider direct-action theories.  International Journal of Quantum Foundations, 1:56–64.
  * [Klaczynski, 2016] Klaczynski, L. (2016).  Haag’s theorem in renormalised quantum field theories.  PhD thesis, Humboldt U., Berlin.
  * [Lomon, 1976] Lomon, E. L. (1976).  Summation of two-pion inserts to the $\sigma$-meson propagator and nucleon forces.  Phys. Rev. D, 14:2402–2418.
  * [Lupher, 2005] Lupher, T. (2005).  Who proved haag’s theorem?  International Journal of Theoretical Physics, 44(11):1995–2005.
  * [Maiezza and Vasquez, 2020] Maiezza, A. and Vasquez, J. C. (2020).  On haag’s theorem and renormalization ambiguities.  arXiv preprint arXiv:2011.08875.
  * [Maiezza and Vasquez, 2021a] Maiezza, A. and Vasquez, J. C. (2021a).  Non-Wilsonian ultraviolet completion via transseries.  Int. J. Mod. Phys. A, 36(02):2150016.
  * [Maiezza and Vasquez, 2021b] Maiezza, A. and Vasquez, J. C. (2021b).  Resurgence of the QCD Adler function.  Phys. Lett. B, 817:136338.
  * [Malament, 1996] Malament, D. B. (1996).  In defense of dogma: Why there cannot be a relativistic quantum mechanics of (localizable) particles.  In Perspectives on quantum reality, pages 1–10. Springer.
  * [Miller, 2018] Miller, M. E. (2018).  Haag’s theorem, apparent inconsistency, and the empirical adequacy of quantum field theory.  The British Journal for the Philosophy of Science, 69(3):801–820.
  * [Mohr et al., 1998] Mohr, P. J., Plunien, G., and Soff, G. (1998).  Qed corrections in heavy atoms.  Physics Reports, 293(5):227–369.
  * [Narlikar, 1968] Narlikar, J. V. (1968).  On the general correspondence between field theories and the theories of direct particle interaction.  Proceeding of the Cambridge Phil. Soc., 64.
  * [Partovi and Lomon, 1972] Partovi, F. and Lomon, E. L. (1972).  $\sigma$-exchange potential for nucleon-nucleon scattering.  Phys. Rev. D, 5:1192–1205.
  * [Partovi, 1978] Partovi, M. H. (1978).  Derivation of interaction potentials from field theory.  Phys. Rev. D, 18:2861–2871.
  * [Partovi and Lomon, 1969] Partovi, M. H. and Lomon, E. L. (1969).  Field-theoretical nucleon-nucleon potential.  Phys. Rev. Lett., 22:438–441.
  * [Pohlmeyer, 1969] Pohlmeyer, K. (1969).  Jost-schroer theorem for zero-mass fields.  Commun. Math. Phys., 12: 204-11(1969).
  * [Reed and Simon, 1980] Reed, M. and Simon, B. (1980).  Methods of Modern Mathematical Physics, III: Scattering Theory.  Academic Press.
  * [Reeh and Schlieder, 1961] Reeh, H. and Schlieder, S. (1961).  Bemerkungen zur unitäräquivalenz von lorentzinvarianten feldern.  Il Nuovo Cimento (1955-1965), 22(5):1051–1068.
  * [Ruetsche, 2011] Ruetsche, L. (2011).  Interpreting quantum theories.  Oxford University Press.
  * [Salpeter, 1952] Salpeter, E. E. (1952).  Mass corrections to the fine structure of hydrogen-like atoms.  Phys. Rev., 87:328–343.
  * [Salpeter and Bethe, 1951] Salpeter, E. E. and Bethe, H. A. (1951).  A relativistic equation for bound-state problems.  Phys. Rev., 84:1232–1242.
  * [Schwinger, 1948a] Schwinger, J. (1948a).  On quantum-electrodynamics and the magnetic moment of the electron.  Phys. Rev., 73:416–417.
  * [Schwinger, 1948b] Schwinger, J. (1948b).  Quantum electrodynamics. i. a covariant formulation.  Physical Review, 74(10):1439.
  * [Sebens, 2022] Sebens, C. T. (2022).  The fundamentality of fields.  Synthese, 200(5):380.
  * [Seidewitz, 2017] Seidewitz, E. (2017).  Avoiding haag’s theorem with parameterized quantum field theory.  Foundations of Physics, 47(3):355–374.
  * [Teller, 1995] Teller, P. (1995).  An interpretive introduction to quantum field theory.  Princeton University Press.
  * [Tong, 2012] Tong, D. (2012).  Lectures on Quantum Field Theory.  https://physics.whu.edu.cn/dfiles/wenjian/3_02_QFT_Tong.pdf/.  [Online; accessed 5-Aug-2022].
  * [Wallace, 2001] Wallace, D. (2001).  Emergence of particles from bosonic quantum field theory.  arXiv preprint quant-ph/0112149.
  * [Wallace, 2006] Wallace, D. (2006).  In defence of naiveté: The conceptual status of lagrangian quantum field theory.  Synthese, 151(1):33–80.
  * [Wallace, 2011] Wallace, D. (2011).  Taking particle physics seriously: A critique of the algebraic approach to quantum field theory.  Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, 42(2):116–125.
  * [Weatherall, 2019] Weatherall, J. O. (2019).  Part 1: Theoretical equivalence in physics.  Philosophy Compass, 14(5):e12592.  e12592 10.1111/phc3.12592.
  * [Wigner, 1939] Wigner, E. (1939).  On unitary representations of the inhomogeneous lorentz group.  Annals of Mathematics, 40(1):149–204.