Quantum Field Theory (QFT) is the mathematical and conceptualframework for contemporary elementary particle physics. It is also aframework used in other areas of theoretical physics, such ascondensed matter physics and statistical mechanics. In a ratherinformal sense QFT is the extension of quantum mechanics (QM), dealingwith particles, over to fields, i.e., systems with an infinite numberof degrees of freedom. (See the entry onquantum mechanics.)In the last decade QFT has become a more widely discussedtopic in philosophy of science, with questions ranging frommethodology and semantics to ontology. QFT taken seriously in itsmetaphysical implications seems to give a picture of the world whichis at variance with central classical conceptions of particles andfields, and even with some features of QM.

The following sketches how QFT describes fundamental physics and whatthe status of QFT is among other theories of physics. Since there is astrong emphasis on those aspects of the theory that are particularlyimportant for interpretive inquiries, it does not replace anintroduction to QFT as such. One main group of target readers arephilosophers who want to get a first impression of some issues thatmay be of interest for their own work, another target group arephysicists who are interested in a philosophical view upon QFT.

• 1. What is QFT?
• 2. The Basic Structure of the Conventional Formulation
  • 2.1 The Lagrangian Formulation of QFT
  • 2.2 Interaction
  • 2.3 Gauge Invariance
  • 2.4 Effective Field Theories and Renormalization
• 3. Beyond the Standard Model
  • 3.1 Quantum Gravity
  • 3.2 String Theory
• 4. Axiomatic Reformulations of QFT and their Interpretive Significance
  • 4.1 Motive One: Operationalism
  • 4.2 Motive Two: Mathematical Rigour
  • 4.3 Motive Three: Dealing with Inequivalent Representations
  • 4.4 Algebras, Inequivalent Representations, and the Physical-Content Question
  • 4.5 The Interpretive Impact of Accelerated Observers and Interacting Systems
• 5. Further Philosophical Issues
  • 5.1 Setting the Stage: Candidate Ontologies
  • 5.2 Did Wigner Define the Particle Concept?
  • 5.3 Non-Localizability Theorems
  • 5.4 The Role of Symmetries
  • 5.5 Taking Stock: Where do we Stand?
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. What is QFT?

In contrast to many other physical theories there is no canonicaldefinition of what QFT is. Instead one can formulate a number oftotally different explications, all of which have their merits andlimits. One reason for this diversity is the fact that QFT has grownsuccessively in a very complex way. Another reason is that theinterpretation of QFT is particularly obscure, so that even thespectrum of options is not clear. Possibly the best and mostcomprehensive understanding of QFT is gained by dwelling on itsrelation to other physical theories, foremost with respect to QM, butalso with respect to classical electrodynamics, Special RelativityTheory (SRT) and Solid State Physics or more generally StatisticalPhysics. However, the connection between QFT and these theories isalso complex and cannot be neatly described step by step.

If one thinks of QM as the modern theory of one particle (or, perhaps,very few particles), one can then think of QFT as an extension of QMfor the analysis of systems with many particles—and therefore with a large number of degrees of freedom. In this respect going from QM toQFT is not inevitable but rather beneficial for pragmaticreasons. However, a general threshold is crossed when it comes tofields, like the electromagnetic field, which are not merely difficultbut impossible to deal with in the frame of QM. Thus the transitionfrom QM to QFT allows treatment of both particles and fields within auniform theoretical framework. (As an aside, focusing on the number ofparticles, or degrees of freedom respectively, explains why the famousrenormalization group methods can be applied in QFT as well as inStatistical Physics. The reason is simply that both disciplines studysystems with a large or an infinite number of degrees of freedom,either because one deals with fields, as does QFT, or because onestudies the thermodynamic limit, a very useful artifice in StatisticalPhysics.) Moreover, issues regarding the number of particles underconsideration yield yet another reason why we need to extendQM. Neither QM nor its immediate relativistic extension with theKlein-Gordon and Dirac equations can describe systems with avariable number of particles. However, obviously this is essential fora theory that is supposed to describe scattering processes, whereparticles of one kind are destroyed while others are created.

One gets a very different kind of access to what QFT is when focusingon its relation to QM and SRT. Historically, QFT resulted from thesuccessful reconciliation of QM and SRT. In order to understand theinitial problem one has to realize that QM is not only inapotential conflict with SRT, more exactly: the localitypostulate of SRT, because of the famous EPR correlations of entangledquantum systems. There is also a manifest contradiction between QM andSRT on the level of the dynamics. The Schrödinger equation,i.e., the fundamental law for the temporal evolution of the quantummechanical state function, cannot possibly obey the relativisticrequirement that all physical laws of nature be invariant underLorentz transformations. The Klein-Gordon and Dirac equations,resulting from the search for relativistic analogues of theSchrödinger equation in the 1920s, do respect the requirement ofLorentz invariance. Nevertheless, ultimately they are not satisfactorybecause they do not permit a description of fields in a principledquantum-mechanical way.

Fortunately, for various phenomena it is legitimate to neglect thepostulates of SRT, namely when the relevant velocities are small inrelation to the speed of light and when the kinetic energies of theparticles are small compared to their mass energies\(mc^2\). And this is the reason why non-relativisticQM, although it cannot be the correct theory in the end, has itsempirical successes. But it can never be the appropriate framework forelectromagnetic phenomena because electrodynamics, which prominentlyencompasses a description of the behavior of light, is alreadyrelativistically invariant and therefore incompatible withnon-relativistic QM. Relativistic scattering experiments are another context in which QMfails. Since the involved particles often travel close tothe speed of light, relativistic effects can no longer beneglected. For that reason high-energy scattering experiments can only becorrectly confronted by QFT.

Unfortunately, the catchy characterization of QFT as the successfulmerging of QM and SRT has its limits. On the one hand, as alreadymentioned above, there also is a relativistic QM, with theKlein-Gordon- and the Dirac-equation among their most famousresults. On the other hand, and this may come as a surprise, it ispossible to formulate a non-relativistic version of QFT (see Bain2011). The nature of QFT thus cannot simply be that it reconciles QMwith the requirement of relativistic invariance. Consequently, for adiscriminating criterion it is more appropriate to say that only QFT,and not QM, allows describing systems with an infinite number ofdegrees of freedom, i.e., fields (and systems in the thermodynamiclimit). According to this line of reasoning, QM would be the modern(as opposed to classical) theory of particles and QFT the moderntheory of particlesand fields. Unfortunately however, andthis shall be the last turn, even this gloss is not untarnished.There is a widely discussed no-go theorem by Malament (1996) with thefollowing proposed interpretation: Even the quantum mechanics of onesingle particle can only be consonant with the locality principle ofspecial relativity theory in the framework of a field theory, such asQFT. Hence ultimately, the characterization of QFT, on the one hand,as the quantum physical description of systems with an infinite numberof degrees of freedom, and on the other hand, as the only way ofreconciling QM with special relativity theory, are intimatelyconnected with one another.

Figure 1.

The diagram depicts the relations between different theories, whereNon-Relativistic Quantum Field Theory is not a historical theory butrather an ex post construction that is illuminating for conceptualpurposes. Theoretically, [(i), (ii), (iii)], [(ii), (i), (iii)] and[(ii), (iii), (i)] are three possible ways to get from ClassicalMechanics to Relativistic Quantum Field Theory. But note that this ismeant as a conceptual decomposition; history didn’t go all these stepsseparately. On the one hand, by good luck, so to say, Maxwell’s equations of classicalelectrodynamics were relativistically covariant from inception. Thesuccessful quantization of that theory lead directly to the early Relativistic Quantum Field Theories. On the other hand, some would argue (e.g., Malament 1996) thatthe only way to reconcile QM and SRT is in terms of a field theory, sothat (ii) and (iii) would coincide. Note that the steps (i), (ii) and(iii), i.e., quantization, transition to an infinite number of degreesof freedom, and reconciliation with SRT, are all ontologicallyrelevant. In other words, by these steps the nature of the physicalentities the theories talk about may change fundamentally. See Huggett2003 for an alternative three-dimensional “map oftheories”.

Further Reading on QFT and Philosophy of QFT. Mandland Shaw (2010), Peskin and Schroeder (1995), Weinberg (1995) andWeinberg (1996) are standard textbooks on QFT. Teller (1995) andAuyang (1995) are the first systematic monographs on the philosophy ofQFT. The anthologies Brown and Harré (1988), Cao (1999) andKuhlmann et al. (2002) are anthologies with contributions byphysicists and philosophers (of physics), the last of whichfocuses on ontological issues. The literature on the philosophy of QFThas increased significantly in the last decade. Besides several papers there are a few new monographs, Cao (2010), Kuhlmann (2010), Ruetsche (2011) and Duncan (2012) and a special issue (May 2011) ofStudies in Historyand Philosophy of Modern Physics. Bain (2011), Huggett (2000), Ruetsche (2002) and Swanson (2017) provide article length discussions on a number ofissues in the philosophy of QFT.

See also the following supplementary document:

The History of QFT.

2. The Basic Structure of the Conventional Formulation

2.1 The Lagrangian Formulation of QFT

The crucial step towardsquantum field theory is in somerespects analogous to the corresponding quantization in quantummechanics, namely by imposing commutation relations, which leads tooperator valued quantum fields. The starting point is the classicalLagrangian formulation of mechanics, which is a so-called analyticalformulation as opposed to the standard version of Newtonian mechanics.A generalized notion of momentum (theconjugate orcanonical momentum) is defined bysetting \(p = \partial L/\partial\dot{q}\), where \(L(q, \dot{q})\) isthe Lagrange function, or Lagrangian. Here \(\dot{q}\equiv dq/dt\).The Lagrangian defines the theory, so it has no a-prioridefinition. The special case of the Newtonian theory is described by\(L = T - V\) (\(T\) is the Newtonian kinetic energy and \(V\) thepotential). This fact can be motivated by looking at the special caseof a Lagrange function with a potential \(V\) which depends only onthe position so that (using Cartesian coordinates) \(\partialL/\partial\dot{x} = (\partial/\partial\dot{x})(m\dot{x}^2 /2) =m\dot{x} = p_x\). Under these conditions the generalized momentumcoincides with the Newtonian mechanical momentum. In classicalLagrangianfield theory one associates with the given field\(\phi\) a second field, namely the conjugate field

where \(\mathcal{L}\) is a Lagrangian density. The field \(\phi\) andits conjugate field \(\pi\) are the direct analogues of the canonicalcoordinate \(q\) and the generalized (canonical or conjugate) momentum\(p\) in classical mechanics of point particles.

In both cases, QM and QFT, requiring that the canonical variablessatisfy certain commutation relations implies that the basicquantities become operator valued. From a physical point of view thisshift implies a restriction of possible measurement values forphysical quantities some (but not all) of which can have their valuesonly in discrete steps now. In QFT the canonical commutationrelations for a field \(\phi\) and the corresponding conjugate field \(\pi\)are

Note these are equal-time commutation relations, i.e., these commutatorsrefer to fields at the same time. It is not obvious that theequal-time commutation relations are Lorentz invariant but one canformulate a manifestly covariant form of the canonical commutationrelations. The relations above apply to a bosonic field, likethe Klein-Gordon field or the electromagnetic field. For a fermionicfield, like the Dirac field for electrons, one has to useanticommutation relations.

While there are close analogies between quantization in QM and in QFTthere are also important differences. Whereas the commutationrelations in QM refer to a quantum object with three degrees offreedom, so that one has a set of 15 equations, the commutationrelations in QFT do in fact comprise an infinite number of equations,namely for each of the infinitely many space-time 4-tuples\((\mathbf{x},t)\) there is a new set of commutationrelations. This infinite number of degrees of freedom embodies thefield character of QFT.

It is important to realize that the operator valued field\(\phi(\mathbf{x},t)\) in QFT isnot analogousto the wavefunction \(\psi(\mathbf{x},t)\) in QM, i.e.,the quantum mechanical state in its position representation. While thewavefunction in QM is acted upon by observables/operators, in QFT itis the (operator valued) field itself which acts on the space ofstates. In a certain sense the single particle wave functions havebeen transformed, via their reinterpretation as operator valuedquantum fields, into observables. This step is sometimes called‘second quantization’ because the single particle wave equations in relativistic QM already came about by a quantizationprocedure, e.g., in the case of the Klein-Gordon equation by replacingposition and momentum by the corresponding quantum mechanicaloperators. Afterwards the solutions to these single particle waveequations, which are states in relativistic QM, are considered asclassical fields, which can be subjected to the canonical quantizationprocedure of QFT. The term ‘second quantization’ has oftenbeen criticized partly because it blurs the important fact that thesingle particle wave function \(\phi\) in relativistic QM and theoperator valued quantum field \(\phi\) are fundamentally different kindsof entities despite their connection in the context of discovery.

In conclusion, it must be emphasized that both in QM and QFTstatesand observables are equally important. However, tosome extent their roles are switched. While states in QM can have aconcrete spatio-temporal meaning in terms of probabilities forposition measurements, in QFT states are abstract entities and it isthe quantum field operators that seem to allow for a spatio-temporalinterpretation. See the section on the field interpretation of QFT fora critical discussion.

2.2 Interaction

Up to this point, the aim was to develop a free field theory. Doingso does not only neglect interaction with other particles (fields), itis even unrealistic for one free particle because it interacts withthe field that it generates itself. For the description ofinteractions—such as scattering in particle colliders—we need certain extensions and modifications of the formalism. Theimmediate contact between scattering experiments and QFT is given bythe scattering or S-matrix which contains all the relevant predictiveinformation about, e.g., scattering cross sections. There are many schemes to calculate the S-matrix, among which one introduces theHamiltonian formalism. The Hamiltonian density can be derived from the Lagrangian density by means of a Legendre transformation.

To discuss interactions it is convenient to introduce a newrepresentation, theinteraction picture, which is analternative to the Schrödinger and the Heisenberg picture. Forthe interaction picture one splits up the Hamiltonian, which is thegenerator of time-translations, into two parts \(H = H_0 +H_{\textit{int}}\), where \(H_0\) describes the free system, i.e.,without interaction. It is solved exactly, and gets absorbed in are-definition of the fields. Then \(H_{\textit{int}}\) is theinteraction part of the Hamiltonian, or short the ‘interactionHamiltonian.’ Using the interaction picture is advantageousbecause the equations of motion as well as, under certain conditions,the commutation relations are the same for interacting fields as forfree fields. Therefore, various results that were established for freefields can still be used in the case of interacting fields. Thecentral instrument for the description of scattering is again theS-matrix, which expresses the connection between in and out states byspecifying the transition amplitudes. In QED, for instance, a state\(|\textit{in}\rangle\) describes one particular configuration of electrons,positrons and photons, i.e., it describes how many of these particlesthere are and which momenta, spins and polarizations they have beforethe interaction. The S-matrix supplies the transition amplitude thatthis state goes over to a particular \(|\textit{out}\rangle\) state,e.g., that a particular counter responds after the interaction. Thetransition amplitude is squared to form a probability, and suchprobabilities are checked in experiments.

The canonical formalism of QFT as introduced in the previous sectionis only applicable in the case of free fields since the inclusion ofinteraction leads to infinities (see the historical part). Sincelittle of realistic models can be solved exactly, perturbation theorymakes up a large part of most publications on QFT. The importance ofperturbative methods is understandable realizing that they establishthe immediate contact between theory and experiment. Although thetechniques of perturbation theory have become ever more sophisticatedit is somewhat disturbing that perturbative methods are difficult toavoid. Some have argued this is a matter of principle. (And on theother hand, physicists have been astonishingly creative in developing“toy models— that can be solved exactly, precisely toescape perturbation theory.) One reason for unease is thatperturbation theory is felt to be rather a matter of (highlysophisticated) craftsmanship than of understandingnature. Accordingly, the corpus of perturbative methods plays a smallrole in philosophical investigations of QFT. Two recent exceptions areFraser (2018) and Passon (2019). What does matter, however, is inwhich sense the consideration of realistic interactions affects thegeneral framework of QFT. An overview about perturbation theory isgiven in section 4.1 (“Perturbation Theory—Philosophy andExamples”) of Peskin & Schroeder (1995).

2.3 Gauge Invariance

Some theories, in particular those of current particle physics, aredistinguished by beinggauge invariant, which meansthatgauge transformations of certain terms do not change anyobservable quantities. A gauge transformation is a local symmetrytransformation, with parameters that are smooth functions of theposition \(\mathbf{x}\) and time \(t\). Requiring gauge invarianceprovides an elegant and systematic way of introducing models forinteracting fields. Moreover, gauge invariance plays an important rolein selecting theories. The prime example of an intrinsically gaugeinvariant theory is electrodynamics. In the potential formulation ofMaxwell’s equations one introduces the vector potential\(\mathbf{A}\) and the scalar potential \(\phi\), which are linked tothe magnetic field \(\mathbf{B}(\mathbf{x},t)\) and the electric field\(\mathbf{E}(\mathbf{x},t)\) by

or covariantly

where \(F^{\mu \nu}\) is the electromagnetic fieldtensor and \(A^{\mu} = (\phi , \mathbf{A})\) the4-vector potential. The important point in the present context isthat given the identification (2.3), or (2.4), there remains a certainflexibility or freedom in the choice of \(\mathbf{A}\) and \(\phi\),or \(A^{\mu}\). In order to see that, consider theso-calledgauge transformations

or covariantly

where \(\chi\) is a scalar function (of space and time or of space-time)which can be chosen arbitrarily. Inserting the transformedpotential(s) into equation(s) (2.3), or (2.4), one can see that theelectric field \(\mathbf{E}\) and the magnetic field\(\mathbf{B}\), or covariantly the electromagnetic field tensor\(F^{\mu \nu}\), are not affected by a gauge transformation of the potential(s). Since only the electric field\(\mathbf{E}\) and the magnetic field \(\mathbf{B}\), andquantities constructed from them, are observable, whereas the vectorpotential itself is not, nothing physical seems to be changed by agauge transformation because it leaves \(\mathbf{E}\) and\(\mathbf{B}\) unaltered. Note that gauge invariance is a kind ofsymmetry that does not come about by space-time transformations.

In order to link the notion of gauge invariance to the Lagrangianformulation of QFT one needs a more general form of gaugetransformations which applies to the field operator \(\phi\) and whichis supplied by

where \(\Lambda\) is an arbitrary real constant. Equations (2.7)describe aglobal gauge transformation whereas alocal gauge transformation

varies with \(x\).

It turned out that requiring invariance under local gaugetransformations supplies a systematic way for finding the equationsdescribing fundamental interactions. For instance, starting with theLagrangian for a free electron, the requirement of local gaugeinvariance can only be fulfilled by introducing additional terms,namely those for the electromagnetic field. Gauge invariance can becaptured by certain symmetry groups: \(\mathrm{U}(1)\) forelectromagnetic, \(\mathrm{SU}(2)\otimes\mathrm{U}(1)\) forelectroweak and \(\mathrm{SU}(3)\) for strong interaction. This hasbecome an important basis for unification programs, as is the analogyto general relativity where a local gauge symmetry (generalcovariance) leads to the Einstein-Hilbert theory of the gravitationalfield. Moreover, gauge invariant quantum field theories aregenerically renormalizable. (There are many renormalizable theories,but finding theories with massive vector fields was stopped fordecades before gauge theories and the Higgs mechanism came into thefore.) All this can be taken to show that a mathematically richframework, with surplus structures, can be very valuable in theconstruction of theories.

Auyang (1995) emphasizes the general conceptual significance ofinvariance principles; Redhead (2002) and Martin (2002) focusspecifically on gauge symmetries. Healey (2007) and Lyre (2004 and2012) discuss the ontological significance of gauge theories, amongother things concerning the Aharanov-Bohm effect and ontic structuralrealism.

2.4 Effective Field Theories and Renormalization

In the 1970s a program emerged in which the theories of the standardmodel of elementary particle physics are considered as effective fieldtheories (EFTs) which have a common quantum field theoreticalframework. EFTs describe relevant phenomena only in a certain domainsince the Lagrangian contains only those terms that describe particleswhich are relevant for the respective range of energy. EFTs areinherently approximative and change with the range of energyconsidered. EFTs are only applicable on a certain energy scale, i.e.,they only describe phenomena in a certain range of energy. Influencesfrom higher energy processes contribute to average values but theycannot be described in detail. This procedure has no severeconsequences since the details of low-energy theories are largelydecoupled from higher energy processes. Both domains are onlyconnected by altered coupling constants and the renormalization groupdescribes how the coupling constants depend on the energy.

The main idea of EFTs is that theories, i.e., in particular theLagrangians, depend on the energy of the phenomena which areanalysed. The physics changes by switching to a different energyscale, e.g., new particles can be created if a certain energythreshold is exceeded. The dependence of theories on the energyscale distinguishes QFT from, e.g., Newton’s theory of gravitationwhere the same law applies to an apple as well as to the moon.Nevertheless, laws from different energy scales are not completelyindependent of each other. A central aspect of considerations aboutthis dependence are the consequences of higher energy processes on thelow-energy scale.

On this background a new attitude towards renormalization developed inthe 1970s, which revitalizes earlier ideas that divergences resultfrom neglecting unknown processes of higher energies. Low-energybehavior is thus affected by higher energy processes. Since higherenergies correspond to smaller distances this dependence is to beexpected from an atomistic point of view. According to thereductionist program the dynamics of constituents on the microlevelshould determine processes on the macrolevel, i.e., here thelow-energy processes. However, as, for instance hydrodynamics shows,in practice theories from different levels are not quite as closelyconnected because a law which is applicable on the macrolevel can belargely independent of microlevel details. For this reason analogieswith statistical mechanics play an important role in the discussionabout EFTs. The basic idea of this new story about renormalization isthat the influences of higher energy processes are localizable in afew structural properties which can be captured by an adjustment ofparameters. “In this picture, the presence of infinities inquantum field theory is neither a disaster, nor an asset. It is simplya reminder of a practical limitation—we do not know what happensat distances much smaller than those we can look at directly”(Georgi 1989: 456). This new attitude supports the view thatrenormalization is the appropriate answer to the change of fundamentalinteractions when the QFT is applied to processes on different energyscales. The price one has to pay is that EFTs are only valid in alimited domain and should be considered as approximations to bettertheories on higher energy scales. This prompts the important questionwhether there is a last fundamental theory in this tower of EFTs whichsupersede each other with rising energies. Some people conjecturethat this deeper theory could be a string theory, i.e., a theory whichis not a field theory any more. Or should one ultimately expect fromphysics theories that they are only valid as approximations and in alimited domain? Hartmann (2001) and Castellani (2002) discuss the fateof reductionism vis-à-vis EFTs. Williams (2019) argues that EFTs by no means undermine a realist interpretation of QFT, provided one adopts a more refined notion of scientific realism. Wallace (2011) and Fraser(2011) discuss what the successful application of renormalizationmethods in quantum statistical mechanics means for their role in QFT,reaching very different conclusions. Egg et al. (2017) review that debate by comparing it to Bohmian QFT.

3. Beyond the Standard Model

The “standard model of elementary particle physics” issometimes used almost synonymously with QFT. However, there is acrucial difference. While the standard model is a theory with a fixedontology (understood in a prephilosophical sense), i.e., threefundamental forces and a certain number of elementary particles, QFTis rather a frame, the applicability of which is open. Thus whilequantum chromodynamics (or ‘QED’) is apart of the standardmodel, it is aninstance of a quantum field theory, or short“a quantum field theory” and not a part of QFT. Thissection deals with only some particularly important proposals that gobeyond the standard model, but which do not necessarily break up thebasic framework of QFT.

3.1 Quantum Gravity

The standard model of particle physics covers the electromagnetic, theweak and the strong interaction. However, the fourth fundamental forcein nature, gravitation, has defied quantization so far. Althoughnumerous attempts have been made in the last 80 years, and inparticular very recently, there is no commonly accepted solution up tothe present day. One basic problem is that the mass, length and timescales quantum gravity theories are dealing with are so extremelysmall that it is almost impossible to test the differentproposals.

The most important extant versions of quantum gravity theories are canonical quantumgravity, loop theory and string theory. Canonical quantum gravityapproaches leave the basic structure of QFT untouchedandjust extend the realm of QFT by quantizing gravity.Other approaches try to reconcile quantum theory and generalrelativity theory not by supplementing the reach of QFT but rather bychanging QFT itself. String theory, for instance, proposes acompletely new view concerning the most fundamental building blocks:It does not merely incorporate gravitation but it formulates a newtheory that describes all four interactions in a unified way, namelyin terms of strings (see next subsection).

While quantum gravity theories are very complicated and even moreremote from classical thinking than QM, SRT and GRT, it is not sodifficult to see why gravitation is far more difficult to deal withthan the other three forces. Electromagnetic, weak and strong forceall act in a given space-time. In contrast, gravitation is, accordingto GRT, not an interaction that takes placein time, butgravitational forces are identified with the curvature of space-timeitself. Thus quantizing gravitation could amount to quantizingspace-time, and it is not at all clear what that could mean. Onecontroversial proposal is to deprive space-time of its fundamentalstatus by showing how it “emerges ” in somenon-spatio-temporal theory. The “emergence” of space-timethen means that there are certain derived terms in the new theory thathave some formal features commonly associated with space-time. SeeKiefer (2007) for physical details, Rickles (2008) for an accessibleand conceptually reflected introduction to quantum gravity andWüthrich (2005) for a philosophical evaluation of the allegedneed to quantize the gravitational field. Also, see the entry onquantum gravity.

3.2 String Theory

String theory is one of the most promising candidates for bridgingthe gap between QFT and general relativity theory by supplying aunified theory of all natural forces, including gravitation. Thebasic idea of string theory is not to take particles as fundamentalobjects but strings that are very small but extended in onedimension. This assumption has the pivotal consequence that stringsinteract on an extended distance and not at a point. This differencebetween string theory and standard QFT is essential because it is thereason why string theory also encompasses the gravitational forcewhich is very difficult to deal with in the framework of QFT.

It is so hard to reconcile gravitation with QFT because the typicallength scale of the gravitational force is very small, namely atPlanck scale, so that the quantum field theoretical assumption ofpoint-like interaction leads to untreatable infinities. To put itanother way, gravitation becomes significant (in particular incomparison to strong interaction) exactly where QFT is most severelyendangered by infinite quantities. The extended interaction ofstrings brings it about that such infinities can be avoided. Incontrast to the entities in standard quantum physics strings are notcharacterized by quantum numbers but only by their geometrical anddynamical properties. Nevertheless, “macroscopically”strings look like quantum particles with quantum numbers. A basicgeometrical distinction is the one between open strings, i.e., stringswith two ends, and closed strings which are like bracelets. Thecentral dynamical property of strings is their mode of excitation,i.e., how they vibrate.

Reservations about string theory are mostly due to the lack oftestability since it seems that there are no empirical consequenceswhich could be tested by the methods which are, at least up to now,available to us. The reason for this “problem” is that thelength scale of strings is in the average the same as the one ofquantum gravity, namely the Planck length of approximately\(10^{-33}\) centimeters which lies far beyond theaccessibility of feasible particle experiments. But there are alsoother peculiar features of string theory which might be hard toswallow. One of them is the fact that preferred models of string theory needspace-time with 10, 11 or even 26 dimensions. In order to explain theappearance of only four space-time dimensions string theory assumesthat the other dimensions are somehow folded away or“compactified” so that they are no longer visible. Anintuitive idea can be gained by thinking of a macaroni which is atube, i.e., a two-dimensional piece of pasta rolled together, butwhich looks from the distance like a one-dimensional string.

Due to the problems of string theory, many physicists have abandoned it, but not all. Some think that, among the numerous alternative proposals for reconciling quantum physics and general relativity theory, string theory is still the best candidate, with“loop quantum gravity” as its strongest rival (see theentry onquantum gravity). Correspondingly, string theory has also received some attention withinthe philosophy of physics community in recent years. Probably thefirst philosophical investigation of string theory is Weingard (2001)in Callender & Huggett (2001), an anthology with further relatedarticles. Dawid (2003) (see Other Internet Resources below) arguesthat string theory has significant consequences for the philosophicaldebate about realism, namely that it speaks against the plausibilityof anti-realistic positions. Also see Dawid (2009). Johansson andMatsubara (2011) assess string theory from various differentmethodological perspectives, reaching conclusions in disagreement withDawid (2009). Standard introductory monographs on string theory arePolchinski (2000) and Kaku (1999). Greene (1999) is a very successfulpopular introduction. An interactive website with a nice elementaryintroduction is ‘Stringtheory.com’ (see the Other InternetResources section below).

4. Axiomatic Reformulations of QFT and their Interpretive Significance

There are three main motives for reformulating conventional QFT. Thefirst motive is operationalism, the second one mathematical rigour andthe third one finding a way to deal with the availability ofinequivalent Hilbert space representations for systems with aninfinite number of degrees of freedom, such as fields. While inprinciple the three motives are independent of one another there aremultiple interconnections in their actual implementation. One way inwhich the three motives are connected is the following: In QFT thequests for operationalism and mathematical rigour seem to go hand inhand, i.e., the best means to achieve one is also the best way toachieve the other. Moreover, it leads to an algebraic formulation thatavoids privileging one amoung various available inequivalentrepresentations, which tacitly happens in conventional QFT.

4.1 Motive One: Operationalism

The first motive–operationalism–is not so higly valued anymore today, and for good reasons (see entry on Operationalism).Nevertheless, it was, not only in physics, very strong in and aroundthe 1950s, when axiomatic reformulations of QFT entered the scene.Accordingly, the impact of operationalism must not by overlooked.Already in the 1930s the problem of perturbative infinities (seeSupplement “The History of QFT”) as well as thepotentially heuristic status of the Lagrangian formulation of QFTstimulated the search for concise and ideally axiomaticreformulations. About a dacade later it became clear thatoperationalism and mathematical rigour may go hand in hand, becausethe setting of conventional QFT—where quantum fields are basic,with field values being assigned to points in spacetime—is bothmathematically ill-defined and in conflict with the“operational” idea that the core elements of an empiricaltheory should be observable quantities, which can be measured by meansof certain physical operations.

The mathematical aspect of the problem is that a field at a point,\(\phi (x)\), is not an operator on a Hilbert space. The physicalcounterpart of the problem is that it would require an infinite amountof energy to measure a field at a point of space-time. One way tohandle this situation—and one of the starting points foraxiomatic reformulations of QFT—is not to consider fields at apoint but instead fields which are smeared out in the vicinity of thatpoint using certain functions, so-called test functions. The result isa smeared field \(\phi (f) = \int \phi (x) f(x) dx\) withsupp\((f)\subset \mathcal{O}\), where supp(\(f\)) is the support ofthe test function \(f\) and \(\mathcal{O}\) is a bounded open regionin Minkowski space-time. In Wightman’s field axiomatics from theearly 1950s, the basic entities are then polynomial algebras\(P(\mathcal{O})\) of smeared fields, i.e., sums of products ofsmeared fields in finite space-time regions \(\mathcal{O}\). Thus itreplaces the mapping \(x \rightarrow \phi (x)\) in the conventionalformulation of QFT by \(\mathcal{O} \rightarrow P(\mathcal{O})\).

From an operationalist perspective equally troublesome as point-likequantities are global quantities, like total charge, total energy ortotal momentum of a field. They are unobservable since theirmeasurement would have to take place in the whole universe.Accordingly, quantities which refer to infinitely extended regions ofspace-time should not appear among the observables of the theory, asthey do in the standard formulation of QFT. In the discussion of such“parochial observables” below we will see that it is notso clear in the end whether this is really a good argument (seeRuetsche 2011 and Feintzeig 2018). In any case, however, it has beenimportant in the formation of axiomatic reformulations of QFT.

Another operationalist reason for favouring algebraic formulationsderives from the fact that two quantum fields are physicallyequivalent when they generate the same algebras of local observables.Such equivalent quantum field theories belong to the same so-calledBorchers class which entails that they lead to the same \(S\)-matrix.As Haag (1996) stresses,fields are only an instrument in order to “coordinatize”observables, more precisely: sets of observables, with respect todifferent finite space-time regions. The choice of a particular fieldsystem is to a certain degree conventional, namely as long as itbelongs to the same Borchers class. Thus it is more appropriate toconsider these algebras, rather than quantum fields, as thefundamental entities in QFT. The resulting operationalistic view ofQFT is that it is a statistical theory aboutlocal measurement outcomes, expressed in termsoflocal algebras of observables. Thus, it is no surprisethat Haag’s (1996) famous textbook on “AlgebraicQFT”, the most successful axiomatic reformulation, bears thetitle “Local Quantum Physics.”

4.2 Motive Two: Mathematical Rigour

So far, we focussed on the operationalist motives for reformulatingQFT and some of its consequences. Now we will distinguish different,partly competing ways of implementing these general ideas. The secondmotive—mathematical rigour—consists foremost in the questtowards a conciseaxiomatic formulation, insteadof the grab bag of conventional QFT, with its numerous mathematicallydubious, even though successful, approximation techniques. This questcomprises three parts, namely, first, the choice of those entitiesupon which the axioms are to be imposed, second, the choice ofappropriate axioms, and, third, the proof that one has actuallyachieved an axiomaticreformulation ofconventional QFT, which can reproduce all the established empiricaland theoretical successes. While axiomatic approaches are clear andsharp on the first two counts, their success is more limited withrespect to the third. In general, one can say there are valuablesuccesses with respect to very general theoretical insights, such asthe connection of spin and statistics as well as non-localizability,while the weak point is the lack of realistic models for interactingquantum field theories. Since the fundamental entities in axiomaticreformulations of QFT arealgebras (of smearedfield operators or of observables) instead of quantumfields, reformulating QFT in algebraic terms andin axiomatic terms are enterprises with a large factual overlap.However, only one attempt to reformulate QFT axiomatically bears thename “Algebraic Quantum FieldTheory,” because it is here that the algebraic structure of thefundamental entities was first fully realized and is explicity at thecentre. However, trying to do this in a strictly axiomatic way, oneonly gets ‘reformulations’ which are not as rich asstandard QFT. As Haag (1996) concedes, the “algebraic approach[…] has given us a frame and a language not a theory.”

Wightman’s “field axiomatics”,already mentioned above, is–besides AQFT–one of the twomost prominent proposals, for an axiomatisation of QFT. Bothoriginated in the 1950s and influenced each other in theirformation. In Wightman’s field axiomatics, the entities uponwhich the axioms are imposed are smeared field operators, in factpolynomial algebras \(P(\mathcal{O})\) thereof. The crucial axioms arecovariance,microcausality(spacelike separated field operators required to either commute oranticommute), andspectrum condition (positiveenergy in all Lorentz frames, so that the vacuum is a stable groundstate). One shortcoming of this approach is that field operators aregauge-dependent and thereby arguably not qualified as directlyrepresenting physical quantities. Moreover, the use of unboundedfield operators makes Wightman’s approach mathematicallycumbersome.

In contrast,Algebraic Quantum Field Theory(AQFT)—the most successful attempt toreformulate QFT axiomatically—employs only bounded operators. Itbuilds upon work in the 1940s by Gelfand, Neumark, and in particularSegal, who tried to describe quantum physics in terms of \(C\)*-algebras[section 4.1. in the entryQuantum Theory and MathematicalRigour has a more detailed account]. The notion of a \(C\)*-algebrageneralizes the notion ofthe algebra \(\mathcal{B(H)}\) of allbounded operators on a Hilbert space \(\mathcal{H}\),which is also the most importantexample for a \(C\)*-algebra. Infact, it can be shown that any \(C\)*-algebrais isomorphic to a (norm-closed,self-adjoint) algebra of bounded operators on a Hilbert space. Theboundedness (and self-adjointness) of the operators is the reason why\(C\)*-algebras are considered asideal for representing physical observables. The ‘C’indicates that one is dealing with a complex vector space and the‘*’ refers to the operation that maps an element \(A\)of an algebra to itsinvolution (or adjoint) \(A\)*,which generalizes the conjugate complex ofcomplex numbers to operators. This involution is needed in order todefine the crucial norm property of \(C\)*-algebras,which is of central importance for theproof of the above isomorphism claim.

AQFT takes so-callednets of algebras as basic for themathematical description of quantum systems, i.e., the mapping \(\mathcal{O}\rightarrow\mathcal{A}(\mathcal{O})\)fromfinite space-time regions to algebras of local observables. Theinsight behind this apporoach is that the net structure of algebras,i.e., the very way how algebras of local observables are linked tospace-time regions, supplies observables with physical significance.In this rather abstract setting, physical states are identified aspositive, linear, normalized functionals which map elements of localalgebras to real numbers. States can thus be understood as assignmentsof expectation values to observables. Via the so-calledGelfand-Neumark-Segal construction, one can recover the concreteHilbert space representations in the conventional formalism. Thus,“all the Hilbert spaces we will ever need are hidden inside thealgebra itself” (Halvorson & Müger 2007, section7).

AQFT then imposes a whole list of axioms on the abstract algebraicstructure, namely relativistic axioms (in particular locality andcovariance), general physical assumptions (e.g., the spectrumcondition), and finally some technical assumptions concerning themathematical formulation. The principle of (Einstein)causality is the main relativistic ingredient of AQFT: Allobservables of a local algebra connected with a space-time region\(\mathcal{O}\) are required tocommute with all observables of another algebra which is associatedwith a space-time region \(\mathcal{O}'\)that is space-like separated from\(\mathcal{O}\). As areformulation of QFT, AQFT is expected toreproduce the main features of QFT, like the existence ofantiparticles, internal quantum numbers, the relation of spin andstatistics, etc. That this aim could not be achieved on a purelyaxiomatic basis is partly due to the fact that the connection betweenthe respective key concepts of AQFT and QFT, i.e., algebras ofobservables and quantum fields, is not sufficiently clear. One mainlink aresuperselection rules, which put restrictions on theset of all observables and allow for classification schemes in termsof permanent or essential properties.

Today, many philosophers of physics who work on QFT rest most of theirconsiderations on AQFT. This predominance of AQFT for foundationalstudies about QFT becomes problematic, however, when AQFT is seen as aphysical theory in competition to Conventional QFT (CQFT). Wallace(2006, 2011) even urges that, seen from today, CQFT succeeded, whereasAQFT failed, so that “to be lured away from the Standard Modelby [AQFT] is sheer madness” (Wallace 2011:124). Contra Wallace,Fraser (2011) questions Wallace’s crucial point in defense ofCQFT, namely that the application of renormalization group techniquesin QFT finally solved the problem of ultraviolet divergences thattroubled CQFT in the 1950s. The empirical success of renormalizationin CQFT leaves the physical reasons for this success in the dark,argues Fraser, unlike in condensed matter physics, where its successis due to the fact that matter is discrete at atomic lengthscales.

Ultimately, the most reasonable position may be a liberal one,according to which neither of the three, AQFT, Wightman’s fieldaxiomatics and CQFT, should be regarded as rival research programs,only one of which can and should survive (Kuhlmann 2010b, Swanson2017). Wightman’s Axiomatic QFT is fruitful in the constructionof concrete models, AQFT is advantageous for ontologicalconsiderations because it clearly separates fundamental and derivedentities, and CQFT is very good for actual calculations of the highenergy physicist.

The monographs Haag (1996) and Horuzhy (1990) and the articles Haag& Kastler (1964), Roberts (1990), Buchholz (1998) areintroductions to AQFT. Halvorson & Müger (2007), Ruetsche(2011) and Ruetsche (2012a,b) are tailored for philosophers ofphysics, where the first emphasizes technical and the latterinterpretive issues. Streater & Wightman (1964) is an earlypioneering monograph on axiomatic QFT. Bratteli & Robinson (1997)is a classic on the mathematical theory of operator algebras,emphasising physical applications. Philosophical studies on AQFT canbe found, among many others, in Baker (2009), Baker & Halvorson(2010), Earman & Fraser (2006), Fraser (2008, 2009, 2011),Feintzeig (2018), Feintzeig et al. (2019), Feintzeig and Weatherall(2019), Kronz & Lupher (2005), Kuhlmann (2010a, 2010b), Lupher(2018), Miller (2018), Rédei & Valente (2010), and Ruetsche(2002, 2003, 2006, 2011).

4.3 Motive Three: Dealing with Inequivalent Representations

The third important problem for standard QFT which promptedreformulations is the existence ofinequivalentrepresentations. In the context of quantum mechanics,Schrödinger, Dirac, Jordan and von Neumann realized thatHeisenberg’s matrix mechanics and Schrödinger’s wavemechanics are just two (unitarily) equivalent representations of thesame underlying abstract structure, i.e., an abstract Hilbert space\(\mathcal{H}\) and linear operatorsacting on this space. We are merely dealing with two different waysfor representing the same physical reality, and it is possible toswitch between these different representations by means of a unitarytransformation, i.e., an operation that is analogous to an innocuousrotation of the frame of reference.Representations of some given algebra or groupare sets of mathematical objects, like numbers, rotations or moreabstract transformations (e.g., differential operators) together with abinary operation (e.g., addition or multiplication) that combines anytwo elements of the algebra or group, such that the structure of thealgebra or group to be represented is preserved. This means that thecombination of any two elements in the representation space, say \(a\)and \(b\),leads to a third element which corresponds tothe element that results when you combine the elements correspondingto \(a\) and \(b\)in the algebra or group that is represented. In1931 von Neumann gave a detailed proof (of a conjecture by Stone) thatthe canonical commutation relations (CCRs) for position coordinatesand their conjugate momentum coordinates in configuration space fixthe representation of these two sets of operators in Hilbert space upto unitary equivalence (von Neumann’s uniqueness theorem). Thismeans that the specification of the purely algebraic CCRs suffices todescribe a particular physical system.

In quantumfield theory, however, vonNeumann’s uniqueness theorem loses its validity since here oneis dealing with an infinite number of degrees of freedom. Now one isconfronted with a multitude of unitarilyinequivalent representations (UIRs) of the CCRs and it is not obvious whatthis means physically and how one should cope with it. Since thetroublesome inequivalent representations of the CCRs that arise in QFTare allirreducible their inequivalence is notdue to the fact that some are reducible while others are not (arepresentation isreducible if there is aninvariant subrepresentation, i.e., a subset which alone represent theCCRs already). Since unitarilyinequivalent representations seem to describe different physical states of affairs it would no longer be legitimate to simply choosethe most convenient representation, just like choosing the mostconvenient frame of reference. In principle all but one of the UIRscould be physically irrelevant, i.e., mathematical artefacts of aredundant formalism. However, it seems that at least some irreduciblerepresentations of the CCRs are inequivalentandphysically relevant.

These considerations motivate the algebraic point of view thatalgebras of observables rather than observablesthemselves in a particular representation should be taken as the basicentities in the mathematical description of QFT, so that theabove-mentioned problems are to some degree avoided from the outset.However, obviously this cannot just be the end of the story. Even ifUIRs are not basic, it is still necessary to say what the availabilityof different UIRs means, physically and thereby ontologically.

4.4 Algebras, Inequivalent Representations, and the Physical-Content Question

One of the most fundamental interpretative obstacles concerning QFT isthe question which formalism to consider and to then identify whichparts of the respective formalism carry the physical content, andwhich parts are surplus structure, from an ontological point of view.Roughly, one may distingush “Hilbert space conservatists”and “algebraic imperialists” (Arageorgis 1995 and Ruetsche2002 coined these terms).While Hilbert space conservativism seems to be the default position, often adopted without further justification, algebraic imperialism usually comes with an explicitjustification.

Hilbert space conservatism dismisses the availability of a plethora ofUIRs as a mathematical artifact with no physical relevance. In contrast, algebraic imperialism argues that instead of choosing a particularHilbert space representation, one should stay on the abstract algebraic level.Haag & Kastler (1964) try to give an operationalistjustification for their claim that no diverging physical content isencoded by the different unitarily inequivalent representations. Thecore point of the argument is that the topology used to distinguishthese representations as different, namely the uniform operatortopology or “norm topology”, is inappropriately fine-grained (where a topology defines what is meant by theneighborhood of an element). Therefore, it classifies representationsas inequivalent, which are in fact empirically and therefore“physically equivalent.” The whole argument depends decisively on a theorem by Fell (1960), according to which a finitenumber of measurements, performed with some inaccuracy, can neverdistinguish unitarily inequivalent representations. They are“weakly equivalent,” a notion introduced by Fell (1960), using a weak operator topology. Invoking Fell’s theorem and equatingweak equivalence andphysical equivalence,Haag & Kastler (1964) reason that the “relevant object is the abstract algebra and not the representation. The selection of a particular (faithful) representation is a matter of conveniencewithout physical implications. It may provide a more or less handyanalytical apparatus.”

Arageorgis (1995), Ruetsche (2003, 2011), Lupher & Kronz (2005),and most recently, Lupher (2018) attack algebraic imperialists forusing Fell’s theorem in order to denigrate UIRs.Ruetsche’s most important point is that UIRs do real explanatorywork in physics, e.g., in quantum statistical mechanics (see Bratteli& Robinson 1997 and Ruetsche 2003) and in particular when it comesto spontaneous symmetry breaking. The coexistence of UIRs can bereadily understood by looking at ferromagnetism for infinite spinchains (see Ruetsche 2006). At high temperatures the atomic dipoles inferromagnetic substances fluctuate randomly. Below a certaintemperature the atomic dipoles tend to align to each other in somedirection. Since the basic laws governing this phenomenon arerotationally symmetrical, no direction is preferred. Thus once thedipoles have “chosen” one particular direction, thesymmetry is broken. Since there is a different ground state for eachdirection of magnetization, one needs different Hilbertspace representations—each containing a unique ground state—in order todescribe symmetry breaking systems. Correspondingly, one has to employinequivalent representations.

Already Lupher & Kronz (2005) point out that an attractivealternative to the strict \(C^{\ast}\)-algebraicimperialism is to extend thephysical content to von Neumann algebras, the elements of which aregenerated by representations of algebra \(\mathcal{A}\).Lupher (2018) analyzes the algebraicimperialists’ use of Fell’s theorem in detail and pointsout that there are important representations in the algebraic approachto which Fell’s theorem does not apply, in particular many \(W^{\ast}\)-algebras,the abstractcounterparts of von Neumann algebras. Moreover, he argues that it is a\(W^{\ast}\)-algebra, namely thebidual \(\mathcal{A}^{\ast \ast}\),which is the appropriate locus of physical content for the algebraicimperialist, to which Lupher subscribes in the form of what he dubs“bidualism.” One crucial advantage of taking the \(W^{\ast}\)-algebra\(\mathcal{A}^{\ast\ast}\) is that it is larger thanthe \(C^{\ast}\)-algebra \(\mathcal{A}\).In particular it alsocontains what Ruetsche (2011) calls “parochialobservables” (or the abstract counterparts thereof), such as netmagnetization and particle number. Also against Ruetsche, Feintzeig(2018) argues that not only the universalist (or bidualist) but also\(C^{\ast}\)-algebraic imperialismhas access to “parochial observables,” since it hasadditional tools to represent such observables by means ofidealizations from the observables in the abstract algebra.

To conclude, it is difficult to say how the availability of UIRsshould be interpreted in general. Clifton and Halvorson (2001b)propose seeing this as a form of complementarity. Ruetsche (2003)advocates a “Swiss army approach”, according to which theavailability of UIRs shows that physical possibilities in differentdegrees must be included into our ontology. In addition, Ruetsche(2011: 119) argues that both Hilbert space conservatism and algebraicimperialism are extremist, or “pristine”, positions in theend. Accordingly, she advocates taking UIRs more seriously than inthese extremist approaches. As we have just seen, however, recentadherents of algebraic imperialism and universalism/bidualism haveaccepted the challenge.

4.5 The Interpretive Impact of Accelerated Observers and Interacting Systems

One important interpretive issue where unitarily inequivalentrepresentations (UIRs) play a crucial role is theUnruheffect: a uniformly accelerated observer in a Minkowskivacuum should detect a thermal bath of particles, the so-calledRindler quanta (Unruh 1976, Unruh & Wald 1984). The Unruh effect constitutes a severe challenge to a particle interpretation ofQFT, because it seems that the very existence of the basic entities of an ontology should not depend on the state of motion of the detectors. Teller (1995: 110–113) tries to dispel this problem by pointing out that while the Minkowski vacuum has the definite value zero for the Minkowski number operator, the particle number is indefinite forthe Rindler number operator, since one has a superposition of Rindlerquanta states. This means that there are only propensities fordetecting different numbers of Rindler quanta but no actual quanta.However, this move is problematic since it seems to suggest thatquantum physical propensities in general don’t need to be takenfully for real.

Clifton and Halvorson (2001b) argue, contra Teller, that it isinapproriate to give priority to either the Minkowski or the Rindlerperspective. Both are needed for a complete picture. The Minkowski aswell as the Rindler representation are true descriptions of the world,namely in terms of objective propensities. Arageorgis, Earman andRuetsche (2003) argue that Minkowski and Rindler (or Fulling)quantization donot constitute a satisfactory case ofphysically relevant UIRs. First, there are good reasons to doubt thatthe Rindler vacuum is a physically realizable state. Second, theauthors argue, the unitary inequivalence in question merely stems fromthe fact that one representation is reducible and the other oneirreducible: The restriction of the Minkowski vacuum to a Rindlerwedge, i.e., what the Minkowski observer says about the Rindler wedge,leads to a mixed state (a thermodynamic KMS state) and therefore areducible representation, whereas the Rindler vacuum is a pure stateand thus corresponds to an irreducible representation. Therefore, theUnruh effect does not cause distress for the particleinterpretation—which the authors see to be fighting a losingbattle anyhow—because Rindler quanta are not real and theunitary inequivalence of the representations in question has nothingspecific to do with conflicting particle ascriptions.

The occurrence of UIRs is also at the core of an analysis by Fraser(2008). She restricts her analysis to inertial observers but comparesthe particle notion for free and interacting systems. Fraser argues,first, that the representations for free and interacting systems areunavoidably unitarily inequivalent, and second, that therepresentation for an interacting system does not have the minimalproperties that are needed for any particle interpretation—e.g.Teller’s (1995) quanta version—namely the countabilitycondition (quanta are aggregable) and a relativistic energy condition.Note that for Fraser’s negative conclusion about the tenabilityof the particle (or quanta) interpretation for QFT there is no need toassume localizability.

Bain (2000) has a diverging assessment of the fact that onlyasymptotically free states, i.e., states very long before or after ascattering interaction, have a Fock representation that allows for aninterpretation in terms of countable quanta. For Bain, the occurrenceof UIRs without a particle (or quanta) interpretation for interveningtimes, i.e., close to scattering experiments, is irrelevant because thedata that are collected from those experiments always refer to systemswith negligible interactions. Bain concludes that although theinclusion of interactions does in fact lead to the break-down of thealleged duality of particles and fields it does not undermine thenotion of particles (or fields) as such.

Fraser (2008) rates this as an unsuccessful “last ditch”attempt to save a quanta interpretation of QFT because it is ad hocand can’t even show that at least something similar to the freefield total number operator exists for finite times, i.e., between theasymptotically free states. Moreover, Fraser (2008) points out that,contrary to what some authors suggest, the main source of theimpossibility to interpret interacting systems in terms of particlesisnot that many-particle states are inappropriatelydescribed in the Fock representation if one deals with interactingfields but rather that QFT obeys special relativity theory (also seeEarman and Fraser (2006) and Miller (2018) on Haag’s theorem). As Fraserconcludes, “[F]or a free system, special relativity and thelinear field equation conspire to produce a quantainterpretation.” In his reply Bain (2011) points out that thereason why there is no total number operator in interactingrelativistic quantum field theories is that this would require anabsolute space-time structure, which in turn is not an appropriaterequirement.

Baker (2009) points out that the main arguments against the particleinterpretation—concerning non-localizability (e.g., Malament1996) and failure for interacting systems (Fraser 2008)—may alsobe directed against the wave functional version of the fieldinterpretation (see field interpretation (iii) above). Mathematically,Baker’s crucial point is that wave functional space is unitarilyequivalent to Fock space, so that arguments against the particleinterpretation that attack the choice of the Fock representation maycarry over to the wave functional interpretation. First, a Minkowskiand a Rindler observer may also detect different field configurations.Second, if the Fock space representation is not apt to describeinteracting systems, then the unitarily equivalent wave functionalrepresentation is in no better situation: Interacting fields areunitarily inequivalent to free fields, too.

5. Further Philosophical Issues

5.1 Setting the Stage: Candidate Ontologies

Ontology is concerned with the most general features, entities andstructures of being. One can pursue ontology in a very general senseor with respect to a particular theory or a particular part or aspectof the world. With respect to the ontology of QFT one is tempted tomore or less dismiss ontological inquiries and to adopt the followingstraightforward view. There are two groups of fundamental fermionicmatter constituents, two groups of bosonic force carriers and four(including gravitation) kinds of interactions. As satisfying as thisanswer might first appear, the ontological questions are, in a sense,not even touched. Saying that, for instance the down quark is afundamental constituent of our material world is the starting pointrather than the end of the (philosophical) search for an ontology ofQFT. The main question is what kind of entity, e.g., the down quarkis. The answer does not depend on whether we think of down quarks ormuon neutrinos since the sought features are much more general thanthose ones which constitute the difference between down quarks or muonneutrinos. The relevant questions are of a different type. What areparticles at all? Can quantum particles be legitimately understood asparticles any more, even in the broadest sense, when we take, e.g.,their localization properties into account? How can one spell outwhat a field is and can “quantum fields” in fact beunderstood as fields? Could it be more appropriate not to think of,e.g., quarks, as the most fundamental entities at all, but rather ofproperties or processes or events?

5.1.1 The Particle Interpretation

Many of the creators of QFT can be found in one of the two campsregarding the question whether particles or fields should be givenpriority in understanding QFT. While Dirac, the later Heisenberg,Feynman, and Wheeler opted in favor of particles, Pauli, the earlyHeisenberg, Tomonaga and Schwinger put fields first (see Landsman1996). Today, there are a number of arguments which prepare theground for a proper discussion beyond mere preferences.

5.1.1.1 The Particle Concept

It seems almost impossible to talk about elementaryparticlephysics, or QFT more generally, without thinking of particles whichare accelerated and scattered in colliders. Nevertheless, it is thisvery interpretation which is confronted with the most fully developedcounter-arguments. There still is the option to say that our classicalconcept of a particle is too narrow and that we have to loosen some ofits constraints. After all, even in classical corpuscular theories ofmatter the concept of an (elementary) particle is not as unproblematicas one might expect. For instance, if the whole charge of a particlewas contracted to a point, an infinite amount of energy would bestored in this particle since the repulsive forces become infinitelylarge when two charges with the same sign are brought together. Theso-calledself energy of a point particle is infinite.

Probably the most immediate trait of particles is theirdiscreteness. Particles are countable or ‘aggregable’entities in contrast to a liquid or a mass. Obviously thischaracteristic alone cannot constitute a sufficient condition forbeing a particle since there are other things which are countable aswell without being particles, e.g., money or maxima and minima of thestanding wave of a vibrating string. It seems that one alsoneedsindividuality, i.e., it must be possible to say that itis this or that particle which has been counted in order to accountfor the fundamental difference between ups and downs in a wave patternand particles. Teller (1995) discusses a specific conception ofindividuality,primitive thisness, as well as other possiblefeatures of the particle concept in comparison to classical conceptsof fields and waves, as well as in comparison to the concept of fieldquanta, which is the basis for the interpretation that Telleradvocates. A critical discussion of Teller’s reasoning can be foundin Seibt (2002). Moreover, there is an extensive debate onindividuality of quantum objects in quantum mechanical systems of‘identical particles’. Since this discussion concerns QMin the first place, and not QFT, any further details shall be omittedhere. French and Krause (2006) offer a detailed analysis of thehistorical, philosophical and mathematical aspects of the connectionbetween quantum statistics, identity and individuality. See Dieks andLubberdink (2011) for a critical assessment of the debate. Alsoconsult the entry onquantum theory: identity and individuality.

There is still another feature which is commonly taken to be pivotalfor the particle concept, namely that particles are localizable inspace. While it is clear from classical physics already that therequirement oflocalizability need not refer to point-like localization, wewill see that even localizability in an arbitrarily large but stillfinite region can be a strong condition for quantum particles. Bain(2011) argues that the classical notions of localizability andcountability are inappropriate requirements for particles if one isconsidering a relativistic theory such as QFT.

Eventually, there are some potential ingredients of the particleconcept which are explicitly opposed to the corresponding (andtherefore opposite) features of the field concept. Whereas it is acore characteristic of a field that it is a system with an infinitenumber of degrees of freedom, the very opposite holds forparticles. A particle can for instance be referred to by thespecification of the coordinates \(\mathbf{x}(t)\)that pertain, e.g., to its center of mass—presupposingimpenetrability. A further feature of the particle concept isconnected to the last point and again explicitly in opposition to thefield concept. In a pure particle ontology the interaction betweenremote particles can only be understood as anaction at adistance. In contrast to that, in a field ontology, or acombined ontology of particles and fields,local action isimplemented by mediating fields. Finally, classical particles aremassive and impenetrable, again in contrast to (classical) fields.

5.1.1.2 Why QFT Seems to be About Particles

The easiest way to quantize the electromagnetic (or: radiation) fieldconsists of two steps. First, one Fourier analyses the vectorpotential of the classical field into normal modes (using periodicboundary conditions) corresponding to an infinite but denumerablenumber of degrees of freedom. Second, since each mode is describedindependently by a harmonic oscillator equation, one can apply theharmonic oscillator treatment from non-relativistic quantum mechanicsto each single mode. The result for the Hamiltonian of the radiationfield is

where \(a_{r}^{\dagger}(\mathbf{k})\)and \(a_r (\mathbf{k})\) are operators whichsatisfy the following commutation relations

with the index \(r\) labeling the polarisation. Thesecommutation relations imply that one is dealing with a bosonic field.

The operators \(a_{r}^{\dagger}(\mathbf{k})\)and \(a_r (\mathbf{k})\)have interesting physical interpretations as so-called particlecreation and annihilation operators. In order to see this, one has toexamine the eigenvalues of the operators

which are the essential parts in \(H_{rad}\). Due tothe commutation relations (5.2) one finds that the eigenvalues of\(N_r (\mathbf{k})\) are the integers\(n_r (\mathbf{k}) = 0, 1, 2,\ldots\) and thecorresponding eigenfunctions (up to a normalisation factor) are

where the right hand side means that \(a_{r}^{\dagger}(\mathbf{k})\)operates \(n_r (\mathbf{k})\) times on \(|0\rangle\), the state vectorof the vacuum with no photons present. The interpretation of theseresults is parallel to the one of the harmonicoscillator. \(a_{r}^{\dagger}(\mathbf{k})\) is interpreted asthecreation operator of a photon with momentum \(\hslash\mathbf{k}\) and energy \(\hslash \omega_{\mathbf{k}}\) (and apolarisation which depends on \(r\) and \(\mathbf{k})\). That is,equation (5.4) can be understood in the following way. One gets astate with \(n_r (\mathbf{k})\) photons of momentum \(\hslash\mathbf{k}\) and energy \(\hslash \omega_{\mathbf{k}}\) when thecreation operator \(a_{r}^{\dagger}(\mathbf{k})\) operates \(n_r(\mathbf{k})\) times on the vacuum state \(|0\rangle\). Accordingly,\(N_r (\mathbf{k})\) is called thenumber operator and \(n_r(\mathbf{k})\) the ‘occupation number’ of the mode that isspecified by \(\mathbf{k}\) and \(r\), i.e., this mode is occupied by\(n_r (\mathbf{k})\) photons. Note that Pauli’s exclusionprinciple is not violated since it only applies to fermions and not tobosons like photons. The corresponding interpretation fortheannihilation operator \(a_r (\mathbf{k})\) is parallel:When it operates on a state with a given number of photons this numberis lowered by one.

It is a widespread view that these results complete “thejustification for interpreting \(N(k)\) as the numberoperator, and hence for the particle interpretation of the quantizedtheory” (Ryder 1996: 131). This is a rash judgement,however. For instance, the question of localizability is not eventouched while it is certain that this is a pivotal criterion forsomething to be a particle. All that is established so far is thatcertain mathematical quantities in the formalism arediscrete. However, countability is merely one feature of particles andnot at all conclusive evidence for aparticle interpretation ofQFT yet. It is not clear at this stage whether we are in factdealing with particles or with fundamentally different objects whichonly have this one feature of discreteness in common withparticles.

Teller (1995) argues that the Fock space or “occupation number” representation does support a particle ontology in termsoffield quanta since these can be counted or aggregated,although not numbered. The degree of excitation of a certain mode ofthe underlying field determines the number of objects, i.e., theparticles in the sense of quanta. Labels for individual particles likein the Schrödinger many-particle formalism do not occur any more,which is the crucial deviation from the classical notion ofparticles. However, despite of this deviation, says Teller, quantashould be regarded as particles: Besides their countability anotherfact that supports seeing quanta as particles is that they have thesame energies as classical particles. Teller has been criticized fordrawing unduly far-reaching ontological conclusions from oneparticular representation, in particular since the Fock spacerepresentation cannot be appropriate in general because it is onlyvalid for free particles (see, e.g., Fraser 2008). In order to avoidthis problem Bain (2000) proposes an alternative quanta interpretationthat rests on the notion of asymptotically free states in scatteringtheory. For a further discussion of the quanta interpretation see thesubsection on inequivalent representations below.

The vacuum state \(|0\rangle\) is the energy ground state, i.e., theeigenstate of the energy operator with the lowest eigenvalue. It is aremarkable result in ordinary non-relativistic QM that the groundstate energy of e.g., the harmonic oscillator isnot zero incontrast to its analogue in classical mechanics. In addition to this,the relativisticvacuum of QFT has the even more strikingfeature that the expectation values for various quantities do notvanish, which prompts the question what it is that has these values orgives rise to them if the vacuum is taken to be the state with noparticles present. If particles were the basic objects of QFT how canit be that there are physical phenomena even if nothing is thereaccording to this very ontology? Eventually, studies of QFT in curvedspace-time indicate that the existence of a particle number operatormight be a contingent property of the flat Minkowski space-time,because Poincaré symmetry is used to pick out a preferredrepresentation of the canonical commutation relations which isequivalent to picking out a preferred vacuum state (see Wald1994).

Before exploring whether other (potentially) necessary requirementsfor the applicability of the particle concept are fulfilled let us seewhat the alternatives are. Proceeding this way makes it easier toevaluate the force of the following arguments in a more balancedmanner.

5.1.2 The Field Interpretation

Since various arguments seem to speak against a particleinterpretation, the allegedly only alternative, namely a fieldinterpretation, is often taken to be the appropriate ontology ofQFT. So let us see what a physical field is and why QFT may beinterpreted in this sense. A classical point particle can bedescribed by its position \(\mathbf{x}(t)\) and itsmomentum \(\mathbf{p}(t)\), which change as thetime \(t\) progresses. So there are six degrees of freedom forthe motion of a point particle corresponding to the three coordinatesof the particle’s position and three more coordinates for itsmomentum. In the case of a classical field one has an independentvalue for each single point \(\mathbf{x}\) in space, where thisspecification changes as time progresses. The field value \(\phi\) can bea scalar quantity, like temperature, a vectorial one as for theelectromagnetic field, or a tensor, such as the stress tensor for acrystal. A field is therefore specified by a time-dependent mappingfrom each point of space to a field value\(\phi(\mathbf{x},t)\). Thus a field is a system withan infinite number of degrees of freedom, which may be restrained bysome field equations. Whereas the intuitive notion of a field is thatit is something transient and fundamentally different from matter, itcan be shown that it is possible to ascribe energy and momentum to apure field even in the absence of matter. This somewhat surprisingfact shows how gradual the distinction between fields and mattercan be.

The transition from a classical field theory to a quantum field theoryis characterized by the occurrence ofoperator-valued quantumfields \(\hat{\phi}(\mathbf{x},t)\), and corresponding conjugatefields, for both of which certain canonical commutation relationshold. Thus there is an obvious formal analogy between classical andquantum fields: in both cases field values are attached to space-timepoints, where these values are specified by real numbers in the caseof classical fields and operators in the case of quantum fields. Thatis, the mapping \(\mathbf{x} \mapsto \hat{\phi}(\mathbf{x},t)\) in QFTis analogous to the classical mapping \(\mathbf{x} \mapsto\phi(\mathbf{x},t)\). Due to this formal analogy it appears to bebeyond any doubt that QFT is a field theory.

But is a systematic association of certain mathematical terms with allpoints in space-time really enough to establish a field theory in aproper physical sense? Is it not essential for a physical field theorythat some kind of real physicalproperties are allocated tospace-time points? This requirement seems not fulfilled in QFT,however. Teller (1995: ch. 5) argues that the expressionquantumfield is only justified on a “perverse reading” ofthe notion of a field, since no definite physical values whatsoeverare assigned to space-time points. Instead, quantum field operatorsrepresent the whole spectrum of possible values so that they ratherhave the status of observables (Teller: “determinables”)or general solutions. Only a specificconfiguration, i.e., anascription of definite values to the field observables at all pointsin space, can count as a proper physical field.

There are at least four proposals for a field interpretation of QFT,all of which respect the fact that the operator-valuedness of quantumfields impedes their direct reading as physical fields.

(i) Teller (1995) argues that definite physical quantities emerge whennot only the quantum field operators but also the state of the systemis taken into account. More specifically, for a given state \(|\psi \rangle\)one can calculate the expectation values\(\langle \psi |\phi(x)|\psi \rangle\) which yields an ascription ofdefinite physical values to all points x in space and thusaconfiguration of the operator-valued quantum field that maybe seen as a proper physical field. The main problem with proposal(i), and possibly with (ii), too, is that an expectation value is theaverage value of a whole sequence of measurements, so that it does notqualify as the physical property of any actual single field system, nomatter whether this property is a pre-existing (or categorical) valueor a propensity (or disposition).

(ii) The vacuum expectation value orVEV interpretation,advocated by Wayne (2002), exploits a theorem by Wightman(1956). According to this reconstruction theorem all the informationthat is encoded in quantum field operators can be equivalentlydescribed by an infinite hierarchy of \(n\)-point vacuumexpectation values, namely the expectation values of all products ofquantum field operators at \(n\) (in general different)space-time points, calculated for the vacuum state. Since thiscollection of vacuum expectation values comprises only definitephysical values it qualifies as a proper field configuration, and,Wayne argues, due to Wightman’s theorem, so does the equivalent set ofquantum field operators. Thus, and this is the upshot of Wayne’sargument, an ascription of quantum field operators to all space-timepoints does by itself constitute a field configuration, namely for thevacuum state, even if this is not the actual state.

But this is also a problem for the VEV interpretation: While it showsnicely that much more information is encoded in the quantum fieldoperators than just unspecifically what could be measured, it stilldoes not yield anything like anactual fieldconfiguration. While this last requirement is likely to be too strongin a quantum theoretical context anyway, the next proposal may come atleast somewhat closer to it.

(iii) In recent years the termwave functional interpretationhas been established as the name for the default field interpretationof QFT. Correspondingly, it is the most widely discussed extantproposal; see, e.g., Huggett (2003), Halvorson and Müger (2007),Baker (2009) and Lupher (2010). In effect, it is not very differentfrom proposal (i), and with further assumptions for (i) evenidentical. However, proposal (ii) phrases things differently and in avery appealing way. The basic idea is that quantized fields should beinterpreted completely analogously to quantized one-particle states,just as both result analogously from imposing canonical commutationrelations on the non-operator-valued classical quantities. In the caseof a quantum mechanical particle, its state can be described by a wavefunction \(\psi\)(x), which maps positions to probability amplitudes,where \(|\psi(x)|^2\) can be interpreted as theprobability for the particle to be measured atposition \(x\). For a field, the analogue of positions areclassical field configurations \(\phi(x)\), i.e., assignments offield values to points in space. And so, the analogy continues, justas a quantum particle is described by a wave function that mapspositions to probabilities (or rather probability amplitudes) for theparticle to be measured at \(x\), quantum fields can beunderstood in terms ofwave functionals\(\psi[\phi(x)\)] that map functions to numbers, namelyclassical field configurations \(\phi(x)\) to probabilityamplitudes, where \(|\psi[\phi(x)]|^2\) can beinterpreted as the probability for a given quantum field system to befound in configuration \(\phi(x)\) when measured. Thus just asa quantum state in ordinary single-particle QM can be interpreted as asuperposition of classical localized particle states, the state of aquantum field system, so says the wave functional approach, can beinterpreted as a superposition of classical field configurations. Andwhat superpositions mean depends on one’s general interpretation ofquantum probabilities (collapse with propensities, Bohmian hiddenvariables, branching Everettian many-worlds,…). In practice,however, QFT is hardly ever represented in wave functional spacebecause usually there is little interest in measuring fieldconfigurations. Rather, one tries to measures ‘particle’ states andtherefore works in Fock space.

(iv) For a modification of proposal (iii), indicated in Baker (2009:sec. 5) and explicitly formulated as an alternative interpretation byLupher (2010), see the end of the section “Non-LocalizabilityTheorems” below.

5.1.3 Ontic Structural Realism

The multitude of problems for particle as well as fieldinterpretations prompted a number of alternative ontologicalapproaches to QFT. Auyang (1995) and Dieks (2002) propose differentversions of event ontologies. Seibt (2002) and Hättich (2004)defend process-ontological accounts of QFT, which are scrutinized inKuhlmann (2002, 2010a: ch. 10). In recent years, however, onticstructural realism (OSR) has become the most fashionable ontologicalframework for modern physics. While so far the vast majority ofstudies concentrates on ordinary QM and General Relativity Theory, itseems to be commonly believed among advocates of OSR that their caseis even stronger regarding QFT, in light of the paramount significanceof symmetry groups (also see below)—hence the namegroupstructural realism (Roberts 2010). Explicit arguments are fewand far between, however.

One of the rare arguments in favor of OSR that deal specificallywith QFT is due to Kantorovich (2003), who opts for a Platonic versionof OSR; a position that is otherwise not very popular amongOSRists. Kantorovich argues that directly after the big bang“the world was baryon-free, whereas the symmetry of grandunification existed as an abstract structure” (p. 673). Cao(1997b) points out that the best ontological access to QFT is gainedby concentrating on structural properties rather than on anyparticular category of entities. Cao (2010) advocates a“constructive structural realism” on the basis of adetailed conceptual investigation of the formation of quantumchromodynamics. However, Kuhlmann (2011) shows that Cao’s positionhas little to do with what is usually taken to be ontic structuralrealism, and that it is not even clear whether it should at least berated as an epistemic variant of structural realism.

The central significance of gauge theories in modern physics maysupport structural realism. Lyre (2004) offers a case study concerningthe \(\mathrm{U}(1)\) gauge symmetry group, which characterizesQED. Recently Lyre (2012) has been advocating an intermediate form ofOSR, which he calls “Extended OSR (ExtOSR)”, according towhich there are not only relational structural properties but alsostructurally derived intrinsic properties, namely the invariants ofstructure: mass, spin, and charge. Lyre claims that only ExtOSR is ina position to account for gauge theories. Moreover, it can make senseof zero-value properties, such as the zero mass of photons.

Category theory could be a promising framework for OSR in general and QFT in particular, because the main reservation against the radical but also seemingly incoherent idea of relations without relata may depend on the common set theoretic framework. Bain (2013) offers a category theoretic formulation of such a radical OSR, Lam and Wütrich (2015) a critique and Eva (2016) a defense. See SEP entries onstructural realism (4.2 on OSR and QFT) and oncategory theory.

5.1.4 Trope Ontology

Kuhlmann (2010a) proposes aDispositional Trope Ontology (DTO)as the most appropriate ontological reading of the basic structure ofQFT, in particular in its algebraic formulation, AQFT. The term‘trope’ refers to a conception of properties that breakswith tradition by regarding properties as particulars rather thanrepeatables (or ‘universals’). This new conception ofproperties permits analyzing objects as pure bundles ofproperties/tropes without excluding the possibility of havingdifferent objects with (qualitatively but not numerically) exactly thesame properties. One of Kuhlmann’s crucial points is that (A)QFTspeaks in favor of a bundle conception of objects because the netstructure of observable algebras alone (see section “Basic Ideasof AQFT” above) encodes the fundamental features of a givenquantum field theory, e.g., its charge structure.

In the DTO approach, the essential properties/tropes of a trope bundleare then identified with the defining characteristics of asuperselection sector, such as different kinds of charges, mass andspin. Since these properties cannot change by any state transitionthey guarantee the object’s identity over time. Superselectionsectors are inequivalent irreducible representations of the algebra ofall quasi-local observables. While the essential properties/tropes ofan object are permanent, its non-essential ones may change. Since weare dealing with quantum physical systems many properties aredispositions (or propensities); hence the namedispositionaltrope ontology.

A trope bundle is not individuated via spatio-temporal co-localizationbut because of the particularity of its constitutive tropes. Morganti(2009) also advocates a trope-ontological reading of QFT, which refersdirectly to the classification scheme of the Standard Model.

5.2 Did Wigner Define the Particle Concept?

Wigner’s (1939) famous analysis of the Poincaré group is oftenassumed to provide a definition of elementary particles. The mainidea of Wigner’s approach is the supposition that each irreducible(projective) representation of the relevant space-time symmetry groupyields the state space of one kind of elementary physical system,where the prime example is an elementary particle which has the morerestrictive property of being structureless. The physicaljustification for linking up irreducible representations withelementary systems is the requirement that “there must be norelativistically invariant distinction between the various states ofthe system” (Newton & Wigner 1949). In other words thestate space of an elementary system shall have no internal structurewith respect to relativistic transformations. Put more technically,the state space of an elementary system must not contain anyrelativistically invariant subspaces, i.e., it must be the state spaceof an irreducible representation of the relevant invariance group. Ifthe state space of an elementary system had relativistically invariantsubspaces then it would be appropriate to associate these subspaceswith elementary systems. The requirement that a state space has to berelativistically invariant means that starting from any of its statesit must be possible to get to all the other states by superposition ofthose states which result from relativistic transformations of thestate one started with. The main part of Wigner’s analysis consistsin finding and classifying all the irreducible representations of thePoincaré group. Doing that involves finding relativisticallyinvariant quantities that serve to classify the irreduciblerepresentations. Wigner’s pioneering identification of types ofparticles with irreducible unitary representations of thePoincaré group has been exemplary until the present, as it isemphasized, e.g., in Buchholz (1994). For an alternative perspectivefocusing on “Wigner’s legacy” for ontic structural realismsee Roberts (2011).

Regarding the question whether Wigner has supplied a definition ofparticles, one must say that although Wigner has in fact found ahighly valuable and fruitfulclassification of particles, hisanalysis does not contribute very much to the question what a particleis and whether a given theory can be interpreted in terms ofparticles. What Wigner has given is rather a conditionalanswer.If relativistic quantum mechanics can be interpretedin terms of particlesthen the possible types of particlesand their invariant properties can be determined via an analysis ofthe irreducible unitary representations of the Poincarégroup. However, the question whether, and if yes in what sense, atleast relativistic quantum mechanics can be interpreted as a particletheory at all is not addressed in Wigner’s analysis. For this reasonthe discussion of the particle interpretation of QFT is not finishedwith Wigner’s analysis as one might be tempted to say. For instance,the pivotal question of the localizability of particle states, to bediscussed below, is still open. Moreover, once interactions areincluded, Wigner’s classification is no longer applicable (see Bain2000). Kuhlmann (2010a: sec. 8.1.2) offers an accessible introductionto Wigner’s analysis and discusses its interpretive relevance.

5.3 Non-Localizability Theorems

The observed ‘particle traces’, e.g., on photographicplates of bubble chambers, seem to be a clear indication for theexistence of particles. However, the theory which has been built onthe basis of these scattering experiments, QFT, turns out to haveconsiderable problems to account for the observed ‘particletrajectories’. Not only are sharp trajectories excluded byHeisenberg’s uncertainty relations for position and momentumcoordinates, which hold for non-relativistic quantum mechanicsalready. More advanced examinations in AQFT show that ‘quantumparticles’ which behave according to the principles ofrelativity theory cannot be localized in any bounded region ofspace-time, no matter how large, a result which excludes eventube-like trajectories. It thus appears to be impossible that ourworld is composed of particles when we assume that localizability is anecessary ingredient of the particle concept. So far there is nosingle unquestioned argument against the possibility of a particleinterpretation of QFT but the problems are piling up. Reeh &Schlieder, Hegerfeldt, Malament and Redhead all gained mathematicalresults, or formalized their interpretation, which prove that certainsets of assumptions, which are taken to be essential for the particleconcept, lead to contradictions.

TheReeh-Schlieder theorem (1961) is a central result inAQFT. It asserts that acting on the vacuum state \(\Omega\) with elementsof the von Neumann observable algebra \(R(O)\) for openspace-time region \(O\), one can approximate as closely as onelikes any state in Hilbertspace\(\mathcal{H}\), in particular one that is verydifferent from the vacuum in some space-like separatedregion \(O'\). The Reeh-Schlieder theorem is thusexploiting long distance correlations of the vacuum. Or one canexpress the result by saying that local measurements do not allow fora distinction between an N-particle state and the vacuum state.Redhead’s (1995a) take on the Reeh-Schlieder theorem is that localmeasurements can never decide whether one observes an N-particlestate, since a projection operator \(P_{\Psi}\) whichcorresponds to an N-particle state \(\Psi\) can never be an element of alocal algebra\(R(O)\). Clifton & Halvorson (2001) discuss whatthis means for the issue of entanglement. Halvorson (2001) shows thatan alternative “Newton-Wigner” localization scheme failsto evade the problem of localization posed by the Reeh-Schliedertheorem.

Malament (1996) formulates ano-go theorem to the effectthat a relativistic quantum theory of a fixed number of particlespredicts a zero probability for finding a particle in any spatial set,provided four conditions are satisfied, namely concerning translationcovariance, energy, localizability and locality.Thelocalizability condition is the essential ingredient ofthe particle concept: A particle—in contrast to afield—cannot be found in two disjoint spatial sets at the sametime. Thelocality condition is the main relativistic partof Malament’s assumptions. It requires that the statistics formeasurements in one space-time region must not depend on whether ornot a measurement has been performed in a space-like related secondspace-time region. Malament’s proof has the weight of a no-go theoremprovided that we accept his four conditions as natural assumptions fora particle interpretation. A relativistic quantum theory of a fixednumber of particles, satisfying in particular the localizability andthe locality condition, has to assume a world devoid of particles (orat least a world in which particles can never be detected) in ordernot to contradict itself. Malament’s no-go theorem thus seems to showthat there is no middle ground between QM and QFT, i.e., no theorywhich deals with a fixed number of particles (like in QM) and which isrelativistic (like QFT) without running into the localizabilityproblem of the no-go theorem. One is forced towards QFT which, asMalament is convinced, can only be understood as a fieldtheory. Nevertheless, whether or not a particle interpretation of QFTis in fact ruled out by Malament’s result is a point of debate. Atleast prima facie Malament’s no-go theorem alone cannot supply a finalanswer since it assumes a fixed number of particles, an assumptionthat is not valid in the case of QFT.

The results about non-localizability which have been explored abovemay appear to be not very astonishing in the light of the followingfacts about ordinary QM: Quantum mechanical wave functions (inposition representation) are usually smeared out over all \(\Re^3\), so that everywhere in space there is a non-vanishing probability forfinding a particle. This is even the case arbitrarily close after asharp position measurement due to the instantaneous spreading of wavepackets over all space. Note, however, that ordinary QM isnon-relativistic. A conflict with SRT would thus not be verysurprising although it is not yet clear whether the above-mentionedquantum mechanical phenomena can actually be exploited to allow forsuperluminal signalling. QFT, on the other side, has been designed tobe in accordance with special relativity theory (SRT). The localbehavior of phenomena is one of the leading principles upon which thetheory was built. This makes non-localizability within the formalismof QFT a much severer problem for a particle interpretation.

Malament’s reasoning has come under attack in Fleming &Butterfield (1999) and Busch (1999). Both argue to the effect thatthere arealternatives to Malament’s conclusion. The main lineof thought in both criticisms is that Malament’s ‘mathematicalresult’ might just as well be interpreted as evidence that theassumed concept of a sharp localization operator is flawed and has tobe modified either by allowing for unsharp localization (Busch 1999)or for so-called “hyperplane dependent localization”(Fleming & Butterfield 1999). In Saunders (1995) a differentconclusion from Malament’s (as well as from similar) results isdrawn. Rather than granting Malament’s four conditions and deriving aproblem for a particle interpretation Saunders takes Malament’s proofas further evidence that one can not hold on to all fourconditions. According to Saunders it is the localizability conditionwhich might not be a natural and necessary requirement on secondthought. Stressing that “relativity requires the language ofevents, not of things” Saunders argues that the localizabilitycondition loses its plausibility when it is applied to events: Itmakes no sense to postulate that the same event can not occur at twodisjoint spatial sets at the same time. One can only require for thesamekind of event not to occur at both places. For Saundersthe particle interpretation as such is not at stake in Malament’sargument. The question is rather whether QFT speaks about things atall. Saunders considers Malament’s result to give a negative answer tothis question. A kind of meta paper on Malament’s theorem isHalvorson & Clifton (2002). Various objections to the choice ofMalament’s assumptions and his conclusion are considered andrebutted. Moreover, Halvorson and Clifton establish two further no-gotheorems which preserve Malament’s theorem by weakening tacitassumptions and showing that the general conclusion still holds. Onething seems to be clear. Since Malament’s ‘mathematicalresult’ appears to allow for various different conclusions itcannot be taken as conclusive evidence against the tenability of aparticle interpretation of QFT and the same applies to Redhead’sinterpretation of the Reeh-Schlieder theorem. For a more detailedexposition and comparison of the Reeh-Schlieder theorem and Malament’stheorem see Kuhlmann (2010a: sec. 8.3).

Does thefield interpretation also suffer from problemsconcerning non-localizability? In the section “Deficiencies ofthe Conventional Formulation of QFT” we already saw that,strictly speaking, field operators cannot be defined at points butneed to be smeared out in the (finite and arbitrarily small) vicinityof points, giving rise to smeared fieldoperators \(\hat{\phi}(f)\),which represent the weighted average field value in the respectiveregion. This procedure leads to operator-valued distributions insteadof operator-valued fields. The lack of field operators at pointsappears to be analogous to the lack of position operators in QFT,which troubles the particle interpretation. However, for positionoperators there is no remedy analogous to that for field operators:while even unsharply localized particle positions do not exist in QFT(see Halvorson and Clifton 2002, theorem 2), the existence of smearedfield operators demonstrates that there are at least point-like fieldoperators. On this basis Lupher (2010) proposes a “modifiedfield ontology”.

5.4 The Role of Symmetries

Symmetries play a central role in QFT. In order to characterize aspecial symmetry one has to specify transformations T and featuresthat remain unchanged during these transformations: invariants I.Symmetries are thus pairs \(\{\)T, I\(\}\). The basic idea is that thetransformations change elements of the mathematical description (theLagrangians for instance) whereas the empirical content of the theoryis unchanged. There are space-time transformations and so-calledinternal transformations. Whereas space-time symmetries are universal,i. e., they are valid for all interactions, internal symmetriescharacterize special sorts of interaction (electromagnetic, weak orstrong interaction). Symmetry transformations define properties ofparticles/quantum fields that are conserved if the symmetry is notbroken. The invariance of a system defines a conservation law, e.g.,if a system is invariant under translations the linear momentum isconserved, if it is invariant under rotation the angular momentum isconserved. Inner transformations, such as gauge transformations, areconnected with more abstract properties.

Symmetries are not only defined for Lagrangians but they can also befound in empirical data and phenomenological descriptions. Symmetriescan thus bridge the gap between descriptions which are close toempirical results (‘phenomenology’) and the more abstractgeneral theory which is a most important reason for their heuristicforce. If a conservation law is found one has some knowledge about thesystem even if details of the dynamics are unknown. The analysis ofmany high energy collision experiments led to the assumption ofspecial conservation laws for abstract properties like baryon numberor strangeness. Evaluating experiments in this way allowed for aclassification of particles. This phenomenological classification wasgood enough to predict new particles which could be found in theexperiments. Free places in the classification could be filled even ifthe dynamics of the theory (for example the Lagrangian of stronginteraction) was yet unknown. As the history of QFT for stronginteraction shows, symmetries found in the phenomenologicaldescription often lead to valuable constraints for the construction of thedynamical equations. Arguments from group theory played a decisive rolein the unification of fundamental interactions. In addition,symmetries bring about substantial technical advantages. For example,by using gauge transformations one can bring the Lagrangian into aform which makes it easy to prove the renormalizability of the theory.See also the entry onsymmetry and symmetry breaking.

In many cases symmetries are not only heuristically useful but supplysome sort of ‘justification’ by being used in thebeginning of a chain of explanation. To a remarkable degree thepresent theories of elementary particle interactions can be understoodby deduction from general principles. Under these principles symmetryrequirements play a crucial role in order to determine theLagrangian. For example, the only Lorentz invariant and gaugeinvariant renormalizable Lagrangian for photons and electrons isprecisely the original Dirac Lagrangian. In this way symmetryarguments acquire an explanatory power and help to minimize theunexplained basic assumptions of a theory. Heisenberg concludes thatin order “to find the way to a real understanding of thespectrum of particles it will therefore be necessary to look for thefundamental symmetries and not for the fundamental particles.”(Blumet al. 1995: 507).

Since symmetry operations change the perspective of an observer butnot the physics an analysis of the relevant symmetry group can yieldvery general information about those entities which are unchanged bytransformations. Such an invariance under a symmetry group is anecessary (but not sufficient) requirement for something to belong tothe ontology of the considered physical theory. Hermann Weylpropagated the idea that objectivity is associated with invariance(see, e.g., his authoritative work Weyl 1952: 132). Auyang (1995)stresses the connection between properties of physically relevantsymmetry groups and ontological questions. Kosso argues thatsymmetries help to separate objective facts from the conventions ofdescriptions; see his article in Brading & Castellani (2003), ananthology containing numerous further philosophical studies aboutsymmetries in physics.

Symmetries are typical examples of structures that show morecontinuity in scientific change than assumptions about objects. Forthat reason structural realists consider structures as “the bestcandidate for what is ‘true’ about a physicaltheory” (Redhead 1999: 34). Physical objects such as electronsare then taken to be similar to fiction that should not be takenseriously, in the end. In the epistemic variant of structural realismstructure is all we know about nature whereas the objects which arerelated by structures might exist but they are not accessible tous. For the extreme ontic structural realist there is nothing butstructures in the world (Ladyman 1998).

5.5 Taking Stock: Where do we Stand?

Aparticle interpretation of QFT answers most intuitively whathappens in particle scattering experiments and why we seem to detectparticle trajectories. Moreover, it would explain most naturally whyparticle talk appears almost unavoidable. However, the particleinterpretation in particular is troubled by numerous seriousproblems. There are no-go theorems to the effect that, in arelativistic setting, quantum “particle” states cannot belocalized in any finite region of space-time no matter how large itis. Besides localizability, another hard core requirement for theparticle concept that seems to be violated in QFT iscountability. First, many take the Unruh effect to indicate that theparticle number is observer or context dependent. And second,interacting quantum field theories cannot be interpreted in terms ofparticles because their representations are unitarily inequivalent toFock space (Haag’s theorem), which is the only known way to representcountable entities in systems with an infinite number of degrees offreedom.

At first sight thefield interpretation seems to be much betteroff, considering that a field is not a localized entity and that itmay vary continuously—so no requirements for localizability andcountability. Accordingly, the field interpretation is often taken tobe implied by the failure of the particle interpretation. However, oncloser scrutiny the field interpretation itself is not above reproach.To begin with, since “quantum fields” are operator valuedit is not clear in which sense QFT should be describing physicalfields, i.e., as ascribing physical properties to points in space. Inorder to get determinate physical properties, or even justprobabilities, one needs a quantum state. However, since quantumstates as such are not spatio-temporally defined, it is questionablewhether field values calculated with their help can still be viewed aslocal properties. The second serious challenge is that the arguablystrongest field interpretation—the wave functionalversion—may be hit by similar problems as the particleinterpretation, since wave functional space is unitarily equivalent toFock space.

The occurrence ofunitarily inequivalent representations(UIRs), which first seemed to cause problems specifically for theparticle interpretation but which appears to carry over to the fieldinterpretation, may well be a severe obstacle for any ontologicalinterpretation of QFT. However, it is controversial whether the twomost prominent examples, namely the Unruh effect and Haag’s theorem,really do cause the contended problems in the first place. Thus one ofthe crucial tasks for the philosophy of QFT is further unmasking theontological significance of UIRs.

The two remaining contestants approach QFT in a way that breaks moreradically with traditional ontologies than any of the proposedparticle and field interpretations.Ontic Structural Realism(OSR) takes the paramount significance of symmetry groups toindicate that symmetry structures as such have an ontological primacyover objects. However, since most OSRists are decidedly againstPlatonism, it is not altogether clear how symmetry structures could beontologically prior to objects if they only exist in concreterealizations, namely in those objects that exhibit thesesymmetries.

Dispositional Trope Ontology (DTO) deprives both particlesand fields of their fundamental status, and proposes an ontology whosebasic elements are properties understood as particulars, called‘tropes’. One of the advantages of the DTO approach is itsgreat generality concerning the nature of objects which it analyzes asbundles of (partly dispositional) properties/tropes: DTO is flexibleenough to encompass both particle and field like features withoutbeing committed to either a particle or a field ontology.

In conclusion one has to recall that one reason why the ontologicalinterpretation of QFT is so difficult is the fact that it isexceptionally unclear which parts of the formalism should be taken torepresent anything physical in the first place. And it looks as ifthat problem will persist for quite some time.

Bibliography

• Arageorgis, A., 1995,Fields, Particles, and Curvature:Foundations and Philosophical Aspects of Quantum Field Theory inCurved Spacetime, Ph.D. thesis, University ofPittsburgh.
• Auyang, S. Y., 1995,How is Quantum Field TheoryPossible?, Oxford-New York: Oxford University Press.
• Bain, J., 2000, “Against particle/field duality: Asymptoticparticle states and interpolating fields in interacting QFT (or:Who’s afraid of Haag’stheorem?)”,Erkenntnis, 53: 375–406.
• –––, 2011, “Quantum fieldtheories in classical spacetimes and particles”,Studies inHistory and Philosophy of Modern Physics, 42: 98–106.
• –––, 2013, “Category-theoretic structure and radical ontic structural realism”,Synthese, 190: 1621–1635.
• Baker, D. J., 2009, “Against fieldinterpretations of quantum field theory”,British Journal for the Philosophy of Science, 60: 585–609.
• Baker, D.J. and H. Halvorson, 2010,“Antimatter”,British Journal for the Philosophy ofScience, 61: 93–121.
• Born, M., with W. Heisenberg, and P. Jordan, 1926, “ZurQuantenmechanik II”,Zeitschr. für Physik 35, 557.
• Brading, K. and E. Castellani (eds.), 2003,Symmetries inPhysics: Philosophical Reflections, Cambridge: CambridgeUniversity Press.
• Bratteli, O. and D. W. Robinson, 1979,Operator Algebras andQuantum Statistical Mechanics 1: \(C^*\) and\(W^*\)-Algebras, Symmetry Groups, Decomposition ofStates, New York et al.: Springer
• Brown, H. R. and R. Harré (eds.), 1988,Philosophical Foundations of Quantum Field Theory,Oxford: Clarendon Press.
• Buchholz, D., 1994, “On the manifestations ofparticles”, in R. N. Sen and A. Gersten, eds.,Mathematical Physics Towards the 21st Century, Beer-Sheva:Ben-Gurion University Press.
• –––, 1998, “Current trends in axiomatic qantumfield theory”, in P. Breitenlohner and D. Maison, eds,Quantum Field Theory. Proceedings of the Ringberg Workshop1998, pp. 43–64, Berlin-Heidelberg: Springer.
• Busch, P., 1999, “Unsharp localization and causality inrelativistic quantum theory”,Journal of Physics A:Mathematics General, 32: 6535.
• Butterfield, J. and H. Halvorson (eds.), 2004,QuantumEntanglements — Selected Papers — Rob Clifton,Oxford: Oxford University Press.
• Butterfield, J. and C. Pagonis (eds.), 1999,From Physics toPhilosophy, Cambridge: Cambridge University Press.
• Callender, C. and N. Huggett (eds.), 2001,Physics MeetsPhilosophy at the Planck Scale, Cambridge: Cambridge UniversityPress.
• Cao, T. Y., 1997a,Conceptual Developments of 20thCentury Field Theories, Cambridge: Cambridge University Press.
• –––, 1997b, “Introduction: Conceptual issues inQFT”, in Cao 1997a, pp. 1–27.
• –––, (ed.), 1999,Conceptual Foundations of QuantumField Theories, Cambridge: Cambridge University Press.
• –––, 2010,From Current Algebra toQuantum Chromodynamics: A Case for Structural Realism, Cambridge:Cambridge University Press.
• Castellani, E., 2002, “Reductionism, emergence, andeffective field theories”,Studies in History and Philosophyof Modern Physics, 33: 251–267.
• Clifton, R. (ed.), 1996,Perspectives on QuantumReality: Non-Relativistic, Relativistic, and Field-Theoretic,Dordrecht et al.: Kluwer.
• Clifton, R. and H. Halvorson, 2001, “Entanglement and opensystems in algebraic quantum field theory”,Studies inHistory and Philosophy of Modern Physics, 32: 1–31; reprinted in Butterfield & Halvorson 2004.
• Davies, P. (ed.), 1989,The New Physics,Cambridge: Cambridge University Press.
• Dawid, R., 2009, “On the conflictingassessments of string theory”,Philosophy of Science, 76:984–996.
• Dieks, D., 2002, “Events and covariance in theinterpretation of quantum field theory”, in Kuhlmannet al. 2002, pp. 215–234.
• Dieks, D. and A. Lubberdink, 2011,“How classical particles emerge from the quantum world”,Foundations of Physics, 41: 1051–1064.
• Dirac, P. A. M., 1927, “The quantum theory of emission andabsorption of radiation”,Proceedings of the Royal Societyof London, A 114: 243–256.
• Earman, J., 2011, “The Unruh effect forphilosophers”,Studies In History and Philosophy of ModernPhysics, 42: 81 – 97.
• Earman, J. and D. Fraser, 2006,“Haag’s theorem and its implications for the foundations ofquantum field theory”,Erkenntnis, 64: 305–344.
• Egg, M., Lam, V. and A. Oldofredi, 2017, “Particles, cutoffs and inequivalent representations — Fraser and Wallace on quantum field theory”,Foundation of Physics, 47: 453–466.
• Eva, B., 2016, “Category theory and physical structuralism”,European Journal for Philosophy of Science, 6: 231–246.
• Feintzeig, B., 2018, “Toward an understanding of parochialobservables”,British Journal for the Philosophy ofScience, 69: 161–191.
• Feintzeig, B., Le Manchak,, J.B., Rosenstock, S., and J. Weatherall,2019, “Why Be regular?, part I”,Studies in Historyand Philosophy of Modern Physics, 65: 122–132
• Feintzeig, B., and J. Weatherall, 2019, “Why Be regular?, partII”,Studies in History and Philosophy of ModernPhysics, 65: 133–144.
• Fell, J., 1960, “The dual spaces of C*-algebras”,Transactions of the American Mathematical Society, 94:365–403.
• Fleming, G. N. and J. Butterfield, 1999, “Strangepositions”, in Butterfield & Pagonis 1999, pp. 108–165.
• Fraser, D., 2008, “The fate of“particles” in quantum field theories withinteractions”,Studies in History and Philosophy of ModernPhysics, 39: 841–59.
• –––, 2009, “Quantum field theory:Underdetermination, inconsistency, and idealization”,Philosophyof Science, 76: 536–567.
• –––, 2011, “How to take particlephysics seriously: A further defence of axiomatic quantum fieldtheory”,Studies in History and Philosophy of Modern Physics, 42:126–135.
• Fraser, J. D., 2018, “The real problem with perturbative quantum field theory”,The British Journal for the Philosophy of Science,first online 28 June 2018. doi:10.1093/bjps/axx042
• Georgi, H., 1989, “Effective quantum field theories”,in Davies 1989, pp. 446–457.
• Greene, B., 1999,The Elegant Universe. Superstrings, HiddenDimensions and the Quest for the Ultimate Theory, New York:W. W. Norton and Company.
• Haag, R., 1996,Local Quantum Physics: Fields, Particles,Algebras, 2nd edition, Berlin et al.:Springer.
• Haag, R. and D. Kastler, 1964, “An algebraic approach toquantum field theory”,Journal of Mathematical Physics,5: 848–861.
• Halvorson, H., 2001, “Reeh-Schliederdefeats Newton-Wigner: On alternative localization schemes inrelativistic quantum field theory”,Philosophy of Science, 68:111–133.
• Halvorson, H. and R. Clifton, 2002, “No place for particlesin relativistic quantum theories?”Philosophy ofScience, 69: 1–28; reprinted in Butterfield and Halvorson 2004and in Kuhlmannet al. 2002.
• Halvorson, H. and M. Müger, 2007,“Algebraic quantum field theory (with an appendix by MichaelMüger)”, inHandbook of the Philosophy of Physics —Part A, Jeremy Butterfield and John Earman (eds.), Amsterdam:Elsevier, 731–922.
• Hartmann, S., 2001, “Effective field theories, reductionism,and explanation”,Studies in History and Philosophy of ModernPhysics, 32: 267–304.
• Hättich, F., 2004,Quantum Processes — A WhiteheadianInterpretation of Quantum Field Theory, Münster: agenda Verlag.
• Healey, R., 2007,Gauging What’s Real:The Conceptual Foundations of Contemporary Gauge Theories, Oxford:Oxford University Press.
• Heisenberg, W. and W. Pauli, 1929, “Zur Quantendynamik derWellenfelder”,Zeitschrift für Physik, 56: 1–61.
• Hoddeson, L., with L. Brown, M. Riordan, and M. Dresden (eds.),1997,The Rise of the Standard Model: A History of Particle Physicsfrom 1964 to 1979, Cambridge: Cambridge University Press.
• Horuzhy, S. S., 1990,Introduction to Algebraic QuantumField Theory, 1st edition, Dordrecht et al.: Kluwer.
• Huggett, N., 2000, “Philosophicalfoundations of quantum field theory”,The British Journal for thePhilosophy Science, 51: 617–637.
• –––, 2003, “Philosophicalfoundations of quantum field theory”, inPhilosophy of ScienceToday, P. Clark and K. Hawley, eds., Oxford: Clarendon Press,617?37.
• Johansson, L. G. and K. Matsubara, 2011, “String theory andgeneral methodology: A mutual evaluation”,Studies inHistory and Philosophy of Modern Physics, 42: 199–210.
• Kadison, R., and J. Ringrose, 1997,Fundamentals of the Theory ofOperator Algebras, Providence, RI: American MathematicalSociety.
• Kaku, M., 1999,Introduction to Superstrings and M-Theory,New York: Springer.
• Kantorovich, A., 2003, “The priority ofinternal symmetries in particle physics”,Studies in History andPhilosophy of Modern Physics, 34: 651–675.
• Kastler, D. (ed.), 1990,The Algebraic Theory ofSuperselection Sectors: Introduction and Recent Results,Singapore et al.: World Scientific.
• Kiefer, C., 2007,Quantum Gravity, Oxford: OxfordUniversity Press. Second edition.
• Kronz, F. and T. Lupher, 2005, “Unitarily inequivalentrepresentations in algebraic quantum theory”,InternationalJournal of Theoretical Physics, 44: 1239–1258.
• Kuhlmann, M., 2010a,The UltimateConstituents of the Material World – In Search of an Ontology forFundamental Physics, Frankfurt: ontos Verlag.
• –––, 2010b, “Why conceptualrigour matters to philosophy: On the ontological significance ofalgebraic quantum field theory”,Foundations of Physics, 40:1625–1637.
• –––, 2011, “Review of“From Current Algebra to Quantum Chromodynamics: A Case forStructural Realism” by T. Y. Cao”,Notre Dame PhilosophicalReviews,available online.
• Kuhlmann, M. with H. Lyre and A. Wayne (eds.),2002,Ontological Aspects of Quantum Field Theory, London:World Scientific Publishing.
• Ladyman, J., 1998, “What is structural realism?”Studies in History and Philosophy of Science, 29:409–424.
• Lam, V. and Wütrich, C., 2015, “No categorical support for radical ontic structural realism”,British Journal for the Philosophy of Science, 66:605–634.
• Landsman, N. P., 1996, “Local quantum physics”,Studies in History and Philosophy of Modern Physics, 27:511–525.
• Lupher, T., 2010, “Not particles, notquite fields: An ontology for quantum field theory”,HumanaMente, 13: 155–173.
• –––, 2018, “The limits of physical equivalence inalgebraic quantum field theory”,British Journal for thePhilosophy of Science, 69: 553–576.
• Lyre, H., 2004, “Holism and structuralism inU(1) gauge theory”,Studies in History and Philosophy ofModern Physics, 35/4: 643–670.
• –––, 2012, “Structural invariants, structuralkinds, structural laws”, inProbabilities, Laws, andStructures, Dordrecht: Springer, 179–191.
• Malament, D., 1996, “In defense of dogma: Why there cannotbe a relativistic quantum mechanics of (localizable) particles”,in Clifton 1996, pp. 1–10.
• Mandl, F. and G. Shaw, 2010,Quantum FieldTheory, Chichester (UK): John Wiley & Sons, second ed.
• Martin, C. A., 2002, “Gauge principles, gauge argumentsand the logic of nature”,Philosophy of Science,69/3: 221–234.
• Miller, M.E., 2018, “Haag’s Theorem, apparent inconsistency, and the empirical adequacy of Quantum FieldTheory”,British Journal for the Philosophy of Science,69: 801–820.
• Morganti, M., 2009, “Tropes andphysics”,Grazer Philosophische Studien, 78:185–205.
• Newton, T. D. and E. P. Wigner, 1949, “Localizedstates for elementary particles”,Reviews of ModernPhysics, 21/3: 400–406.
• Passon, O., 2019, “On the interpretation of Feynmandiagrams, or, did the LHC experiments observe \(H \rightarrow \gamma\gamma\)?”,European Journal for the Philosophy ofScience, 9: 20.
• Peskin, M. E. and D. V. Schroeder,1995,Introduction to Quantum Field Theory, Cambridge (MA):Perseus Books.
• Polchinski, J., 2000,String Theory, 2 volumes, Cambridge: Cambridge University Press.
• Redhead, M. L. G., 1995a, “More ado about nothing”,Foundations of Physics, 25: 123–137.
• –––, 1995b, “The vacuum in relativisticquantum field theory”, in Hullet al. 1994 (vol. 2),pp. 88–89.
• –––, 1999, “Quantum field theory and thephilosopher”, in Cao 1999, pp. 34–40.
• –––, 2002, “The interpretation of gaugesymmetry”, in Kuhlmannet al. 2002, pp. 281–301.
• Reeh, H. and S. Schlieder, 1961, “Bemerkungen zurUnitäräquivalenz von Lorentzinvarianten Feldern”,Nuovo Cimento, 22: 1051–1068.
• Rickles, D., 2008, “Quantum gravity: A primer forphilosophers”, inThe Ashgate Companion to ContemporaryPhilosophy of Physics, Dean Rickles (ed.), Aldershot: Ashgate,262–382.
• Roberts, B. W., 2011, “Groupstructural realism”,The British Journal for the PhilosophyScience, 62: 47?69.
• Roberts, J. E., 1990, “Lectures on algebraic quantum fieldtheory”, in Kastler 1990, pp. 1–112.
• Ruetsche, L., 2002, “Interpreting quantumfield theory”,Philosophy of Science, 69: 348–378.
• –––, 2003, “A matter of degree: Putting unitaryequivalence to work”,Philosophy of Science, 70/5:1329–1342.
• –––, 2006, “Johnny’s solong at the ferromagnet”,Philosophy of Science, 73:473–486.
• –––, 2011, “Why benormal?”,Studies in History and Philosophy of ModernPhysics, 42: 107–115.
• –––, 2011,Interpreting QuantumTheories, New York: Oxford University Press.
• –––, 2012a, “Philosophical aspects of quantum field theory: I”,Philosophy Compass, 7/8: 559–570.
• –––, 2012b, “Philosophical aspects of quantum field theory: II”,Philosophy Compass, 7/8: 571–584.
• –––, 2018, “Renormalization group realism:The ascent of pessimism”,Philosophy of Science, 85:1176–1189.
• Ryder, L. H., 1996,Quantum Field Theory, 2nd edition,Cambridge: Cambridge University Press.
• Saunders, S., 1995, “A dissolution of the problem oflocality”, in Hull, M. F. D., Forbes, M., and Burian, R. M.,eds., 1995,Proceedings of the Biennial Meeting of the Philosophyof Science Association: PSA 1994, East Lansing, MI: Philosophy ofScience Association, vol. 2, pp. 88–98.
• Saunders, S. and H. R. Brown (eds.), 1991,The Philosophy ofVacuum, Oxford: Clarendon Press.
• Schweber, S. S., 1994,QED and the Men Who Made It”,Princeton: Princeton University Press.
• Segal, I. E., 1947, “Postulates for general quantummechanics”,Annals of Mathematics, 48/4:930–948.
• Seibt, J., 2002, “The matrix of ontological thinking:Heuristic preliminaries for an ontology of QFT”, in Kuhlmannet al. 2002, pp. 53–97.
• Streater, R. F. and A. S. Wightman, 1964,PCT, Spin andStatistics, and all that, New York: Benjamin.
• Swanson, N., 2017, “A philosopher’s guide to thefoundations of quantum field theory”,PhilosophyCompass, 12(5): e12414.
• Teller, P., 1995,An Interpretive Introduction to QuantumField Theory, Princeton: Princeton University Press.
• Unruh, W. G., 1976, “Notes on black hole evaporation”,Physical Review D, 14: 870–92.
• Unruh, W. G. and R. M. Wald, 1984, “What happens when anaccelerating observer detects a Rindler particle?”Physical Review D, 29: 1047–1056.
• Wallace, D., 2006, “In defence ofnaiveté: The conceptual status of Lagrangian quantum fieldtheory”,Synthese, 151: 33–80.
• –––, 2011, “Taking particlephysics seriously: A critique of the algebraic approach to quantumfield theory”,Studies in History and Philosophy of ModernPhysics, 42: 116–125.
• Wayne, Andrew, 2002, “A naive view of the quantumfield”, in Kuhlmann et al. 2002, 127–133.
• –––, 2008, “A trope-bundle ontology for fieldtheory”, inThe Ontology of Spacetime II, Dennis Dieks(ed.), Amsterdam: Elsevier, 1–15.
• Weinberg, S., 1995,The Quantum Theory of Fields –Foundations (Volume 1), Cambridge: Cambridge UniversityPress.
• –––, 1996,The Quantum Theory of Fields– Modern Applications (Volume 2), Cambridge: CambridgeUniversity Press.
• Weingard, R., 2001, “A philosopher looks at string theory”,in Callender & Huggett 2001, pp. 138–151.
• Weyl, H., 1952,Symmetry, Princeton: Princeton UniversityPress.
• Wightman, A. S., 1956, “Quantumfield theory in terms of vacuum expectation values”,PhysicalReview, 101: 860–66.
• Wigner, E. P., 1939, “On unitary representations of theinhomoneneous Lorentz group”,Annals of Mathematics,40: 149–204.
• Williams, P., 2019, “Scientific realism made effective”,British Journal for the Philosophy of Science, 70: 209–237.

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