An ongoing debate in the foundations of quantum physics concerns the role ofmathematical rigor. The contrasting views of von Neumannand Dirac provide interesting and informative insights concerning twosides of this debate. Von Neumann’s contributions often emphasizemathematical rigor and Dirac’s contributions emphasize pragmaticconcerns. The discussion below begins with an assessment of theircontributions to the foundations of quantum mechanics. Theircontributions to mathematical physics beyond quantum mechanics arethen considered, and the focus will be on the influence that thesecontributions had on subsequent developments in quantum theorizing,particularly with regards to quantum field theory and itsfoundations. The entryquantum field theory provides an overview of a variety of approaches todeveloping a quantum theory of fields. The purpose of this article isto provide a more detailed discussion of mathematically rigorousapproaches to quantum field theory, as opposed to conventionalapproaches, such as Lagrangian quantum field theory, which aregenerally portrayed as being more heuristic in character. The currentdebate concerning whether Lagrangian quantum field theory or axiomaticquantum field theory should serve as the basis for interpretiveanalysis is then discussed.

• 1. Introduction
• 2. Von Neumann and the Foundations of Quantum Theory
  • 2.1 The Separable Hilbert Space Formulation of Quantum Mechanics
  • 2.2 Rings of Operators, Quantum Logics, and Continuous Geometries
• 3. Dirac and the Foundations of Quantum Theory
  • 3.1 Dirac’s \(\delta\)-Function,Principles, and Bra-Ket Notation
  • 3.2 The Rigged Hilbert Space Formulation of Quantum Mechanics
• 4 Mathematical Rigor: Two Paths
  • 4.1 Algebraic Quantum Field Theory
  • 4.2 Wightman’s Axiomatic Quantum Field Theory
• 5. Philosophical Issues
  • 5.1 Pragmatics versus Axiomatics
  • 5.2 Middle Grounds
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

There are two competing mathematical strategies that are used inconnection with physical theory, one emphasizes rigor and the otherpragmatics. The pragmatic approach often compromises mathematicalrigor, but offers instead expediency of calculation and elegance ofexpression. A case in point is the notion of an infinitesimal, anon-zero quantity that is smaller than any finitequantity. Infinitesimals were used by Kepler, Galileo, Newton, Leibnizand many others in developing and using their respective physicaltheories, despite lacking a mathematically rigorous foundation, asBerkeley clearly showed in his famous 1734 treatiseTheAnalyst criticizing infinitesimals. Such criticisms did notprevent various 18th Century mathematicians, scientists, and engineerssuch as Euler and Lagrange from using infinitesimals to get accurateanswers from their calculations. Nevertheless, the pull towards rigorled to the development in the 19th century of the concept of a limitby Cauchy and others, which provided a rigorous mathematical frameworkthat effectively replaced the theory of infinitesimals. A rigorousfoundation was eventually provided for infinitesimals by Robinsonduring the second half of the 20th Century, but infinitesimals arerarely used in contemporary physics. For more on the history ofinfinitesimals, see the entry oncontinuity and infinitesimals.

The competing mathematical strategies are manifest in a more recentdiscussion concerning the mathematical foundations of quantummechanics. In the preface to von Neumann’s (1955) treatise on thattopic, he notes that Dirac provides a very elegant and powerful formalframework for quantum mechanics, but complains about the central rolein that framework of an “improper function with self-contradictoryproperties,” which he also characterizes as a “mathematical fiction.”He is referring to the Dirac \(\delta\) function, which has the followingincompatible properties: it is defined over the real line, is zeroeverywhere except for one point at which it is infinite, and yieldsunity when integrated over the real line. Von Neumann promotes analternative framework, which he characterizes as being “just as clearand unified, but without mathematical objections.” He emphasizes thathis framework is not merely a refinement of Dirac’s; rather, it is aradically different framework that is based on Hilbert’s theory ofoperators.

Dirac is of course fully aware that the \(\delta\) function is not awell-defined expression. But he is not troubled by this for tworeasons. First, as long as one follows the rules governing the \(\delta\)function (such as using the \(\delta\) function only under an integralsign, meaning in part not asking the value of a \(\delta\) function at agiven point), then no inconsistencies will arise. Second, the \(\delta\)function can be eliminated, meaning that it can be replaced with awell-defined mathematical expression. However, the drawback in thatcase is, according to Dirac, that the substitution leads to a morecumbersome expression that obscures the argument. In short, whenpragmatics and rigor lead to the same conclusion, pragmatics trumpsrigor due to the resulting simplicity, efficiency, and increase inunderstanding.

As in the case of the notion of an infinitesimal, the Dirac \(\delta\)function was eventually given a mathematically rigorousfoundation. That was done within Schwartz’s theory of distributions,which was later used in developing the notion of a rigged Hilbertspace. The theory of distributions was used to provide a mathematicalframework for quantum field theory (Wightman 1964). The rigged Hilbertspace was used to do so for quantum mechanics (Böhm 1966) andthen for quantum field theory (Bogoluliubov et al. 1975).

The complementary approaches, rigor and pragmatics, which areexhibited in the development of quantum mechanics, later came about ina more striking way in connection with the development of quantumelectrodynamics (QED) and, more generally, quantum field theory(QFT). The emphasis on rigor emerges in connection with twoframeworks, algebraic QFT and Wightman’s axiomatic QFT. Algebraic QFThas its roots in the work of von Neumann on operator algebras, whichwas developed by him in an attempt to generalize the Hilbert spaceframework. Wightman’s axiomatic QFT has its roots in Schwartz’s theoryof distributions, and it was later developed in the rigged Hilbertspace framework. Roughly, the basic distinction between the twoapproaches is that the algebra of operators is the basic mathematicalconcept in algebraic QFT, while operator-valued distributions (thequantum analogues of field quantities) are fundamental in Wightman’saxiomatic QFT. It is worth noting that algebraic QFT is generallyformulated axiomatically, and that it is just as deserving of the name“axiomatic” QFT. However, that term is often taken to referspecifically to the approach based on operator-valueddistributions. To avoid any possible confusion, that approach isreferred to here as “Wightman’s axiomatic” QFT. The emphasis onpragmatics arises most notably in Lagrangian QFT, which usesperturbation theory, path integrals, and renormalizationtechniques. Although some elements of the theory were eventuallyplaced on a firmer mathematical foundation, there are still seriousquestions about its being a fully rigorous approach on a par withalgebraic and Wightman’s axiomatic QFT. Nevertheless, it has beenspectacularly successful in providing numerical results that areexceptionally accurate with respect to experimentally determinedquantities, and in making possible expedient calculations that areunrivaled by other approaches.

The two approaches to QFT continue to develop in parallel. Fleming(2002, 135–136) brings this into focus in his discussion ofdifferences between Haag’sLocal Quantum Physics (1996) andWeinberg’sQuantum Field Theory (1995); Haag’s book presentsalgebraic QFT, and Weinberg’s book presents Lagrangian QFT. While bothbooks are ostensibly about the same subject, Haag gives a preciseformulation of QFT and its mathematical structure, but does notprovide any techniques for connecting with experimentally determinedquantities, such as scattering cross sections. Weinberg gives apragmatic formulation that engages with physical intuition andprovides heuristics that are important for performing calculations;however, it is not as mathematically rigorous. Moreover, there are anumber of important topics that are examined in one book while noteven mentioned in the other. For example, unitarily inequivalentrepresentations are discussed by Haag, but not by Weinberg. Bycontrast, Weinberg discusses Feynman’s rules for path integrals, whichare not mentioned at all by Haag. There is also the issue ofdemographics. Most particle and experimental physicists will read andstudy Weinberg’s book, but very few will read Haag’s book. Because ofthese differences, Fleming (2002, 136) suggests that one mightquestion whether the two books are really about the same subject. Thisgives rise to the question whether any formulation of QFT is worthy ofphilosophical attention to its foundations. In particular, there is adebate between Wallace (2006, 2011) and Fraser (2009, 2011) overwhether an interpretation of QFT should be based on the standardtextbook treatment of QFT or an axiomatic formulation of QFT.

2. Von Neumann and the Foundations of Quantum Theory

In the late 1920s, von Neumann developed the separable Hilbert spaceformulation of quantum mechanics, which later became the definitiveone (from the standpoint of mathematical rigor, at least). In themid-1930s, he worked extensively on lattice theory (see the entryonquantum logic), rings of operators, and continuous geometries. Part of hisexpressed motivation for developing these mathematical theories was todevelop an appropriate framework for QFT and a better foundation forquantum mechanics. During this time, he noted two closely relatedstructures, modular lattices and finite type-II factors (a specialtype of ring of operators), that have what he regarded as desirablefeatures for quantum theory. These observations led to his developinga more general framework, continuous geometries, for quantumtheory. Matters did not work out as von Neumann had expected. He soonrealized that such geometries must have a transition probabilityfunction, if they are to be used to describe quantum mechanicalphenomena, and that the resulting structure is not a generalization atall beyond the operator rings that were already available. Moreover,it was determined much later that the type-III factors are the mostimportant type of ring of operators for quantum theory. In addition, asimilar verdict was delivered much later with regards to hisexpectations concerning lattice theory. The lattices that areappropriate for quantum theory are orthomodular — a lattice isorthomodular only if it is modular, but the converse is false. Of thethree mathematical theories, it is the rings of operators that haveproven to be the most important framework for quantum theory. It ispossible to use a ring of operators to model key features of physicalsystems in a purely abstract, algebraic setting (this is discussed insection 4.1). A related issue concerns whether it is necessary tochoose a representation of the ring in a Hilbert space; see Haag andKastler (1964), Ruetsche (2003), and Kronz and Lupher (2005) forfurther discussion of this issue. In any case, the separable Hilbertspace remains a crucial framework for quantum theory. The simplestexamples of separable Hilbert spaces are the finite dimensional ones,in which case the algebra of operators is atype-I\(_n\) factor (n is a positive integer). Theoperators are n-by-n complex matrices, which are typically used todescribe internal degrees of freedom such as spin. Readers wanting tofamiliarize themselves with these basic examples should consult theentry onquantum mechanics.

2.1 The Separable Hilbert Space Formulation of Quantum Mechanics

Matrix mechanics and wave mechanics were formulated roughly around thesame time between 1925 and 1926. In July 1925, Heisenberg finished hisseminal paper “On a Quantum Theoretical Interpretation ofKinematical and Mechanical Relations”. Two months later, Bornand Jordan finished their paper, “On Quantum Mechanics”,which is the first rigorous formulation of matrix mechanics. Twomonths after this, Born, Heisenberg, and Jordan finished “OnQuantum Mechanics II”, which is an elaboration of the earlierBorn and Jordan paper; it was published in early 1926. These threepapers are reprinted in van der Waerden (1967). Meanwhile,Schrödinger was working on what eventually became his four famouspapers on wave mechanics. The first was received byAnnalen derPhysik in January 1926, the second was received in February, andthen the third in May and the fourth in June. All four are reprintedin Schrödinger (1928).

Schrödinger was the first to raise the question of therelationship between matrix mechanics and wave mechanics inSchrödinger (1926), which was published inAnnalen inspring 1926 between the publication of his second and third papers ofthe famous four. This paper is also reprinted in Schrödinger(1928). It contains the germ of a mathematical equivalence proof, butit does not contain a rigorous proof of equivalency: the mathematicalframework that Schrödinger associated with wave mechanics is aspace of continuous and normalizable functions, which is too small toestablish the appropriate relation with matrix mechanics. Shortlythereafter, Dirac and Jordan independently provided a unification ofthe two frameworks. But their respective approaches required essentialuse of \(\delta\) functions, which were suspect from the standpoint ofmathematical rigor. In 1927, von Neumann published three papersinGöttinger Nachrichten that placed quantum mechanicson a rigorous mathematical foundation and included a rigorous proof(i.e., without the use of \(\delta\) functions) of the equivalence ofmatrix and wave mechanics. These papers are reprinted in vonNeumann(1961–1963, Volume I, Numbers 8–10). In the prefaceto his famous 1932 treatise on quantum mechanics (von Neumann 1955),which is an elegant summary of the separable Hilbert space formulationof quantum mechanics that he provided in the earlier papers, heacknowledges the simplicity and utility of Dirac’s formulationof quantum mechanics, but finds it ultimately unacceptable. Heindicates that he cannot endure the use of what could then only beregarded as mathematical fictions. Examples of these fictions includeDirac’s assumption that every self-adjoint operator can be putin diagonal form and his use of \(\delta\) functions, which vonNeumann characterizes as “improper functions withself-contradictory properties”. His stated purpose is toformulate a framework for quantum mechanics that is mathematicallyrigorous.

What follows is a brief sketch of von Neumann’s strategy. First, herecognized the mathematical framework of matrix mechanics as whatwould now be characterized as an infinite dimensional, separableHilbert space. Here the term “Hilbert space” denotesa complete vector space with an inner product; von Neumann imposed theadditional requirement of separability (having a countable basis) inhis definition of a Hilbert space. He then attempted to specify a setof functions that would instantiate an (infinite-dimensional)separable Hilbert space and could be identified withSchrödinger’s wave mechanics. He began with the space ofsquare-integrable functions on the real line. To satisfy thecompleteness condition, that all Cauchy sequences of functionsconverge (in the mean) to some function in that space, he specifiedthat integration must be defined in the manner of Lebesgue. To definean inner product operation, he specified that the set of Lebesguesquare-integrable functions must be partitioned into equivalenceclasses modulo the relation of differing on a set of measurezero. That the elements of the space are equivalence classes offunctions rather than functions is sometimes overlooked, and it hasinteresting ramifications for interpretive investigations. It has beenargued in Kronz (1999), for example, that separable Hilbert space isnot a suitable framework for quantum mechanics under Bohm’sontological interpretation (also known asBohmian mechanics).

2.2 Rings of Operators, Quantum Logics, and Continuous Geometries

In a letter to Birkhoff from 1935, von Neumann says: “I wouldlike to make a confession which may seem immoral: I do not believe inHilbert space anymore”; the letter is published in von Neumann(2005). The confession is indeed startling since it comes from thechampion of the separable Hilbert space formulation of quantummechanics and it is issued just three years after the publication ofhis famous treatise, the definitive work on the subject. The irony iscompounded by the fact that less than two years after his confessionto Birkhoff, his mathematical theorizing about the abstractmathematical structure that was to supersede the separable Hilbertspace, continuous geometries with a transition probability, turned outnot to provide a generalization of the separable Hilbert spaceframework. It is compounded again with interest in that subsequentdevelopments in mathematical physics initiated and developed by vonNeumann ultimately served to strengthen the entrenchment of theseparable Hilbert space framework in mathematical physics (especiallywith regards to quantum theory). These matters are explained in moredetail in Section 4.1.

Three theoretical developments come together for von Neumann in histheory of continuous geometries during the seven years following 1932:the algebraic approach to quantum mechanics, quantum logics, and ringsof operators. By 1934, von Neumann had already made substantial movestowards an algebraic approach to quantum mechanics with the help ofJordan and Wigner — their article, “On an AlgebraicGeneralization of the Quantum Mechanical Formalism”,  isreprinted in von Neumann (1961–1963, Vol. II, No. 21). In 1936, hepublished a second paper on this topic, “On an AlgebraicGeneralization of the  Quantum Mechanical Formalism (PartI)”, which is reprinted in von Neumann (1961–1963, Vol. III,No. 9). Neither work was particularly influential, as it turns out. Arelated paper by von Neumann and Birkhoff, “The Logic of QuantumMechanics”, was also published in 1936, and it is reprinted invon Neumann (1961–1963, Vol. IV, No. 7). It was seminal to thedevelopment of a sizeable body of literature onquantum logics.It should be noted, however, that this happens only after modularity,a key postulate for von Neumann, is replaced with orthomodularity (aweaker condition). The nature of the shift is clearly explained inHolland (1970): modularity is in effect a weakening of thedistributive laws (limiting their validity to certain selected triplesof lattice elements), and orthomodularity is a weakening of modularity(limiting the validity of the distributive laws to an even smaller setof triples of lattice elements). The shift from modularity toorthomodularity was first made in (Loomis 1955). Rapid growth ofliterature on orthomodular lattices and the foundations of quantummechanics soon followed. For example, see Pavičić (1992)for a fairly exhaustive bibliography of quantum logic up to 1990,which has over 1800 entries.

Of substantially greater note for the foundations of quantum theoryare six papers by von Neumann (three jointly published with Murray) onrings of operators, which are reprinted in von Neumann (1961–1963,Vol. III, Nos 2–7). The first two, “On Rings of Operators”and a sequel “On Rings of Operators II”, were published in1936 and 1937, and they were seminal to the development of the otherfour. The third, “On Rings of Operators: ReductionTheory”, was written during 1937–1938 but not published until1949. The fourth, “On Infinite Direct Products”, waspublished in 1938. The remaining two, “On Rings of OperatorsIII” and “On Rings of Operators IV” were publishedin 1941 and 1943, respectively. This massive work on rings ofoperators was very influential and continues to have an impact in puremathematics, mathematical physics, and the foundations ofphysics. Rings of operators are now referred to as “von Neumannalgebras” following Dixmier (1981), who first referred to themby this name (stating that he did so following a suggestion made tohim by Dieudonné) in the introduction to his 1957 treatise on operatoralgebras (Dixmier 1981).

A von Neumann algebra is a \(*\)-subalgebra of the set of boundedoperators B(H) on a Hilbert space H that is closed in the weakoperator topology. It is usually assumed that the von Neumann algebracontains the identity operator. A \(*\)-subalgebra contains theadjoint of every operator in the algebra, where the“\(*\)” denotes the adjoint. There are special types ofvon Neumann algebras that are called “factors”. A vonNeumann algebra is a factor, if its center (which is the set ofelements that commute with all elements of the algebra) is trivial,meaning that it only contains scalar multiples of the identityelement. Moreover, von Neumann showed in his reduction-theory paperthat all von Neumann algebras that are not factors can be decomposedas a direct sum (or integral) of factors. There are three mutuallyexclusive and exhaustive factor types: type-I, type-II, andtype-III. Each type has been classified into (mutually exclusive andexhaustive) sub-types: types I\(_n\) \((n = 1,2,\ldots ,\infty),\)II\(_n\) \((n = 1,\infty),\) III\(_z\) \((0\le z\le 1).\) As mentionedabove, type-I\(_n\) correspond to finite dimensional Hilbert spaces,while type-I\(_{\infty}\) corresponds to the infinite dimensionalseparable Hilbert space that provides the rigorous framework for waveand matrix mechanics. Von Neumann and Murray distinguished thesubtypes for type-I and type-II, but were not able to do so for thetype-III factors. Subtypes were not distinguished for these factorsuntil the 1960s and 1970s — see Chapter 3 of Sunder (1987) orChapter 5 of Connes (1994) for details.

As a result of his earlier work on the foundations of quantummechanics and his work on quantum logic with Birkhoff, von Neumanncame to regard the type-II\(_1\) factors as likely to be themost relevant for physics. This is a substantial shift since the mostimportant class of algebra of observables for quantum mechanics wasthought at the time to be the set of bounded operators on aninfinite-dimensional separable Hilbert space, which is atype-I\(_{\infty}\) factor. A brief explanation for this shift isprovided below. See the well-informed and lucid account presented in(Rédei 1998) for a much fuller discussion of von Neumann’sviews on fundamental connections between quantum logic, rings ofoperators (particularly type-II\(_1\) factors), foundations ofprobability theory, and quantum physics. It is worth noting that vonNeumann regarded the type-III factors as a catch-all class for the“pathological” operator algebras; indeed, it took severalyears after the classificatory scheme was introduced to demonstratethe existence of such factors. It is ironic that the predominant viewnow seems to be that the type-III factors are the most relevant classfor physics (particularly for QFT and quantum statisticalmechanics). This point is elaborated further in Section 4.1 afterexplaining below why von Neumann’s program never came to fruition.

In the introduction to the first paper in the series of four entitled“On Rings of Operators”, Murray and von Neumann list tworeasons why they are dissatisfied with the separable Hilbert spaceformulation of quantum mechanics. One has to do with a property of thetrace operation, which is the operation appearing in the definition ofthe probabilities for measurement results (the Born rule), and theother with domain problems that arise for unbounded observableoperators. The trace of the identity is infinite when the separableHilbert space is infinite-dimensional, which means that it is notpossible to define a correctly normalizeda prioriprobability for the outcome of an experiment (i.e., a measurement ofan observable). By definition, thea priori probability foran experiment is that in which any two distinct outcomes are equallylikely. Thus, the probability must be zero for each distinct outcomewhen there is an infinite number of such outcomes, which can occur ifand only if the space is infinite dimensional. It is not clear why vonNeumann believed that it is necessary to have ana prioriprobability for every experiment, especially since von Mises clearlybelieved that a priori probabilities (“uniform distributions” in histerminology) do not always exist (von Mises 1981, pp. 68 ff.) and vonNeumann was influenced substantially by von Mises on the foundationsof probability (von Neumann 1955, p. 198 fn.). Later, von Neumannchanged the basis for his expressed reason for dissatisfaction withinfinite dimensional Hilbert spaces from probabilistic to algebraicconsiderations (Birkhoff and von Neumann 1936, p. 118); namely, thatit violates Hankel’s principle of the preservation of formal law,which leads one to try to preserve modularity — a condition thatholds in finite-dimensional Hilbert spaces but not ininfinite-dimensional Hilbert spaces. The problem with unboundedoperators arises from their only being defined on a merely densesubset of the set elements of the space. This means that algebraicoperations of unbounded operators (sums and products) cannot begenerally defined; for example, it is possible that two unboundedoperators \(A\), \(B\) are such that the range of \(B\)and the domain of \(A\) are disjoint, in which case theproduct \(AB\) is meaningless.

The problems mentioned above do not arise for type-I\(_n\) factors, if\(n\lt \infty\), nor do they arise for type-II\(_1\). That is to say,these factor types have a finite trace operation and are not plaguedwith the domain problems of unbounded operators. Particularlynoteworthy is that the lattice of projections of each of these factortypes (type-I\(_n\) for \(n\lt \infty\) and type-II\(_1)\) ismodular. By contrast, the set of bounded operators on aninfinite-dimensional separable Hilbert space, a type-I\(_{\infty}\)factor, is not modular; rather, it is only orthomodular. Theseconsiderations serve to explain why von Neumann regarded thetype-II\(_1\) factor as the proper generalization of the type-I\(_n\)\((n\lt \infty)\) for quantum physics rather than thetype-I\(_{\infty}\) factors. The shift in the literature from modularto orthomodular lattices that was characterized above is in effect ashift back to von Neumann’s earlier position (prior to hisconfession). But, as was already mentioned, it now seems that this wasnot the best move either.

It was von Neumann’s hope that his program for generalizing quantumtheory would emerge from a new mathematical structure known as“continuous geometry”. He wanted to use this structure tobring together the three key elements that were mentioned above: thealgebraic approach to quantum mechanics, quantum logics, and rings ofoperators. He sought to forge a strong conceptual link between theseelements and thereby provide a proper foundation for generalizingquantum mechanics that does not make essential use of Hilbert space(unlike rings of operators). Unfortunately, it turns out that theclass of continuous geometries is too broad for the purposes ofaxiomatizing quantum mechanics. The class must be suitably restrictedto those having a transition probability. It turns out that there isthen no substantial generalization beyond the separable Hilbert spaceframework. An unpublished manuscript that was finished by von Neumannin 1937 was prepared and edited by Israel Halperin, and then publishedas von Neumann (1981). A review of the manuscript by Halperin waspublished in von Neumann (1961–1963, Vol. IV, No. 16) years before themanuscript itself was published. In that review, Halperin notes thefollowing:

The final result, after 200 pages of deep reasoning is (essentially):every such geometry with transition probability can be identified withthe projection geometry of a finite factor in some finite or infinitedimensional Hilbert space (I\(_m\) or II\(_1)\). This result indicatesthat continuous geometries do not provide new useful mathematicaldescriptions of quantum mechanical phenomena beyond that alreadyavailable from rings of operators.

This unfortunate development does not, however, completely underminevon Neumann’s efforts to generalize quantum mechanics. On thecontrary, his work on rings of operators does provide significantlight to the way forward. The upshot of subsequent developments isthat von Neumann settled on the wrong factor type for the foundationsof physics.

3. Dirac and the Foundations of Quantum Theory

Dirac’s formal framework for quantum mechanics was very usefuland influential despite its lack of mathematical rigor. It was usedextensively by physicists and it inspired some powerful mathematicaldevelopments in functional analysis. Eventually, mathematiciansdeveloped a suitable framework for placing Dirac’s formalframework on a firm mathematical foundation, which is known asarigged Hilbert space (and is also referred to asaGelfand Triplet). This came about as follows. A rigorousdefinition of the \(\delta\) function became possible in distributiontheory, which was developed by Schwartz from the mid-1940s to theearly 1950s. Distribution theory inspired Gelfand and collaboratorsduring the mid-to-late 1950s to formulate the notion of a riggedHilbert space, the firm foundation for Dirac’s formalframework. This development was facilitated by Grothendiek’snotion of a nuclear space, which he introduced in the mid-1950s. Therigged Hilbert space formulation of quantum mechanics was thendeveloped independently by Böhm and by Roberts in 1966. Sincethen, it has been extended to a variety of different contexts in thequantum domain including decay phenomena and the arrow of time. Themathematical developments of Schwartz, Gelfand, and others had asubstantial effect on QFT as well. Distribution theory was takenforward by Wightman in developing the axiomatic approach to QFT fromthe mid-1950s to the mid-1960s. In the late 1960s,  the axiomaticapproach was explicitly put into the rigged Hilbert space framework byBogoliubov and co-workers.

Although these developments were only indirectly influenced by Dirac,by way of the mathematical developments that are associated with hisformal approach to quantum mechanics, there are other elements of hiswork that had a more direct and very substantial impact on thedevelopment of QFT. In the 1930s, Dirac (1933) developed a Lagrangianformulation of quantum mechanics and applied it to quantum fields, and the latter inspired Feynman (1948) to develop thepath-integral approach to QFT. The mathematicalfoundation for path-integral functionals is still lacking (Rivers1987, pp, 109–134), though substantial progress has been made(DeWitt-Moretteet al. 1979). Despite such shortcomings, itremains the most useful and influential approach to QFT to date. Inthe 1940s, Dirac (1943) developed a form of quantum electrodynamics thatinvolved an indefinite metric — see also Pauli(1943) in that connection. This had a substantial influence on laterdevelopments, first in quantum electrodynamics in the early 1950s withthe Gupta-Bluer formalism, and in a variety of QFT models such asvector meson fields and quantum gravity fields by the late 1950s— see Chapter 2 of Nagy (1966) for examples and references.

3.1 Dirac’s \(\delta\) Function,Principles, and Bra-Ket Notation

Dirac’s attempt to prove the equivalence of matrix mechanics and wavemechanics made essential use of the \(\delta\) function, as indicatedabove. The \(\delta\) function was used by physicists before Dirac, but itbecame a standard tool in many areas of physics only after Dirac veryeffectively put it to use in quantum mechanics. It then became widelyknown by way of his textbook (Dirac 1930), which was based on a seriesof lectures on quantum mechanics given by Dirac at CambridgeUniversity. This textbook saw three later editions: the second in1935, the third in 1947, and the fourth in 1958. The fourth editionhas been reprinted many times. Its staying power is due, in part, toanother innovation that was introduced by Dirac in the third edition,his bra-ket formalism. He first published this formalism in (Dirac1939), but the formalism did not become widely used until after thepublication of the third edition of his book. There is no questionthat these tools, first the \(\delta\) function and then the bra-ketnotation, were extremely effective for physicists practicing andteaching quantum mechanics both with regards to setting up equationsand to the performance of calculations. Most quantum mechanicstextbooks use \(\delta\) functions and plane waves, which are key elementsof Dirac’s formal framework, but they are not included in vonNeumann’s rigorous mathematical framework for quantummechanics. Working physicists as well as teachers and students ofquantum mechanics often use Dirac’s framework because of itssimplicity, elegance, power, and relative ease of use. Thus, from thestandpoint of pragmatics, Dirac’s framework is much preferred over vonNeumann’s. The notion of a rigged Hilbert space placed Dirac’sframework on a firm mathematical foundation.

3.2 The Rigged Hilbert Space Formulation of Quantum Mechanics

Mathematicians worked very hard to provide a rigorous foundation forDirac’s formal framework. One key element was Schwartz’s(1945; 1950–1951) theory of distributions. Another key element,the notion of a nuclear space, was developed by Grothendieck(1955). This notion made possible the generalized-eigenvectordecomposition theorem for self-adjoint operators in rigged Hilbertspace — for the theorem see Gelfand and Vilenken (1964,pp. 119–127), and for a brief historical account of theconvoluted path leading to it see Berezanskii (1968,pp. 756–760). The decomposition principle provides a rigorousway to handle observables such as position and momentum in the mannerin which they are presented in Dirac’s formal framework. Thesemathematical developments culminated in the early 1960s with Gelfandand Vilenkin’s characterization of a structure that theyreferred to as arigged Hilbert space (Gelfand and Vilenkin1964, pp. 103–127). It is unfortunate that their chosen name forthis mathematical structure is doubly misleading. First, there is anatural inclination to regard it as denoting a type of Hilbert space,one that isrigged in some sense, but this inclination mustbe resisted. Second, the termrigged has an unfortunateconnotation of illegitimacy, as in the termsrigged electionorrigged roulette table, and this connotation must bedismissed as prejudicial. There is nothing illegitimate about a riggedHilbert space from the standpoint of mathematical rigor (or any otherrelevant standpoint). A more appropriate analogy may be drawn usingthe notion of a rigged ship: the termrigged in this contextmeans fully equipped. But this analogy has its limitations since arigged ship is a fully equipped ship, but (as the first pointindicates) a rigged Hilbert space is not a Hilbert space, though it isgenerated from a Hilbert space in the manner now to be described.

A rigged Hilbert space is a dual pair of spaces\((\Phi , \Phi^x)\) that can generated from aseparable Hilbert space \(\Eta\) using a sequence of norms (orsemi-norms); the sequence of norms is generated using a nuclearoperator (a good approximate meaning is an operator of trace-class,meaning that the trace of the modulus of the operator is finite). Inthe mathematical theory of topological vector spaces, the space \(\Phi\)is characterized in technical terms as anuclear Fréchetspace. To say that \(\Phi\) is aFréchet space meansthat it is a complete metric space, and to say that itisnuclear means that it is the projective limit of asequence of Hilbert spaces in which the associated topologies getrapidly finer with increasing n (i.e., the convergence conditions areincreasingly strict); the termnuclear is used because theHilbert-space topologies are generated using a nuclear operator. Indistribution theory, the space \(\Phi\) is characterized as atest-function space, where a test-function is thought of as a verywell-behaved function (being continuous, n-times differentiable,having a bounded domain or at least dropping off exponentially beyondsome finite range, etc). \(\Phi^x\) is a space ofdistributions, and it is the topological dual of \(\Phi\), meaning thatit corresponds to the complete space of continuous linear functionalson \(\Phi\). It is also the inductive limit of a sequence of Hilbertspaces in which the topologies get rapidly coarser with increasingn. Because the elements of \(\Phi\) are so well-behaved,\(\Phi^x\) may contain elements that are not sowell-behaved, some being singular or improper functions (such asDirac’s \(\delta\) function). \(\Phi\) is the topological anti-dual of\(\Phi^x\), meaning that it is the complete set ofcontinuous anti-linear functionals on \(\Phi^x\); itis anti-linear rather than linear because multiplication by a scalaris defined in terms of the scalar’s complex conjugate.

It is worth noting that neither \(\Phi\) nor \(\Phi^x\)is a Hilbert space in that each lacks an inner product that induces ametric with respect to which the space is complete, though for eachspace there is a topology with respect to which the space iscomplete. Nevertheless, each of them is closely related to the Hilbertspace \(\Eta\) from which they are generated: \(\Phi\) is densely embeddedin \(\Eta\), which in turn is densely embedded in\(\Phi^x\). Two other points are worth noting. First,dual pairs of this sort can also be generated from a pre-Hilbertspace, which is a space that has all the features of a Hilbert spaceexcept that it is not complete, and doing so has the distinctadvantage of avoiding the partitioning of functions into equivalenceclasses (in the case of functions spaces). The termrigged Hilbertspace is typically used broadly to include dual pairs generatedfrom either a Hilbert space or a pre-Hilbert space. Second, thetermGelfand triplet is sometimes used instead of thetermrigged Hilbert space, though it refers to the orderedset \((\Phi , \Eta , \Phi^x)\), where \(\Eta\)is the Hilbert space used to generate \(\Phi\) and\(\Phi^x\).

The dual pair \((\Phi , \Phi^x)\) possesses the means to representimportant operators for quantum mechanics that are problematic in aseparable Hilbert space, particularly the unbounded operators thatcorrespond to the observables position and momentum, and it does so ina particularly effective and unproblematic manner. As already noted,these operators have no eigenvalues or eigenvectors in a separableHilbert space; moreover, they are only defined on a dense subset ofthe elements of the space and this leads to domain problems. Theseundesirable features also motivated von Neumann to seek an alternativeto the separable Hilbert space framework for quantum mechanics, asnoted above. In a rigged Hilbert space, theoperators corresponding to position and momentum can have acomplete set of eigenfunctionals (i.e., generalizedeigenfunctions). The key result is known as the nuclear spectraltheorem (and it is also known as the Gelfand-Maurin theorem). Oneversion of the theorem says that if A is a symmetric linear operatordefined on the space \(\Phi\) and it admits a self-adjoint extensionto the Hilbert space H, then A possesses a complete system ofeigenfunctionals belonging to the dual space \(\Phi^x\) (Gelfand andShilov 1977, chapter 4). That is to say, provided that the statedcondition is satisfied, A can be extended by duality to \(\Phi^x\),its extension \(A^x\) is continuous on \(\Phi^x\) (in the operatortopology in \(\Phi^x)\), and \(A^x\) satisfies a completeness relation(meaning that it can be decomposed in terms of its eigenfunctionalsand their associated eigenvalues). The duality formula for extending\(A\) to \(\Phi^x\) is \(\braket{\phi}{A^x\kappa} =\braket{A\phi}{\kappa}\), for all \(\phi \in \Phi\) and for all\(\kappa \in \Phi^x\). The completeness relation says that for all\(\phi ,\theta \in \Phi\):

where \(v(A)\) is the set of all generalized eigenvalues of \(A^x\)(i.e., the set of all scalars \(\lambda\) for which there is \(\lambda\in \Phi^x\) such that \(\braket{\phi}{A^x\lambda} = \lambda\braket{\phi}{\lambda}\) for all \(\phi \in \Phi)\).

The rigged Hilbert space representation of these observables is aboutas close as one can get to Dirac’s elegant and extremely useful formalrepresentation with the added feature of being placed within amathematically rigorous framework. It should be noted, however, thatthere is a sense in which it is a proper generalization of Dirac’sframework. The rigging (based on the choice of a nuclear operator thatdetermines the test function space) can result in different sets ofgeneralized eigenvalues being associated with an operator. Forexample, the set of (generalized) eigenvalues for the momentumoperator (in one dimension) corresponds to the real line, if the spaceof test functions is the set \(S\) of infinitely differentiablefunctions of \(x\) which together with all derivatives vanishfaster than any inverse power of \(x\) as \(x\) goes toinfinity, whereas its associated set of eigenvalues is the complexplane, if the space of test functions is the set \(D\) ofinfinitely differentiable functions with compact support (i.e.,vanishing outside of a bounded region of the real line). If complexeigenvalues are not desired, then \(S\) would be a moreappropriate choice than \(D\) — see Nagel (1989) for abrief discussion. But there are situations in which it is desirablefor an operator to have complex eigenvalues. This is so, for example,when a system exhibits resonance scattering (a type of decayphenomenon), in which case one would like the Hamiltonian to havecomplex eigenvalues — see Böhm & Gadella (1989). (Ofcourse, it is impossible for a self-adjoint operator to have complexeigenvalues in a Hilbert space.)

Soon after the development of the theory of rigged Hilbert spaces byGelfand and his associates, the theory was used to develop a newformulation of quantum mechanics. This was done independently byBöhm (1966) and Roberts (1966). It was later demonstrated thatthe rigged Hilbert space formulation of quantum mechanics can handle abroader range of phenomena than the separable Hilbert spaceformulation. That broader range includes scattering resonances anddecay phenomena (Böhm and Gadella 1989), as alreadynoted. Böhm (1997) later extended this range to include a quantummechanical characterization of the arrow of time. The Prigogine schooldeveloped an alternative characterization of the arrow of time usingthe rigged Hilbert space formulation of quantum mechanics (Antoniouand Prigogine 1993). Kronz (1998, 2000) used this formulation tocharacterize quantum chaos in open quantum systems. Castagnino andGadella (2003) used it to characterizedecoherencein closed quantum systems.

4 Mathematical Rigor: Two Paths

4.1 Algebraic Quantum Field Theory

In 1943, Gelfand and Neumark published an important paper on animportant class of normed rings, which are now known as abstract\(C^*\)-algebras. Their paper was influenced by Murray and vonNeumann’s work on rings of operators, which was discussed in theprevious section. In their paper, Gelfand and Neumark focus attentionon abstract normed \(*\)-rings. They show that any \(C^*\)-algebra canbe given a concrete representation in a Hilbert space (which need notbe separable). That is to say, there is an isomorphic mapping of theelements of a \(C^*\)-algebra into the set of bounded operators of theHilbert space. Four years later, Segal (1947a) published a paper thatserved to complete the work of Gelfand and Neumark by specifying thedefinitive procedure for constructing concrete (Hilbert space)representations of an abstract \(C^*\)-algebra. It is called the GNSconstruction (after Gelfand, Neumark, and Segal). That same year,Segal (1947b) published an algebraic formulation of quantum mechanics,which was substantially influenced by (though deviating somewhat from)von Neumann’s (1963, Vol. III, No. 9) algebraic formulation ofquantum mechanics, which is cited in the previous section. It is worthnoting that although \(C^*\)-algebras satisfy Segal’spostulates, the algebra that is specified by his postulates is a moregeneral structure known as a Segal algebra. Every \(C^*\)-algebra is aSegal algebra, but the converse is false since Segal’spostulates do not require an adjoint operation to be defined. If aSegal algebra is isomorphic to the set of all self-adjoint elements ofa \(C^*\)-algebra, then it is a special or exceptional Segalalgebra. Although the mathematical theory of Segal algebras has beenfairly well developed, a \(C^*\)-algebra is the most important type ofalgebra that satisfies Segal’s postulates.

The algebraic formulations of quantum mechanics that were developed byvon Neumann and Segal did not change the way that quantum mechanicswas done. Nevertheless, they did have a substantial impact in tworelated contexts: QFT and quantum statistical mechanics. The keydifference leading to the impact has to do with the domain ofapplicability. The domain of quantum mechanics consists of finitequantum systems, meaning quantum systems that have a finite number ofdegrees of freedom. Whereas in QFT and quantum statistical mechanics,the systems of special interest — i.e., quantum fields andparticle systems in the thermodynamic limit, respectively — areinfinite quantum systems, meaning quantum systems that have aninfinite number of degrees of freedom. Dirac (1927) was the first torecognize the importance of infinite quantum systems for QFT, which isreprinted in Schwinger (1958).

Segal (1959, p. 5) was the first to suggest that the beauty and power of thealgebraic approach becomes evident when working with an infinitequantum system . The key advantage of the algebraicapproach, according to Segal (1959, pp. 5–6), is that one may work inthe abstract algebraic setting where it is possible to obtaininteracting fields from free fields by an automorphism on the algebra,one that need not be unitarily implementable. Segal notes (1959, p. 6)that von Neumann (1937) had a similar idea (that field dynamics are to beexpressed as an automorphism on the algebra) in an unpublishedmanuscript. Segal notes this advantage in responseto a result obtained by Haag (1955), that field theory representationsof free fields are unitarily inequivalent to representations ofinteracting fields. Haag mentions that von Neumann (1938) first discovered‘different’ (unitarily inequivalent) representations muchearlier. A different way of approachingunitarily equivalent representations, by contrast with Segal’sapproach, was later presented by Haag and Kastler (1964), who arguedthat unitarilty inequivalent representations are physicallyequivalent. Their notion of physical equivalence was based on Fell’smathematical idea of weak equivalence (Fell 1960).

After indicating important similarities between his and von Neumann’sapproaches to infinite quantum systems, Segal draws an importantcontrast that serves to give the advantage to his approach over vonNeumann’s. The key mathematical difference, according to Segal, isthat von Neumann was working with a weakly closed ring of operators(meaning that the ring of operators is closed with respect to the weakoperator topology), whereas Segal is working with a uniformly closedring of operators (closed with respect to the uniform topology). It iscrucial because it has the following interpretive significance, whichrests on operational considerations:

The present intuitive idea is roughly that the onlymeasurable field-theoretic variables are those that can be expressedin terms of afinite number of canonical operators,oruniformly approximated by such; the technical basis isauniformly closed ring (more exactly, anabstract \(C^*\)-algebra). The crucial difference between the twovarieties of approximation arises from the fact that, in general, weakapproximation has only analytical significance, while uniformapproximation may be defined operationally, two observables beingclose if the maximum (spectral) value of the difference is small(Segal 1959, p. 7).

Initially, it appeared that Segal’s assessment of the relativemerits of von Neumann algebras and \(C^*\)-algebras with respect tophysics was substantiated by a seminal paper, (Haag and Kastler1964). Among other things, Haag and Kastler introduced the key axiomsof the algebraic approach to QFT. They also argued that unitarilyinequivalent representations are “physically equivalent”to each other. However, the use of physical equivalence to show thatunitarily inequivalent representations are not physically significanthas been challenged; see Kronz and Lupher (2005), Lupher (2018), andRuetsche (2011). The prominent role of type-III factor von Neumannalgebras within the algebraic approach to quantum statisticalmechanics and QFT raises further doubts about Segal’sassessment.

The algebraic approach has proven most effective in quantumstatistical mechanics. It is extremely useful for characterizing manyimportant macroscopic quantum effects including crystallization,ferromagnetism, superfluidity, structural phase transition,Bose-Einstein condensation, and superconductivity. A good introductorypresentation is Sewell (1986), and for a more advanced discussion seeBratteli and Robinson (1979, 1981). In algebraic quantum statisticalmechanics, an infinite quantum system is defined by specifying anabstract algebra of observables. A particular state may then be usedto specify a concrete representation of the algebra as a set ofbounded operators in a Hilbert space. Among the most important typesof states that are considered in algebraic statistical mechanics arethe equilibrium states, which are often referred to as “KMSstates” (since they were first introduced by the physicistsKubo, Martin, and Schwinger). There is a continuum of KMS states sincethere is at least one KMS state for each possible temperature value\(\tau\) of the system, for\(0\le \tau \le +\infty\). Given an automorphismgroup, each KMS state corresponds to a representation of the algebraof observables that defines the system, and each of theserepresentations is unitarily inequivalent to any other. It turns outthat each representation that corresponds to a KMS state is a factor:if \(\tau = 0\) then it is a type-I factor, if\(\tau = +\infty\) then it is a type-II factor, and if\(0\lt \tau \lt +\infty\) then it is a type-IIIfactor. Thus, type-III factors play a predominant role in algebraicquantum statistical mechanics.

In algebraic QFT, an algebra of observables is associated with boundedregions of Minkowski spacetime (and unbounded regions including all ofspacetime by way of certain limiting operations) that are required tosatisfy standard axioms of local structure: isotony, locality,covariance, additivity, positive spectrum, and a unique invariantvacuum state. The resulting set of algebras on Minkowski spacetimethat satisfy these axioms is referred to as thenet of localalgebras. It has been shown that special subsets of the net oflocal algebras — those corresponding to various types ofunbounded spacetime regions such as tubes, monotones (a tube thatextends infinitely in one direction only), and wedges — aretype-III factors. Of particular interest for the foundations ofphysics are the algebras that are associated with bounded spacetimeregions, such as a double cone (the finite region of intersection of aforward and a backward light cone). As a result of work done over thelast thirty years, local algebras of relativistic QFT appear to betype III von Neuman algebras see Halvorson (2007, pp. 749–752)for more details.

One important area for interpretive investigation is the existence ofa continuum of unitarily inequivalent representations of an algebra ofobservables. Attitudes towards unitarily inequivalent representationsdiffer drastically in the philosophical literature. In (Wallace 2006)unitarily inequivalent representations are not considered afoundational problem for QFT, while in Ruetsche (2011), Lupher (2018) and Kronz andLupher (2005) unitarily inequivalent representations are consideredphysically significant.

4.2 Wightman’s Axiomatic Quantum Field Theory

In the early 1950s, theoretical physicists were inspired to axiomatizeQFT. One motivation for axiomatizing a theory, not the one for thecase now under discussion, is to express the theory in a completelyrigorous form in order to standardize the expression of the theory asa mature conceptual edifice. Another motivation, more akin to the casein point, is to embrace a strategic withdrawal to the foundations todetermine how renovation should proceed on a structure that isthreatening to collapse due to internal inconsistencies. One thenlooks for existing piles (fundamental postulates) that penetratethrough the quagmire to solid rock, and attempts to drive home othersat advantageous locations. Properly supported elements of thesuperstructure (such as the characterization of free fields,dispersion relations, etc.) may then be distinguished from those thatare untrustworthy. The latter need not be razed immediately, and mayultimately glean supportive rigging from components not yetconstructed. In short, the theoretician hopes that the axiomatizationwill effectively separate sense from nonsense, and that this willserve to make possible substantial progress towards the development ofa mature theory. Grounding in a rigorous mathematical framework can bean important part of the exercise, and that was a key aspect of theaxiomatization of QFT by Wightman.

In the mid-1950s, Schwartz’s theory of distributions was used byWightman (1956) to develop an abstract formulation of QFT, which latercame to be known known asaxiomatic quantum fieldtheory. Mature statements of this formulation are presented inWightman and Gårding (1964) and in Streater and Wightman(1964). It was further refined in the late 1960s by Bogoliubov, whoexplicitly placed axiomatic QFT in the rigged Hilbert space framework(Bogoliubovet al. 1975, p. 256). It is by now standardwithin the axiomatic approach to put forth the following sixpostulates: spectral condition (there are no negative energies orimaginary masses), vacuum state (it exists and is unique), domainaxiom for fields (quantum fields correspond to operator-valueddistributions), transformation law (unitary representation in thefield-operator (and state) space of the restricted inhomogeneousLorentz group — “restricted” means inversions areexcluded, and “inhomogeneous” means that translations areincluded), local commutativity (field measurements at spacelikeseparated regions do not disturb one another), asymptotic completeness(the scattering matrix is unitary — this assumption is sometimesweakened to cyclicity of the vacuum state with respect to thepolynomial algebra of free fields). Rigged Hilbert space entered theaxiomatic framework by way of the domain axiom, so this axiom will bediscussed in more detail below.

In classical physics, a field is is characterized as a scalar- (orvector- or tensor-) valued function \(\phi(x)\) on a domain thatcorresponds to some subset of spacetime points. In QFT, a field ischaracterized by means of an operator rather than afunction. Afield operator may be obtained from a classicalfield function by quantizing the function in the canonical manner— see Mandl (1959, pp. 1–17). For convenience, the fieldoperator associated with \(\phi(x)\) is denoted below by the sameexpression (since the discussion below only concerns fieldoperators). Field operators that are relevant for QFT are too singularto be regarded as realistic, so they are smoothed out over theirrespective domains using elements of a space of well-behaved functionsknown astest functions. There are many differenttest-functions spaces (Gelfand and Shilov 1977, Chapter 4). At first,the test-function space of choice for axiomatic QFT wastheSchwartz space \(\Sigma\), the space of functions whoseelements have partial derivatives of all orders at each point and suchthat each function and its derivatives decreases faster than\(x^{-n}\) for any \(n\in N\) as \(x\rightarrow \infty\). It was laterdetermined that some realistic models require the use of othertest-function spaces. The smoothed field operators \(\phi[f\)] for \(f\in \Sigma\) are known asquantum field operators, and theyare defined as follows

The integral (over the domain of the field operator) of the product ofthe test function \(f(x)\) and the field operator \(\phi(x)\) servesto “smooth out” the field operator over its domain; a morecolloquial description is that the field is “smeared out”over space or spacetime. It is postulated within the axiomaticapproach that a quantum field operator \(\phi[f\)] may be representedas an unbounded operator on a separable Hilbert space \(\Eta\), andthat \(\{\phi[f]: f\in \Sigma \}\) (the set of smoothed fieldoperators associated with \(\phi(x))\) has a dense domain \(\Omega\)in \(\Eta\). The smoothed field operators are often referred toasoperator-valued distributions, and this means that forevery \(\Phi,\Psi \in \Omega\) there is an element of the space ofdistributions \(\Sigma^x\), the topological dual of \(\Sigma\), thatmay be equated to the expression \(\langle \Phi {\mid} \phi[\]{\mid}\Psi\rangle\). If \(\Omega'\) denotes the set of functionsobtained by applying all polynomials of elements of \(\{\phi[f]: f\in\Sigma \}\) onto the unique vacuum state, then the axioms mentionedabove entail that \(\Omega'\) is dense in \(\Eta\) (asymptoticcompleteness) and that \(\Omega'\subset \Omega\) (domainaxiom). The elements of \(\Omega\) correspond to possible states ofthe elements of \(\{\phi[f]: f\in \Sigma \}\). Though only one fieldhas been considered thus far, the formalism is easily generalizable toa countable number of fields with an associated set of countablyindexed field operators \(\phi_k (x)\) — cf. (Streater andWightman 1964).

As noted earlier, the appropriateness of the rigged Hilbert spaceframework enters by way of the domain axiom. Concerning that axiom,Wightman says the following (in the notation introduced above, whichdiffers slightly from that used by Wightman).

At a more advanced stage in the theory it is likely thatone would want to introduce a topology into \(\Omega\) such that\(\phi[f\)] becomes a continuous mapping of \(\Omega\) into\(\Omega\). It is likely that this topology has to be ratherstrong. We want to emphasize that so far we have only required that\(\langle \Phi{\mid}\phi[f]{\mid}\Psi\rangle\) be continuous in \(f\)for \(\Phi ,\Psi\)fixed; continuity in the pair \(\Phi,\Psi\) cannot be expected before we put a suitable strong topology on\(\Omega\) (Wightman and Gårding 1964, p. 137).

In Bogoliubovet al. (1975, p. 256), a topology is introducedto serve this role, though it is introduced on \(\Omega'\) rather thanon \(\Omega\). Shortly thereafter, they assert that it is not hard toshow that \(\Omega'\) is a complete nuclear space with respect to thistopology. This serves to justify a claim they make earlier in theirtreatise:

… it is precisely the consideration of the tripletof spaces \(\Omega \subset \Eta \subset \Omega^*\) which give a natural basisfor both the construction of a general theory of linear operators andthe correct statement of certain problems of quantum field theory(Bogoliubovet al. 1975, p. 34).

Note that they refer to the triplet \(\Omega \subset \Eta \subset\Omega^*\) as a rigged Hilbert space. In the terminology introducedabove, they refer in effect to the Gelfand triplet \((\Omega , \Eta ,\Omega^x )\) or (equivalently) the associated rigged Hilbert space\((\Omega , \Omega^x)\) .

Finally, it is worth mentioning that the status of the field inalgebraic QFT differs from that in Wightman’s axiomatic QFT. In bothapproaches, a field is an abstract system having an infinite number ofdegrees of freedom. Sub-atomic quantum particles are field effectsthat appear in special circumstances. In algebraic QFT, there is afurther abstraction: the most fundamental entities are the elements ofthe algebra of local (and quasi-local) observables, and the field is aderived notion. The termlocal means bounded within a finitespacetime region, and an observable is not regarded as a propertybelonging to an entity other than the spacetime region itself. Thetermquasi-local is used to indicate that we take the unionof all bounded spacetime regions. In short, the algebraic approachfocuses on local (or quasi-local) observables and treats the notion ofa field as a derivative notion; whereas the axiomatic approach (ascharacterized just above) regards the field concept as the fundamentalnotion. Indeed, it is common practice for proponents of the algebraicapproach to distance themselves from the field notion by referring totheir theory as “local quantum physics”. The twoapproaches are mutually complementary — they have developedin parallel and have influenced each other by analogy (Wightman1976). For a discussion of the close connections between these twoapproaches, see Haag (1996, p. 106).

5 Philosophical Issues

5.1 Pragmatics versus Axiomatics

Most physicists use Lagrangian QFT (LQFT) to make predictions thathave been experimentally verified with extraordinary precision in somecases. However, LQFT has been described as a “grab bag of conflictingmathematical ideas” that has not provided a sharp mathematicaldescription of what counts as a QFT model (Swanson 2017, pp. 1–2). Thosecriticisms motivated mathematically inclined physicists to search fora mathematically rigorous formulation of QFT. Axiomatic versions ofQFT have been favored by mathematical physicists and mostphilosophers. With greater mathematical rigor it is possible to proveresults about the theoretical structure of QFT independent of anyparticular Lagrangian. Axiomatic QFT provides clear conceptualframeworks within which precise questions and answers tointerpretational issues can be formulated. There are three mainaxiomatic frameworks for QFT: Wightman QFT, Osterwalder-Schrader QFT,and algebraic QFT. In Wightman QFT, the axioms use functional analysisand operator algebras and is closer to LQFT since its axioms describecovariant field operators acting on a fixed Hilbert space. TheOsterwalder-Schrader axioms use a functional integration approach toQFT. The algebraic QFT axioms use \(C^*\)-algebras to model localobservables. However, axiomatic QFT approaches are sorely lacking withregards to building empirically adequate models. Unlike quantummechanics which has a canonical mathematical framework in terms of vonNeumann’s Hilbert space formulation, QFT has no canonical mathematicalframework. Even though there is a canonical mathematical framework forquantum mechanics, there are many interpretations of that framework,e.g., many-worlds, GRW, Copenhagen, Bohmian, etc... QFT has two levelsthat require interpretation: (1) which QFT framework should be thefocus of these foundational efforts, if any, and (2) how thatpreferred framework should be interpreted. Since (1) involves issuesabout mathematical rigor and pragmatic virtues, it directly bears onthe focus of this article. The lack of a canonical formulation of QFTthreatens to impede any metaphysical or epistemological lessons thatmight be learned from QFT.

One view is that these two approaches to QFT, the mathematicallyrigorous axiomatic approach and the pragmatic / empirically adequateLQFT approach, are rival research programs (see David Wallace (2006,2011) and Doreen Fraser (2009, 2011)), though Swanson (2017) arguesthat they are not rival programs. Fraser (2009, 2011) argues that theinterpretation of QFT should be based on the mathematically rigorousapproach of axiomatic formulations of QFT. By contrast, Wallace (2006,2011) argues that an interpretation of QFT should be based on LQFT.(Wallace, in 2006, calls his preferred QFT framework conventional QFT(CQFT), but changes his terminology to LQFT in Wallace 2011). Swanson(2017) and Egg, Lam, and Oldofedi (2017) are good overviews of thedebate between Fraser and Wallace (for an extended analysis see JamesFraser 2016). The debate covers many different philosophical topics inQFT, which makes it more challenging to pin down exactly what isessential to the arguments for both sides (for one view of what isessential for the debate, see Egg, Lam, and Oldofedi 2017). One issueis the role of internal consistency established by mathematical rigorversus empirical adequacy. Wallace argues that LQFT is empiricallyadequate since it can describe the forces of the Standard Model. LQFThas a collection of calculational techniques including perturbationtheory, path integrals, and renormalization group methods. Onecriticism of LQFT is that the calculational techniques it uses are notmathematically rigorous. Wallace argues that renormalization groupmethods puts perturbative QFT, an approach within LQFT, onmathematically rigorous ground and removes the main motivation foraxiomatic QFT.

5.1.1 Perturbative Quantum Field Theory

What follows is a rough overview of perturbative QFT (see James Fraser2016 for more details). Since exactly solvable free QFT models aremore mathematically tractable than interacting QFT models,perturbative QFT treats interactions as perturbations to the freeLagrangian assuming weak coupling. For strongly coupled theories likequantum chromodynamics that idealization fails. Using perturbationtheory, approximate solutions for interacting QFT models can becalculated by expanding S-matrix elements in a power series in termsof a coupling parameter. However, the higher order terms will oftencontain divergent integrals. Typically, renormalization of the higherorder terms is required to get finite predictions. Two sources ofdivergent integrals are infrared (long distance, low energy) andultraviolet (short distance, high energy) divergences. Infrareddivergences are often handled by imposing a long distance cutoff orputting a small non-zero lower limit for the integral over momentum. Asharp cutoff at low momentum is equivalent to putting the theory in afinite volume box. Imposing asymptotic boundary conditions andrestricting the observables to long distance “friendly” observablesalso help with infrared divergences. Ultraviolet divergences are oftenhandled by imposing a momentum cutoff to remove high momentum modes ofa theory. That is equivalent to freezing out variations in the fieldsat arbitrarily short length scales. Putting the system on a latticewith some finite spacing can also help deal with the highmomentum. Dimensional regularization, where the integral measure isredefined to range over a fractional number of dimensions, can helpwith both infrared and ultraviolet divergences. The last step inrenormalization is to remove the cutoffs by taking the continuum limit(i.e., removing the high momentum cutoff) and the infinite volumelimit (i.e., removing the low momentum cutoff). The hope is that thelimit is well-defined and there are finite expressions of the seriesat each order.

James Fraser (2016) identifies three problems for perturbativeQFT. (1)The rigor problem: perturbative QFT is notmathematically rigorous which makes it difficult to analyze andinterpret. (2)The consistency problem: perturbativecalculations rest on the interaction picture existing, but Haag’stheorem seems to show that the interaction picture does notexist. (3)The justification problem: renormalization lacksphysical motivation and appears ad hoc. James Fraser argues that (1)and (2) do not pose severe problems for perturbative QFT because it isnot attempting to build continuum QFT models. It is buildingapproximate physical quantities – not mathematical structures that areto be interpreted as physical systems.

Baker (2016) and Swanson (2017) note that LQFT makes false or unprovenassumptions such as the convergence of certain infinite sums inperturbation theory. Dyson (1952) gives a heuristic argument thatquantum electrodynamic perturbation series do not converge. Baker andSwanson also argue that the use of long distance cutoffs is at oddswith cosmological theory and astronomical observations which suggestthat the universe is spatially infinite. Even in the weak couplinglimit where perturbation theory can be formally applied, it is notclear when the perturbative QFT gives an accurate approximation of theunderlying physics. In the interacting \(\phi^4\) theory, when thedimension is less than 4 for Minkowski spacetime, the theory isnontrivial, but when the dimension is greater than 4, the renormalizedperturbation series is asymptotic to a free field theory even thoughit appears to describe nontrivial interactions. When there are 4dimensions, the theory is also trivial if additional technicalassumptions hold (see Swanson 2017 (p. 3) for more details).

5.1.2 Path Integrals in Quantum Field Theory

Another area where questions of mathematical rigor arise withinperturbative QFT is the use of path integrals. The S-matrix powerseries expansion contains integrals over momentum space and this iswhere path integrals / Feynman diagrams have been helpful for makingcalculations. The key concept is the partition function \(Z\),which is defined as a functional integral involving the action, whichis itself an integral of the Lagrangian. The following details comemainly from Hancox-Li (2017). More specifically, the action is afunctional of quantum fields. The functional integral over the actionranges over all possible combinations of the quantum fields valuesover spacetime. Informally, the sum is being taken over all possiblefield configurations. As Swanson (2017) notes, the path integralrequires choosing a measure over an infinite dimensional path space,which is only mathematically well-defined in special cases. Forexample, if the system is formulated on a hypercubic lattice, then themeasure can be defined (see section 1.2 of James Fraser2016). Another way of having a well-defined measure is to restrictattention to a finite dimensional subspace. But if functions areallowed to vary arbitrarily on short length scales, then the integralceases to be well-defined (Wallace 2006, p. 42). All of the correlationfunctions (i.e., vacuum state expectation values of the fields atdifferent spacetime points), can be derived from the partitionfunction \(Z\). So, given \(Z\), all empirical quantitiesassociated with the Lagrangian can be calculated, e.g., scatteringcross-sections. Finding \(Z\) amounts to a solution ofLQFT. \(Z\) can be expanded in a Taylor series in the couplingconstant. When this is done, two types of divergences can occur: (1)individual terms of the perturbation series can diverge and/or (2) theperturbation series itself is divergent, though the series may be anasymptotic series. To deal with (1), physicists do the followingprocedures (Hancox-Li 2017, pp. 344-345): (i) regularization, which involvesreducing the number of degrees of freedom via dimensionalregularization, momentum cutoffs, or using a lattice formulation and(ii) add counterterms to compensate for the regularization in (i). Butthis construction is purely formal and not mathematically defined. Therules used to manipulate the Lagrangian, and hence the partitionfunction, are not well-defined.

5.1.3 Renormalization Group Techniques

Wallace (2011) argues that renormalization group techniques haveovercome the mathematical deficiencies of older renormalizationcalculational techniques (for more details on the renormalizationgroup see Butterfield and Bouatta 2015, Fraser 2016, Hancox-Li (2015a,2015b, 2017)). According to Wallace, the renormalization group methodsput LQFT on the same level of mathematical rigor as other areas oftheoretical physics. It provides a solid theoretical framework that isexplanatorily rich in particle physics and condensed matter physics,so the impetus for axiomatic QFT has been resolved. Renormalizationgroup techniques presuppose that QFT will fail at some short lengthscale, but the empirical content of LQFT is largely insensitive to thedetails at such short length scales. Doreen Fraser (2011) argues thatrenormalization group methods help articulate the empirical content ofQFT, but the renormalization group has no significance for thetheoretical content of QFT insofar as it does not tell us whether weshould focus on LQFT or AQFT. James Fraser (2016) and Hancox-Li(2015b) argue that the renormalization group does more than provideempirical predictions in QFT. The renormalization group gives usmethods for studying the behavior of physical systems at differentenergy scales, namely how properties of QFT models depend or do notdepend on small scale structure. The renormalization group provides anon-perturbative explanation of the success of perturbativeQFT. Hancox-Li (2015b) discusses how mathematicians working inconstructive QFT use non-perturbative approximations with wellcontrolled error bounds to prove the existence or non-existence ofultraviolet fixed points. Hancox-Li argues that the renormalizationgroup explains perturbative renormalization non-perturbatively. Therenormalization group can tell us whether certain Lagrangians have anultraviolet limit that satisfies the axioms a QFT shouldsatisfy. Thus, the use of the renormalization group in constructiveQFT can provide additional dynamical information (e.g., whether acertain dynamics can occur in continuous spacetime) that a pureaxiomatic approach does not.

5.2 Middle Grounds

Egg, Lam, and Oldofedi (2017) argue that the main disagreementbetween Doreen Fraser and David Wallace is over the very definition ofQFT. Fraser takes QFT to be the union of quantum theory and specialrelativity. If QFT = QM + SR as Fraser maintains, then LQFT fails tosatisfy that criterion since it employs cutoffs which violate Poincarécovariance. For Wallace, the violation of QFT Poincarécovariance is not as worrisome. QFT is not a trulyfundamental theory since gravity is absent. Wallace is more interestedin what QFT’s approximate truth tells us about the world. LQFT givesus an effective ontology. The renormalization group tell us that QFTcannot be trusted in the high energy regimes where quantum gravity canbe expected to apply, i.e., the Planckian length scale wheregravitational effects cannot be ignored. The violation of Poincarécovariance via cutoffs may not amount to much if the fundamentalquantum theory of gravity imposes some real cutoff, according toWallace. There are, however, other options to consider.

5.2.1 Pluralistic Approaches

Some philosophers have rejected the seemingly either-or nature ofthe debate between Wallace and Fraser to embrace more pluralisticviews. On these pluralistic views, different formulations of QFT mightbe appropriate for different philosophical questions. Baker (2016)advocates that AQFT or LQFT should be trusted in domains of inquirywhere their idealizations are unproblematic. For example, if thedomain to be interpreted is the Standard Model, then LQFT is theappropriate framework. Swanson (2017) analyzes LQFT, AQFT, andWightman QFT and argues that all three approaches are complementaryand have no deep incompatibilities. LQFT supplies various powerfulpredictive tools and explanatory schemas. It can account for gaugetheories, the Standard Model of particle physics, the weak and strongnuclear force, and the electromagnetic force. However, the collectionof calculational techniques are not all mathematicallywell-defined. LQFT provides QFT theories at only certain length scalesand cannot make use of unitarily inequivalent representations sinceLQFT uses cutoffs which renders all representations finite dimensionaland unitarily equivalent by the Stone-von Neumann theorem. AxiomaticQFT is supposed to provide a rigorous description of fundamental QFTat all length scales, but that conflicts with the effective fieldtheory viewpoint where QFT is only defined for certain lengths. But ifaxiomatic QFT capture what all QFTs have in common, then effectivefield theories should be captured by it as well. Axiomatic QFT gives aprecise regimentation of LQFT, but it is unclear if axiomatic QFT isfully faithful to the LQFT picture. Within the axiomatic approach,Wightman QFT has many sophisticated tools for building concrete modelsof QFT in addition to rigorously proving structural results like thePCT theorem and the spin statistics theorem. But Wightman QFT relieson localized gauge-dependent field operators that do not directlyrepresent physical properties. AQFT might provide a more physicallytransparent gauge-free description of QFT. It has topological tools todefine global quantities like temperature, energy, charge, particlenumber which use unitarily inequivalent representations. But AQFT hasdifficulty constructing models. While LQFT is more mathematicallyamorphous, there are recent algebraic constructions of low dimensionalinteracting models with no known Lagrangian, which suggest that AQFTis more general than LQFT (Swanson 2017, p. 5). However, LQFT providesconstructive QFT with guidance on correctly building modelscorresponding to Lagrangians particle physicists use with greatempirical success (Hancox-Li 2017, p. 353).

5.2.2 Constructive Quantum Field Theory

Constructive QFT is an attempt to mediate between LQFT andaxiomatic QFT by rigorously constructing specific interacting modelsof QFT. The nontrivial solutions it constructs are supposed tocorrespond to Lagrangians that particle physicists use. This ensuresthat various axiomatic systems have a physical connection to the worldvia the empirical success of LQFT. While constructive QFT has donethis for some models with dimensions less than 4, it has not yet beenaccomplished for a 4 dimensional Lagrangian that particle physicistsuse. Any model that satisfies the Osterwalder-Schrader axioms willautomatically satisfy the Wightman axioms. Constructive QFT tries toconstruct the functional integral measures for path integrals byshifting from Minkowski spacetime to Euclidean spacetime via a Wickrotation (what follows is based on section four of Hancox-Li(2017)). In Euclidean field theory, the Schwinger functions, which aredefined in terms of \(Z\), must satisfy the Osterwalder-Schraderaxioms. The measure of \(Z\) is a Gaussian measure on the Schwartzspace of rapidly decreasing functions. The Osterwalder-Schrader axiomsare related to the Wightman axioms by the Osterwalder-SchraderReconstruction Theorem which states that any set of functionssatisfying the Osterwalder-Schrader axioms determines a uniqueWightman model whose Schwinger functions form that set. It allows theconstructive field theorists to use the advantages of Euclidean spacefor defining a measure while ensuring that they are constructingmodels that exist in Minkowski spacetime. It still has to be verifiedthat the solution corresponds to a renormalized perturbation seriesthat physicists derive for the corresponding Lagrangian in LQFT. Thechallenge is how to translate something not mathematicallywell-defined into something that is while showing that the“solutions” in LQFT can be reproduced by something that isconsistent with a set of axioms. This is crucial since, as Swanson(2017) points out, it is unclear whether perturbation theory is anaccurate guide for the underlying physics described by LQFT. Thisleads Hancox-Li (2017) to argue that mathematically unrigorous LQFT isrelevant to the rigorous program of constructive QFT in buildingrigorous interacting models of QFT. Those models correspond to theLagrangians of interest to particle physicists. Hence, LQFT can informthe theoretical content of QFT.

Another tool of constructive QFT is the use of asymptotic series,which can tell us which function the perturbative series is asymptoticto, which perturbative QFT does not. Constructive QFT tries todetermine some properties of non-perturbative solutions to theequations of motion which guarantee that certain methods of summingasymptotic expansions will lead to a unique solution (see Hancox-Li2017 (pp. 349–350) for more details). Is the rigorously definedpartition function \(Z\) asymptotic to the renormalized perturbativeseries? Roughly, a function is asymptotic to a series expansion whensuccessive terms of the series provide an increasingly accuratedescription of how quickly the function grows. The difference betweenthe function and each order of the perturbation series isapproximately small. But there are many different functions that havethe same asymptotic expansion. Ideally, we want there to be a uniquefunction because then there is a unique non-perturbative solution. Theconcept of strong asymptoticity requires that the difference betweenthe function and each order of the series is smaller than what wasrequired by asymptoticity. A strongly asymptotic series uniquelydetermines a function. If there is a strong asymptotic series, thenthe function can be uniquely reconstructed from the series by Borelsummation. The Borel transform of the series is given by dividing thecoefficients each term in the series by a factorial of the order ofthat term and then integrating to recover the exact function. Inconstructive QFT, the goal is to associate a unique function with arenormalized perturbation series and some kind of Borel summability isthe main candidate so far, though the Borel transform cannot removelarge-order divergences. The asymptotic behavior of the renormalizedperturbation series can be extremely sensitive to the choice ofregularization and render it asymptotic to a free field theory even ifit appears to describe nontrivial perturbations (see Swanson 2017(p. 11) for more details).

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