The Limits of Physical Equivalence in Algebraic Quantum Field Theory

The Limits of Physical Equivalence in Algebraic Quantum Field Theory Abstract Some physicists and philosophers argue that unitarily inequivalent representations (UIRs) in quantum field theory (QFT) are mathematical surplus structure. Support for that view, sometimes called ‘algebraic imperialism’, relies on Fell’s theorem and its deployment in the algebraic approach to QFT. The algebraic imperialist uses Fell’s theorem to argue that UIRs are ‘physically equivalent’ to each other. The mathematical, conceptual, and dynamical aspects of Fell’s theorem will be examined. Its use as a criterion for physical equivalence is examined in detail and it is proven that Fell’s theorem does not apply to the vast number of representations used in the algebraic approach. UIRs are not another case of theoretical underdetermination, because they make different predictions about ‘classical’ operators. These results are applied to the Unruh effect where there is a continuum of UIRs to which Fell’s theorem does not apply. 1 Introduction 2 Weak Equivalence and Physical Equivalence 3 Mathematical Overview of Algebraic Quantum Field Theory 4 Fell’s Theorem and Philosophical Responses to Weak Equivalence 5 Weak Equivalence in C*-Algebras and W*-Algebras 6 Classical Equivalence and Weak Equivalence 7 Interlude: Is Weak Equivalence Really Physical Equivalence? 8 The Unruh Effect 9 Time Evolution and Symmetries 10 Conclusions Appendix  1 Introduction The interpretive significance of unitarily inequivalent representations (UIRs) in quantum field theory (QFT) has generated intense philosophical discussion. What is the interpretive significance of UIRs in QFT? Does it make any difference which representation we use? Do UIRs make different predictions? Is the existence of UIRs just another case of theoretical underdetermination? Philosophers have radically different views on the significance of UIRs. At one end of the spectrum, UIRs are involved in questions about incommensurable particle ontologies (Arageorgis et al. [2002b], pp. 133–6), Haag’s theorem (Earman and Fraser [2006]), the Unruh effect (Arageorgis et al. [2002a]; Clifton and Halvorson [2001]), and algebraic quantum statistical mechanics (AQSM) (Ruetsche [2003]; Kronz and Lupher [2005]).1 Those authors treat UIRs as physically significant for foundational investigations of QFT. At the other end of the spectrum, Wallace ([2006]) has argued that the appearance of UIRs is not a problem for the foundations of QFT.2 Using Fell’s theorem, Wallace argues that different UIRs are empirically equivalent to each other.3 That argument was originally put forward by two physicists (Haag and Kastler [1964]) and has been influential in the development of algebraic quantum field theory (AQFT), particularly in the 1960s (for example, Robinson [1966], p. 488; Kastler [1964]). Fell’s theorem has been used to argue that different UIRs are, in some sense, ‘physically equivalent’ to each other, and that the choice of a particular representation is underdetermined by any finite amount of evidence we could ever obtain. Fell’s theorem has been used to argue for ‘algebraic imperialism’ (Ruetsche [2003]; Arageorgis [1995]). That position claims that the physical content of AQFT is contained in an abstract C*-algebra—not in UIRs. While some philosophers (Ruetsche [2003]; Arageorgis [1995]) have criticized the use of Fell’s theorem to classify UIRs as physically insignificant, no paper has provided an in-depth examination of the mathematical details surrounding Fell’s theorem in AQFT.4 The applicability of Fell’s theorem depends on the kind of algebraic structure under consideration. In particular, for UIRs of a C*-algebra, Fell’s theorem often cannot be extended to their W*-algebra extensions. If Fell’s theorem is the mathematical embodiment of physical equivalence, then the W*-algebra extensions of those UIRs are not physically equivalent. Another theorem will show that different UIRs cannot be considered empirically equivalent, or even approximately empirically equivalent, for all observables.5 For most UIRs, there exists an observable that will distinguish the UIRs. Thus, UIRs cannot be considered just another case of theoretical underdetermination. They will make different predictions, but the surprising twist is that the observable they will make different predictions about is a classical observable.6 While the results in this article show the limitations of a certain type of algebraic imperialism, they also point toward a new distinction in algebraic imperialism. In short, one can be an algebraic imperialist with respect to an abstract C*-algebra or an abstract W*-algebra. To put my cards on the table, I am an algebraic imperialist of the W*-algebra. For reasons that will become apparent later, this position is called ‘bidualism’. This interpretation will be particularly appealing to people who are sympathetic to algebraic imperialism but think that UIRs are physically significant and cannot be dismissed by appeal to Fell’s theorem. The plan for the article is as follows: Section 2 discusses Haag and Kastler’s interpretation of the mathematical condition of weak equivalence as a kind of physical equivalence. To understand the limitations of weak equivalence, it is crucial to understand how the different algebraic structures are related to each other. A brief overview of the mathematical details of AQFT is provided in Section 3. Fell’s theorem—or, more accurately, the mathematical conditions for the weak equivalence of two representations—the philosophical critique of physical equivalence, and the position of algebraic imperialism are discussed in Section 4. The connections among unitary equivalence, quasi-equivalence, and weak equivalence for representations of C*-algebras and W*-algebras are examined in Section 5. I will also show that most of the physically interesting UIRs are not weakly equivalent in that section. In Section 6, I argue that UIRs do make different empirical predictions for a ‘classical’ observable. The question of whether the mathematical condition of weak equivalence captures the idea of physical equivalence will be examined in Section 7. A brief application of these results is given for the Unruh effect in Section 8, where it is shown that there is a continuum of physically inequivalent representations. In Section 9, I will examine Fell’s theorem under symmetry transformations and time evolution. I will show how Fell's theorem fails in those cases and in the case of spontaneous symmetry breaking. Conclusions are provided in Section 10. 2 Weak Equivalence and Physical Equivalence The condition of unitary equivalence has been important in the history of quantum theory. For example, it follows from the Stone–von Neumann theorem that Schrödinger’s wave mechanics and Heisenberg’s matrix mechanics are unitarily equivalent representations of the canonical commutation relations.7 Unitarily equivalent representations are empirically equivalent to each other, in that they give the same predictions.8 Either formalism can be used since both will give the same result. Using wave mechanics or matrix mechanics is similar to using a particular coordinate system to solve a problem. The loss of unitary equivalence raises the worry of whether different UIRs make different empirical predictions, in which case it matters which representation is chosen. The problem of whether UIRs are physically significant or can be rejected as mathematical surplus structure is particularly acute in QFT because, in many cases, there is a continuum of different UIRs available. The vast number of UIRs might give aid to the intuition that there are far too many possibilities for different UIRs to count as physically significant. The argument that all UIRs are, in a sense, physically equivalent is based on an interpretation of a mathematical result in AQFT known as Fell’s theorem. That theorem introduces the mathematical idea of ‘weak equivalence’ and it is important to make a distinction between the notions of weak equivalence and physical equivalence. Weak equivalence is a mathematical concept introduced by Fell ([1960], p. 375), while physical equivalence, which was introduced by Haag and Kastler ([1964]), is a semantic extrapolation of weak equivalence. To avoid confusion, I will use ‘weak equivalence’ even though Haag and Kastler talk about physical equivalence, because I will argue in Section 7 that weak equivalence does not mathematically encode the concept of physical equivalence. I will explain weak equivalence in Section 4, after introducing some of the mathematical structures in AQFT. Weak equivalence was used by Haag and Kastler ([1964]), among others, in AQFT to elevate the abstract C*-algebra to a position of physical importance while denigrating the significance of representations, in particular that of UIRs, using the following operationalist justification. Since only a finite number of experiments can be carried out and each experiment has finite accuracy, given any state on a Hilbert space associated with one representation, there is a state on a Hilbert space associated with a different representation that is unitarily inequivalent to the first representation such that both states have roughly the same expectation values (within some small ε>0 ?>) for the same observables. Thus, both representations should be considered physically equivalent. UIRs would be mathematical surplus structure. This is how Wallace ([2006], p. 54) explains Fell’s theorem9: Fell’s theorem says that a finite number of measurements, each conducted with finite accuracy, cannot distinguish between representations. Hence we could reproduce any experimental results using whichever representation was most convenient. (Wallace [2006], p. 54) The problem of choosing a particular representation from a continuum of possible choices would evaporate since any choice would be physically equivalent to any other choice. However, as we will see in Section 5, the mathematical condition of weak equivalence, upon which Haag and Kastler’s argument rests, fails to be satisfied by most UIRs and, thus, they will fail to be physically equivalent. The failure of weak equivalence depends on the kind of algebra under consideration. The different algebraic options and their interconnections are explained in the next section. 3 Mathematical Overview of Algebraic Quantum Field Theory What follows is a brief partial overview of the conceptual landscape of AQFT.10 The diagram in Figure 1 is one of the keys for understanding the mathematical limitations of Haag and Kastler’s notion of physical equivalence. Figure 1. View largeDownload slide Figure 1. View largeDownload slide The first thing to notice is that there is an abstract and a concrete level in the algebraic approach on the left side of Figure 1. On a first pass, one can think of the concrete level as involving operators defined on Hilbert spaces and the abstract level as where operators are not defined on a Hilbert space. Beginning in the upper left-hand corner of Figure 1, there is an abstract C*-algebra A ?>.11 An algebra is essentially a set of elements that is closed under linear combinations and products. It is a mathematical structure that has more structure than the more familiar notion of a vector space. Moving to the right, the dual of A ?>, denoted by A* ?>, is the set of bounded linear functionals on A ?>.12 A state is a linear functional that is positive ( ω(A*A)>0 ?> for all A∈A ?>) and normed ( ω(I)=1 ?>, where I is the multiplicative identity in A ?>). The set of states in A* ?> is denoted as A1*+ ?> and A1*+⊂A* ?>. Taking the dual of A* ?> generates a W*-algebra, A** ?>, called the ‘bidual’.13 It is rather surprising that A** ?> has algebraic structure. A** ?> will play a key role in what follows. Moving to the concrete row at the bottom left of Figure 1, Hilbert spaces make their appearance. To get there, we will start at the abstract level in the upper left-hand corner of Figure 1 at A ?> and move downward. A representation, πω ?>, can be built by picking an abstract state, ω, where ω∈A1*+ ?>. πω(A) ?> is a ‘concrete’ C*-algebra and πω ?> is a C*-representation. A representation of a C*-algebra A ?> is a map πω ?> from A ?> into the set of bounded operators B(Hπω) ?> of an associated Hilbert space, Hπω ?>, that preserves the algebraic relations between the elements of A ?>.14 Given an abstract state ω, a Hilbert space Hπω ?> can be constructed from it. This should not be too surprising since A ?> is already a vector space. The standard procedure for generating Hilbert spaces from a C*-algebra A ?> is called the ‘GNS construction’ (named after Gelfand, Naimark, and Segal who showed how to do it). There is also a ‘reverse GNS process’, where states in a Hilbert space have abstract counterparts in A1*+ ?>. For each Hilbert space, Hπω ?>, associated with a representation, πω ?>, there will be a collection of vector states and density matrices defined on Hπω ?>. For each vector state, |Ψπω⟩ ?>, and density matrix, ρπω ?>, there is a corresponding abstract state in A1*+ ?> ( Ψω ?> and ρω ?>, respectively) defined by the following relations: Ψω(A)=⟨Ψπω|πω(A)|Ψπω⟩, ρω(A)=Tr(ρπωπω(A)). In other words, for each concrete state (vector state or density operator) in a Hilbert space, there corresponds an abstract counterpart in A1*+ ?>. The set of all abstract states that are density operators in Hπω ?> is a norm closed convex subset of A1*+ ?> called the folium Fω ?>. Staying at the concrete C*-level and moving to the right, there are two equivalent ways to construct a von Neumann algebra (also called a ‘concrete W*-algebra’) from a concrete C*-algebra, πω(A) ?>: (i) take the bicommutant of πω(A) ?>, that is πω(A)″ ?>,15 or (ii) close πω(A) ?> in the weak operator topology,16 that is, πω(A)− ?>. The most common way to construct a von Neumann algebra is to start with a C*-algebra A ?> and an abstract state ω∈A1*+ ?>, build a representation πω(A) ?> and a Hilbert space (Hπω,πω) ?> via the GNS construction, and then take the bicommutant, πω(A)″ ?>. However, there is an equivalent alternative way to build πω(A)″ ?> using the bidual A** ?>. Since C*-algebras and W*-algebras are both *-algebras, the GNS construction can be used to build Hilbert spaces for both algebras. For any state ω, it can be extended to be a normal state ω∼ ?> on A** ?>.17 If the GNS construction is done using ω∼ ?> and A** ?>, then a von Neumann algebra πω∼(A**) ?> is generated.18 A von Neumann algebra is a concrete W*-algebra; it is a W*-representation of A** ?>.19 All representations (Hπω,πω) ?> of A ?> can be uniquely extended (Hπω,πω∼) ?> to be a W*-representation of A** ?> and this extension is equal to the weak closure of πω(A) ?>, that is, πω(A)″=πω∼(A**)=πω(A)− ?>; see (Bing-Ren [1992], pp. 221-2, Theorem 4.2.7). 4 Fell’s Theorem and Philosophical Responses to Weak Equivalence There are a number of different formulations of Fell’s theorem. To disentangle them, I will begin with Fell’s ([1960]) original paper, which focused on the dual space A* ?> of C*-algebras A ?>. There are two crucial ideas: the topology of the state space and kernels. A topology may be defined on A* ?>, called the weak*-topology on A* ?>, which is sometimes referred to as the σ(A,A*) ?> topology, in which a basis of neighbourhoods N(ϕ,{Ai}i=1k,ε) ?> consists of all sets of the form {ω∈A*||ϕ(Ai)−ω(Ai)|<ε} ?>, where Ai∈A ?> and ε>0 ?>. The weak*-neighbourhoods of an abstract state, ϕ ?>, are indexed by a finite subset, Ai of A ?>, and a positive real number, ε. The second important concept is the kernel of a representation, π, which is defined as the set of elements a representation π maps to the zero operator in its Hilbert space: kerπ={A|π(A)=0,A∈A} ?>. The key theorem of Fell ([1960], p. 367) can now be expressed as follows20: Fell’s Theorem Let A ?> be a C*-algebra and πϕ ?> and πψ ?> be two representations of A ?>.21 The following conditions are equivalent: the kernel of πψ ?> is contained in πϕ ?>, that is, kerπψ⊆kerπϕ ?>, every positive functional η on A ?> associated with πϕ ?> is a weak*-limit of finite sums, γ, of positive functionals associated with πψ ?> for which ||η||≤||γ|| ?>. There are two important aspects of Fell’s theorem: when kerπψ⊆kerπϕ ?>, (i) a vector state associated with πϕ ?> is the weak*-limit of vector states associated with πψ ?> and (ii) the weak*-neighbourhood of an abstract state ρϕ ?> that is a density operator associated with πϕ ?>—that is, ρϕ ?> belongs to the folium Fϕ ?>—contains an abstract state ρψ ?> that is a density operator associated with πψ ?>, that is, ρψ ?> belongs to the folium Fψ ?>. If either (1) or (2) is satisfied, then πψ ?> is ‘weakly contained’ in πϕ ?>. Since the kernel of πϕ ?> is larger than πψ, πϕ ?> maps more elements of A ?> to the zero operator, that is, it is less faithful.22 Fell’s theorem is used to define weak equivalence: two representations πϕ ?> and πψ ?> of A ?> are weakly equivalent if and only if they are weakly contained in each other. The key feature doing work in weak equivalence is the notion of a weak*-neighbourhood at the abstract level of states—not at the concrete level of representations. This is where the limitation of restricting the number of observables, Ai, to a finite set comes from—not from any operationalist scruples about measuring an infinite number of observables. The definition of weak equivalence and Fell’s theorem permits the formulation of another version of Fell’s theorem, which is the basis of a number of presentations of weak equivalence (Haag and Kastler [1964]; Robinson [1966]). Fell’s Theorem 2 Two representations, πϕ ?> and πψ ?>, of a C*-algebra A ?> are weakly equivalent if and only if kerπϕ=kerπψ ?>.23 In Fell’s Theorem 2, the notion of weak equivalence depends on the kernels of the two representations being equal. This provides a simple test for whether two representations are weakly equivalent. If one considers all faithful representations—that is, representations that have trivial kernels—then those representations are all weakly equivalent to each other. This is how weak equivalence is often presented in the literature. Weak Equivalence: All faithful representations of a C*-algebra are weakly equivalent. If the C*-algebra is simple (that is, it has no closed two-sided ideals), then all non-zero representations (that is, π(A)≠0 ?> for some A∈A ?>) of it are faithful (Emch [1972], p. 80). Thus, each non-zero representation is weakly equivalent to every other representation of a simple C*-algebra. Two physically important simple algebras are the Weyl canonical commutation relations C*-algebra (Bratelli and Robinson [1997], pp. 19–22) and the Weyl canonical anti-commutation relations C*-algebra (Bratelli and Robinson [1997], pp. 15–6). Haag and Kastler ([1964]) interpreted the mathematical concept of weak equivalence as a kind of physical equivalence. In setting up an experiment, or a series of experiments, there are three important features: (i) the state, ρπϕ ?>, of the system prior to measurement, (ii) the observable(s), Ai, that are going to be measured, and (iii) the accuracy of each measurement, εi. Let ρπϕ ?> be the density operator in (Hπϕ,πϕ) ?> that is the state of the system prior to a measurement and ρϕ ?> its abstract state counterpart that is a member of the folium Fϕ ?>. Once (ii) and (iii) have been fixed, there will be a weak*-neighbourhood N(ρϕ,Ai,εi) ?>24 around ρϕ ?> in A1*+ ?>, where 1≤i≤k ?>. Let (Hπψ,πψ) ?> be a UIR with respect to (Hπϕ,πϕ) ?>. Will there be an abstract state, ρψ ?>, in the folium of Fψ ?> that is weakly equivalent in the above sense? If kerπϕ=kerπψ ?>, then weak equivalence guarantees that such a state can be found in Fψ ?>, that is, |ρϕ(Ai)−ρψ(Ai)|<εi ?>. As mentioned above, if the algebra is simple, then both representations are weakly equivalent and hence are interpreted as physically equivalent by Haag and Kastler. Various phrases such as ‘algebraic imperialism’ (Arageorgis [1995], p. 132), ‘apologetic imperialism’ (Ruetsche [2011], p. 133), and ‘algebraic chauvinism’ (Ruetsche [2003], p. 1334) have been used to describe the position of many algebraic quantum field theorists in the 1960s. Based on the notion of physical equivalence, it was claimed that Hilbert spaces and representations of a C*-algebra were dispensable and that the abstract C*-algebra contained all of the physical content of a theory.25 […] the important thing here is that the observables form some algebra, and not the representation Hilbert space on which they act. (Segal [1967], p. 128) It is in this new notion of equivalence in field theory that the algebraic approach has its greatest justification. All the physical content of the theory is contained in the algebra itself; nothing of fundamental significance is added to a theory by its expression in a particular representation. From this point of view it becomes clear that only faithful representations are worth consideration because the existence of a non-trivial kernel for a representation implies that there are redundant elements in the algebra. (Robinson [1966], p. 488) The relevant object is the abstract algebra and not the representation. The selection of a particular (faithful) representation is a matter of convenience without physical implications. It may provide a more or less handy analytical apparatus. (Haag and Kastler [1964], pp. 851–2) […] the specification of a special representation is physically irrelevant, all the physical information being contained in the algebraic structure of the abstract algebra A ?> alone. [Interpreting weak equivalence as physical equivalence] shows indeed that the physically relevant object is not a concrete realization of A ?> but the algebra A ?> itself, since any two different concrete realizations (i.e., faithful *-representations, or representations with zero kernel) will be physically equivalent. (Kastler [1964], p. 180–1) Some philosophers of physics have criticized weak equivalence as a sufficient reason to dismiss UIRs as mere mathematical possibilities.26 Explanations of representation dependent features such as symmetry breaking (Arageorgis [1995], pp. 159, 165-6) and phase transitions (Ruetsche [2003], p. 1138) use UIRs.27 Hence, it has been argued that these different UIRs cannot be considered physically equivalent. That argument is unlikely to convince Wallace that different UIRs must be used to explain phase transitions since ‘it is provably impossible for us to ever discover’ (Wallace [2006], p. 54) which representation is the correct representation for describing a particular phase. Since Fell’s theorem is taken to show that a finite number of measurements each carried out with finite accuracy cannot distinguish different representations, an infinite number of measurements is needed to find the correct representation. Rather than focus on semantic or methodological arguments against the algebraic imperialist’s use of Fell’s theorem, I am going to examine the mathematical heart of the matter: the notion of weak equivalence. Using weak equivalence to argue for the physical insignificance of UIRs would be particularly telling if the only type of representation used in algebraic quantum physics is a C*-representation. However, as I will explain below, the most commonly used concrete algebra in both AQFT and AQSM is a von Neumann algebra, which can be viewed as a W*-representation of the bidual A** ?>. If weak equivalence could be shown to hold for W*-representations as well, then a strong case could be made for the algebraic imperialist position on UIRs. However, the mere weak equivalence of C*-representations is not enough to guarantee the weak equivalence of their W*-representations. 5 Weak Equivalence in C*-Algebras and W*-Algebras To evaluate weak equivalence for von Neumann algebras, a new type of equivalence for C*-representations must be introduced. Quasi-equivalence is a weaker notion than unitary equivalence, but it is stronger than weak equivalence28: Unitary Equivalence ⇒ Quasi-equivalence ⇒ Weak Equivalence For C*-representations and W*-representations, the results are summarized below.29 Surprisingly, the weak equivalence of two C*-representations is not sufficient to guarantee the weak equivalence of their respective W*-representations. The stronger condition of quasi-equivalence for C*-representations must be satisfied for their W*-representations to be weakly equivalent. One response to Table 1 might be that C*-representations are more useful for modelling various physical situations and that W*-representations are not needed. However, it is the W*-representations, or von Neumann algebras, that are used more for modelling different systems. Every von Neumann algebra can be written as a direct sum, or direct integral, of types I, II, or III factors.30 There are three factors types: type I, type II, and type III. The rough idea behind the classification is the following31: Every von Neumann algebra is generated from its projections. A dimension function, d, can be defined to classify the range of the projections as follows: Type In ?>: d: {0, 1,…,n} where n is a natural number Physical example: Non-relativistic finite dimensional quantum mechanics (n × n complex matrices) Type I∞ ?>: d: {0, 1,…, ∞ ?>} Physical example: Free boson Fock space Type II1 ?>: d: [0, 1] Physical examples: Free fermion Fock space Infinite temperature maximally chaotic KMS state Type II∞ ?>: d: [0, ∞ ?>) Physical example: The tensor product of the free boson and fermion Fock spaces Type III: d: {0, ∞ ?>} (two-element set) Physical examples: Typically local algebras in AQFT32 KMS states with finite non-zero temperature in AQSM While the type III factor seems to be pathological, ‘[t]o the malicious delight of mathematicians’ (Thirring [1983], p. 41), type III factors are the most useful factor for modelling many physical systems. In particular, local algebras, which are defined on open regions of Minkowski spacetime, are usually type III factors (see Footnote 32). A KMS state (named after Kubo, Martin, and Schwinger) that corresponds to a pure phase of an infinite system at a non-zero finite temperature induces a representation that is also a type III factor.33 Table 1. C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇒ ?> Unitary Equivalence ⇓ ?> ⇓ ?> Quasi-Equivalence ⇔ ?> Weak Equivalence ⇓ ?> Weak (Physical) Equivalence C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇒ ?> Unitary Equivalence ⇓ ?> ⇓ ?> Quasi-Equivalence ⇔ ?> Weak Equivalence ⇓ ?> Weak (Physical) Equivalence Table 1. C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇒ ?> Unitary Equivalence ⇓ ?> ⇓ ?> Quasi-Equivalence ⇔ ?> Weak Equivalence ⇓ ?> Weak (Physical) Equivalence C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇒ ?> Unitary Equivalence ⇓ ?> ⇓ ?> Quasi-Equivalence ⇔ ?> Weak Equivalence ⇓ ?> Weak (Physical) Equivalence As the examples above show, W*-representations, and in particular type III factors, are necessary to model many of the important physical examples in AQFT and AQSM. One might still hope that quasi-equivalence could be used to banish UIRs to the realm of mathematical annoyances. If UIRs could be shown to be quasi-equivalent as C*-representations, then their W*-representations would still be weakly equivalent. However, the distinction between unitary equivalence and quasi-equivalence collapses for many physically important representations such as irreducible representations,34 KMS representations (that is, representations generated by abstract states that satisfy the KMS condition35), and type III factors. In those cases, mere unitary inequivalence is enough to guarantee the weak inequivalence of the W*-representations. For irreducible representations, KMS representations, and type III factors it can be proven that quasi-equivalence implies unitary equivalence, in which case the distinction between quasi-equivalence and unitary equivalence collapses (Table 2).36 Table 2. Irreducible Representations, KMS Representations, and Type III Factors Quasi-equivalence ⇒ Unitary Equivalence ?> Irreducible Representations, KMS Representations, and Type III Factors Quasi-equivalence ⇒ Unitary Equivalence ?> Table 2. Irreducible Representations, KMS Representations, and Type III Factors Quasi-equivalence ⇒ Unitary Equivalence ?> Irreducible Representations, KMS Representations, and Type III Factors Quasi-equivalence ⇒ Unitary Equivalence ?> Table 3. C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇔ ?> Unitary Equivalence ⇓ ?> Weak (Physical) Equivalence C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇔ ?> Unitary Equivalence ⇓ ?> Weak (Physical) Equivalence Table 3. C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇔ ?> Unitary Equivalence ⇓ ?> Weak (Physical) Equivalence C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇔ ?> Unitary Equivalence ⇓ ?> Weak (Physical) Equivalence This modifies the relationships from Table 1 for irreducible representations, KMS representations, and type III factors. Irreducibility can intuitively be thought of as a representation that is as ‘small’ as possible. Some physicists view them as the most important kind of representation since they can serve as the building blocks for other representations. Whether one is looking for representations of the Lorentz group or of the algebra of the canonical commutation relations (Weyl algebra), irreducibility has been a desideratum.37 Irreducibility collapses the distinction between unitary equivalence and quasi-equivalence, as well as the distinction between disjointness and unitary inequivalence38; see (Kadison and Ringrose [1997], p. 740).39 For irreducible representations, KMS representations, and type III factors, two representations are either quasi-equivalent or disjoint. If they are disjoint, then they are not quasi-equivalent. That implies that they are not unitarily equivalent and that their associated W*-representations are not weakly equivalent. Thus, for those three cases mere unitary inequivalence is sufficient to show that their W*-representations are weakly inequivalent. In those cases, the situation is even worse since W*-representations are only going to be weakly equivalent when their C*-representations are unitarily equivalent! Here are some specific physical examples of UIRs that do not satisfy the condition of weak equivalence for W*-representations. Theorem 1 Let ϕ ?> and ψ be (αt,β) ?> and (αt,β′) ?>-KMS states with automorphism group αt, inverse temperatures β and β′ ?>, respectively, such that β≠β′ ?> and at least one temperature is non-zero and finite. Then πϕ∼ ?> and πψ∼ ?> are not weakly equivalent.40 Theorem 1 is a rich source of UIRs that are not weakly equivalent W*-representations. Given any two KMS states with different non-zero finite temperatures, the representations they generate will be unitarily inequivalent and their W*-representations will not be weakly equivalent no matter how small the difference is between their temperatures. Thus, there will be a continuum of UIRs whose W*-representations will not be weakly equivalent—one representation for each temperature! The next theorem proves that UIRs with different chemical potentials are weakly inequivalent. Theorem 2 Let ϕ ?> and ψ be KMS states with chemical potentials μ and μ′ ?>, respectively, such that μ≠μ′ ?>. Then πϕ∼ ?> and πψ∼ ?> are not weakly equivalent.41 As in the case of temperature, Theorem 2 provides a continuum of UIRs whose W*-representations are weakly inequivalent, regardless of how small the difference is between chemical potentials μ and μ′ ?>. These theorems show that KMS states are extremely sensitive to variations in temperature or chemical potential. Though these examples show how to generate UIRs whose W*-representations are weakly inequivalent in AQSM, there are similar examples in AQFT. Later in this article, it will be shown that there is a continuum of such cases for the Unruh effect. How does the failure of weak equivalence for W*-representations connect with the notion of faithful representations that is key to many presentations of Fell’s theorem? Recall that πϕ∼ ?> and πψ∼ ?> are weakly inequivalent if and only if kerπϕ∼≠kerπψ∼ ?>. First, notice that for von Neumann algebras, the faithfulness of a representation is no longer with respect to the original C*-algebra A ?>, but rather to the W*-algebra A** ?>. If kerπϕ∼≠kerπψ∼ ?>, then there are three possible cases: (i) πϕ∼ ?> is faithful and πψ∼ ?> is not faithful, (ii) πϕ∼ ?> is not faithful and πψ∼ ?> is faithful, and (iii) both πϕ∼ ?> and πψ∼ ?> are not faithful. In any of these cases, there is an A∈A** ?> such that A≠0 ?> and, for example, in (ii), πϕ∼(A)=0 ?>. It follows that for some density operator, ρπϕ∼ ?>, associated with πϕ∼(A**) ?>, there does not exist a density operator, ρπψ∼ ?>, associated with πψ∼(A**) ?> that can approximate ρπϕ∼ ?> for A within a given ε > 0, that is, for some ρπϕ∼ ?>, A, and ε that for every ρπψ∼ ?>. Thus, there will always be some observable, A in A** ?>, that will distinguish any pair of weakly inequivalent W*-representations. What is rather surprising is that this observable has classical properties. 6 Classical Equivalence and Weak Equivalence The mathematical basis for the algebraic imperialist argument against UIRs has now been shown to apply only in very exceptional circumstances. For most UIRs, it can be proven that they are not weakly equivalent to each other. Someone might hold onto the hope that different UIRs do not make different predictions. If that is the case, then the appearance of UIRs is just another case of theoretical underdetermination. It would then make no difference which representation is used since they would all make the same, or approximately the same, predictions. What is the difference, if any, between W*-representations that are weakly inequivalent? Does weak equivalence have any usefulness for W*-representations? To borrow a phrase from Ruetsche ([2003]), weak equivalence for W*-representations can be ‘put to work’. In fact, it shows us where to locate the physical differences between any two UIRs. But there is a surprising twist. Two W*-representations that are factors are weakly equivalent if and only if they assign the same expectation values to all classical—not quantum—properties. Theorem 3 Let ϕ ?> and ψ be states that generate factor representations πϕ ?> and πψ ?>, respectively. The W*-representations πϕ∼ ?> and πψ∼ ?> are weakly equivalent if and only if the expectation values are the same for the extensions of ϕ ?> and ψ to be states on A** ?>, that is, ϕ∼(Z)=ψ∼(Z) ?> for every observable Z in the centre of the bidual, Z∈ℨ(A**) ?>.42 Since Z commutes with every observable in A** ?>, it can be considered a ‘classical’ observable.43 The weak equivalence of two W*-representations is sufficient for their being classically equivalent to each other since their states will have exactly the same expectation values for every element in ℨ(A**) ?>. Since every element in ℨ(A**) ?> commutes with every element in A** ?> and A** ?> includes every element in the original C*-algebra A ?>—that is, A⊆A** ?>—this is the largest collection of classical observables. Thus, the weak equivalence of W*-representations is more appropriately a mathematical condition for classical equivalence than physical equivalence. The converse of Theorem 3, namely that πϕ∼ ?> and πψ∼ ?>, are weakly inequivalent if and only if there exists an observable Z∈ℨ(A**) ?> such that ϕ∼(Z)≠ψ∼(Z) ?>, shows that there exists an observable that will distinguish different UIRs, or more precisely, it will distinguish C*-representations that are not quasi-equivalent. It is the classical observables that are members of the much larger algebra A** ?> that serve to distinguish different UIRs. Thus, UIRs are not just another example of theoretical underdetermination, they make different predictions; but these predictions are with respect to classical observables in a W*-algebra A** ?>, not the original C*-algebra A ?>. In the cases discussed in Section 5, the classical properties that distinguished different UIRs were temperature and chemical potential. These results point to the limitations of the original C*-algebra A ?>, namely, that it is not large enough. UIRs can be distinguished by observables in a larger algebra: the bidual A** ?>.44 The argument that UIRs are not another case of undetermination can be strengthened. Weak equivalence says that for any abstract state, ϕ ?>, any finite collection of abstract observables, Ai, any corresponding finite collection of positive real numbers, εi, and any representation, there is a state ψ in a different representation for which the expectation value for each Ai lies within ±εi ?> from its expectation value in ϕ ?>. For a classical observable in a particular representation, all (concrete) states in that representation will have the same value, for example, in πϕ∼ ?> each vector state and density matrix will have the same expectation value for a particular Z∈ℨ(A**) ?>. Given any Z∈ℨ(A**) ?>, it can be assumed without loss of generality that Z either has the value 1 or 0. For any πϕ∼ ?> and πψ∼ ?> that are weakly inequivalent, there exists an observable Z∈ℨ(A**) ?> such that ϕ∼(Z)≠ψ∼(Z) ?>. The only way ϕ∼(Z)≠ψ∼(Z) ?> is that ϕ∼(Z)=1 ?> and ψ∼(Z)=0 ?>, or ϕ∼(Z)=0 ?> and ψ∼(Z)=1 ?>. If ψ∼(Z)=1 ?>, can I find any states in πψ∼ ?> such that ? No, precisely because every density matrix in πψ∼ ?> will have the same expectation value for Z, namely, 0! All of these results show that an algebraic imperialism based on weak equivalence has significant drawbacks. But the type of algebraic imperialism that is being criticized here is of a specific type. A better name for it would be ‘C*-algebraic imperialism’ since the physical content is supposed to be captured by A ?>. These results lend support to a new type of algebraic imperialism, which will be called ‘bidualism’.45 It’s only in the bidual A** ?> that the algebraic imperialist has access to enough observables to distinguish different states, for example, states of different thermodynamic phases (or KMS states of different temperatures). A** ?> contains the abstract counterparts to Ruetsche’s ‘parochial’ observables, as well as all of the observables in the original C*-algebra A ?>.46 For the bidualist, physical content is captured at the abstract algebra level by the observables in A** ?> and the normal states defined on A** ?>. It allows one to be an algebraic imperialist and endorse the physical significance of UIRs.47 7 Interlude: Is Weak Equivalence Really Physical Equivalence? These theorems also show the semantic failure of interpreting weak equivalence as physical equivalence. The interpretation of weak equivalence as physical equivalence was motivated by operationalist considerations, namely, that only a finite number of experiments can be carried out and that each experiment has some finite error associated with it. The problem is that weak equivalence is both too strong and too weak to capture the idea of physical equivalence. To see why it is too weak, consider two KMS states with different temperatures and let the difference in their temperatures be so small that they cannot be distinguished by any experiment. Those two KMS states should satisfy the operationalist’s notion of both systems being physically equivalent to each other since no measurement can distinguish them. However, as shown in the last section, those states are weakly inequivalent to each other, and hence are physically inequivalent to each other. Thus, while two KMS states with slightly different temperatures would satisfy an operationalist understanding of physical equivalence, the mathematical condition of weak equivalence would fail to classify those KMS states as being physically equivalent. The concept of weak equivalence is also too strong to capture the idea of physical equivalence. If the strategy of using weak equivalence to classify all UIRs as physically equivalent and empirically equivalent succeeded, the consequences would be unacceptable. One very rough way of describing Haag’s theorem is that the representations of a free and an interacting system are unitarily inequivalent to each other. If weak equivalence truly captured the concept of physical equivalence and the representations for all free and interacting systems are weakly equivalent to each other, then not only would a free and interacting system be physically equivalent, but they would make roughly the same predictions! It would also classify different phases (for example, a gas and a solid) as physically equivalent. These considerations should indeed make us wary of the semantic extrapolations from weak equivalence to physical equivalence (Emch [1972], p. 108). 8 The Unruh Effect While philosophers have focused on the fate of particles in the Unruh effect and the role of UIRs, there has been no discussion of weak equivalence in the Unruh effect. I will extend the results of the previous sections to show that the Minkowski and Rindler representations are not weakly equivalent as W*-representations. The classical observable that serves to distinguish those representations is temperature. What is even more surprising is that the Unruh effect has more than two UIRs that are not weakly equivalent as W*-representations. A continuum of W*-representations that are not weakly equivalent can be found in the Unruh effect, with one representation for each temperature or acceleration value. There are a number of excellent papers on the Unruh effect (Clifton and Halvorson [2001]; Arageorgis et al. [2002a]), so I will only give a brief overview. The Unruh effect is generated by using Rindler coordinates in Minkowski spacetime.48 There are two spacetime wedges that have two types of trajectories defined on them: inertial motion and uniformly accelerated motion. The Rindler observer in, say, the right wedge is uniformly accelerating while a Minkowski observer’s trajectory is inertial.49 Both observers will associate a special C*-algebra called the Weyl algebra W ?> with the right wedge and build a representation via the GNS construction50. For my purposes, the salient feature of the Unruh effect is that the Rindler representation πωR⊲ ?>, which is only defined on the right wedge, and the Minkowski representation πωM⊲ ?>, which is restricted to the right wedge, are unitarily inequivalent (Clifton and Halvorson [2001], p. 463).51 Given this result, it is straightforward to prove the following theorem: Theorem 4 Let πω∼R⊲ ?> and πω∼M⊲ ?> be the W*-representations of W** ?> associated with πωR⊲ ?> and πωM⊲ ?>, respectively. Then πω∼R⊲ ?> and πω∼M⊲ ?> are weakly inequivalent.52 Thus, the Unruh effect is an example in AQFT of two weakly inequivalent W*-representations. The explanation of their weak inequivalence can be deepened. Since both W*-representations are factors and they are weakly inequivalent, the following theorem immediately follows from Theorem 3 and Theorem 4: Theorem 5 There exists an observable Z∈ℨ(W⊲**) ?> in the centre of the bidual of the Weyl algebra W⊲** ?> on the right Rindler wedge, such that ω∼R⊲(Z)≠ω∼M⊲(Z) ?>. What physical quantity is Z? The Minkowski vacuum state, ωM⊲ ?>, when restricted to the right wedge, satisfies the KMS condition (Arageorgis et al. [2002a], pp. 188–9). Suppressing some technical details53 the temperature of ωM⊲ ?> is proportional to its acceleration in the right wedge. There is a continuum of these ωM⊲ ?> KMS states, one for each acceleration a∈(0,∞) ?> at temperature T=a2π ?>. Using Theorem 1, it then follows that if ωM,T⊲ ?> is a KMS state at temperature T=a2π ?> with acceleration a, ωM,T′⊲ ?> is a KMS state at temperature T′=a′2π ?> with acceleration a′ ?>, T≠T′ ?>, then πω∼M,T⊲ ?> and πω∼M,T′⊲ ?> are weakly inequivalent. Thus, there exists a continuum of UIRs in the Unruh effect (one UIR for each temperature, or acceleration, value), and they are each weakly inequivalent to each other and to πω∼R⊲ ?>! Further, the classical observable that distinguishes them is temperature. 9 Time Evolution and Symmetries One area that has not been explored is whether weak equivalence is preserved under time evolution or symmetry transformations. Symmetries and time evolution are represented in AQFT as automorphisms of the C*-algebra A ?>. An automorphism can be inner or outer depending on whether it can be implemented by a unitary operator. If it can be unitarily implemented, then the automorphism is called an inner automorphism, otherwise it is an outer automorphism. If the automorphism is outer, then there is a spontaneously broken symmetry or non-unitary dynamics. For two states, ω1 and ω2, to be weakly equivalent there must be a finite collection of observables, Ai∈A ?>, where 1≤i≤n ?>, such that |ω1(Ai)−ω2(Ai) |<εi ?>. Let α be an automorphism of A ?>. What does weak equivalence imply about |ω1(α(Ai))−ω2(α(Ai)) | ?>? A result from (Fabri et al. [1967], p. 383) states that for all A∈A, 0≤ |ω(A)−ω(α(A)) |≤2‖A‖ ?> for any pure state ω. Suppose that ω1 and ω2 are both pure and weakly equivalent ( |ω1(Ai)−ω2(Ai) |<εi ?>). Does this imply that ω1 and ω2 are still weakly equivalent under an automorphism α, that is, |ω1(α(Ai))−ω2(α(Ai)) |<εi ?>? No. The inequality becomes |ω1(α(Ai))−ω2(α(Ai)) |<4‖Ai‖+εi ?> (see Appendix for the proof). The only way for the upper bound to be preserved is for ‖Ai‖=0 ?> for all i. As long as ‖Ai‖>0 ?> for some i, weak equivalence will not necessarily hold. As Earman ([2004], p. 185) notes (using a theorem from (Fabri et al. [1967], p. 383), if ω is a pure state and α is an automorphism that is not unitarily implementable with respect to ω, then there exists an X∈A ?> with ‖X‖=1 ?> such that |ω(X)−ω(α(X)) |=2 ?>. Now suppose that α is not unitarily implementable with respect to ω1. If we pick that X as one of the C*-observables in our collection for weak equivalence, then |ω1(α(X))−ω2(α(X))|≤4+ε ?>. The weak equivalence between the representations is blown out of the water by a spontaneously broken symmetry. Even if the algebraic imperialist bashes the detectors with a baseball bat to force the representations to be physically equivalent, detectors that are only 25% efficient could still distinguish the representations. However, two representations, πφ ?> and πψ ?>, are weakly equivalent just in case they have the same kernel. If πφ ?> and πψ ?> have the same kernel, then πα*φ ?> and πα*ψ ?> have the same kernel. Thus, πα*φ ?> and πα*ψ ?> are weakly equivalent. How do we resolve these two results? At the level of representations, two weakly equivalent representations will continue to be weakly equivalent under automorphisms. However, the price is that the state needed to keep the expectation values within εi may change! If the automorphism represents the time evolution of two weakly equivalent representations, a new state may be required at each moment of time to satisfy the inequality! These results also show that weak equivalence is not a strong enough mathematical condition to be interpreted as physical equivalence. Intuitively, if we start with two states that are physically equivalent to each other and rotate the laboratory by a tiny angle, then we should still have two physically equivalent states. However, the weak equivalence between two states is not necessarily preserved even if the rotation is unitarily implementable in both representations. Suppose there are two weakly equivalent representations πφ ?> and πψ ?>. Once we fix the state we want to use in the folium of πφ ?>, the finite set of observables, Ai, and the εi, there will be some state in the folium of πψ ?> that will be weakly equivalent to it. However, once the symmetry transformation is applied, a second state in the folium of πψ ?> may well be needed to maintain weak equivalence. Thus, while only one state is needed from the folium of πφ ?>, two states from the folium of πψ ?> would be needed to maintain weak equivalence. Using two states from the folium of one representation to maintain weak equivalence, does not capture the idea of physical equivalence. 10 Conclusions The algebraic imperialist uses weak equivalence to argue that UIRs are mathematical surplus structure. If it is assumed that Fell’s theorem shows that any two representations of a C*-algebra are weakly equivalent to each other, and hence physically equivalent according to the algebraic imperialist, it does not follow that their extensions to W*-representations are weakly equivalent. However, weak equivalence can be put to use as a criterion for when two W*-representations are classically equivalent. The theorems in this article show that the difference between UIRs that are weakly inequivalent is that they differ in their expectation values for a classical observable such as temperature or chemical potential. These observables are not part of the original C*-algebra A ?>, but a larger W*-algebra: the bidual A** ?>. Thus, UIRs are not just another case of theoretical underdetermination. Nor is the interpretation of weak equivalence as physical equivalence supported. In the case of a unitarily implementable symmetry, different states would be needed before and after the symmetry is applied in order to save the phenomena. The choice of a particular UIR is enforced by the dynamics or macroscopic constraints such as temperature. The lesson of UIRs is that the C*-algebra A ?> was too small to be able to capture all of the possible physical content in either AQFT or AQSM. It also shows the path to a new kind of algebraic imperialism that locates physical content in the bidual A** ?>. Appendix Automorphisms and Weak Equivalence If ω1 and ω2 are both pure and weakly equivalent ( |ω1(Ai)−ω2(Ai) |<εi ?>), then |ω1(α(Ai))−ω2(α(Ai)) |<4‖Ai‖+εi ?> To prove this we need two results: (1) (Fabri et al. [1967], p. 383) for all A∈A,0≤ |ω(A)−ω(α(A)) |≤2‖A‖ ?> and (2) the triangle inequality |φ(A)+ψ(A) |≤ |φ(A) |+ |ψ(A) | ?>. |ω1(α(Ai))−ω2(α(Ai)) |= |ω1(α(Ai))+ω1(Ai)−ω1(Ai)+ω2(Ai)−ω2(Ai)−ω2(α(Ai)) |= |ω1(α(Ai))−ω1(Ai)+ω2(Ai)−ω2(α(Ai))+ω1(Ai)−ω2(Ai) |≤ |ω1(α(Ai))−ω1(Ai) |+ |ω2(Ai)−ω2(α(Ai)) |+ |ω1(Ai)−ω2(Ai) |(A.1)≤ |ω1(Ai)−ω1(α(Ai)) |+ |ω2(Ai)−ω2(α(Ai)) |+ |ω1(Ai)−ω2(Ai) |≤2‖Ai‖+2‖Ai‖+ |ω1(Ai)−ω2(Ai) |(A.2)≤4‖Ai‖+ |ω1(Ai)−ω2(Ai) |<4‖Ai‖+εi.(A.3) The triangle inequality is used in (A.1), the Fabri et al. ([1967]) result is used in (A.2), and the assumption of weak equivalence in (A.3).□ Acknowledgements I would like to thank Aristidis Arageorgis, David Baker, Benjamin Feintzeig, Hans Halvorson, and Fred Kronz for stimulating conversations and comments on earlier drafts of this paper. I would also like to thank the University of Western Ontario, the European Philosophy of Science Association 2013 conference, and the University of Oxford for opportunities to present this material. Footnotes 1Huggett and Weingard ([1996]) and Arageorgis ([1995]) also argue for the significance of UIRs in QFT. 2 After discussing Fell’s theorem, Wallace offers two arguments against the significance of UIRs for the foundations of QFT. Those arguments focus on UIRs that arise from short distance and global features. My discussion of UIRs will focus on showing the limitations of Fell’s theorem. 3Sklar ([2000], p. 20) argues that UIRs are observationally equivalent. Though Sklar does not make specific reference to Fell’s theorem, it seems to be what he has in mind. 4 Though Kronz and Lupher ([2005]) do provide an explicit discussion of how Fell’s theorem fails to apply to certain representations. 5 I will follow the standard terminology of referring to elements of a C*-algebra or W*-algebra as ‘observables’, though many elements of the algebra may not be something we can measure in an experiment. The observables that will distinguish different UIRs belong to an abstract W*-algebra that is the bidual of a C*-algebra (see Section 6 for details). 6 The observable can be considered ‘classical’ because it commutes with all of the other observables in the algebra; see Section 5 for more details. 7 For more information on the Stone–von Neumann theorem, see (Summers [2001]). 8 To see roughly why unitary equivalence of two representations of the canonical commutation relations, or of a C*-algebra or W*-algebra, are empirically equivalent, suppose the representations πφ ?> and πψ ?> are unitarily equivalent. (Precise definitions of what representations are and the unitary equivalence of two representations will be given shortly.) Let ρπφ ?> be a density operator acting on the Hilbert space Hπφ ?> and πφ(A) ?> be a representation of an observable, A, acting on Hπφ ?>. The expectation value of the state ρπφ ?> for the observable πφ(A) ?> is given by Tr (ρπψπψ(A)) ?>. Since πφ ?> and πψ ?> are unitarily equivalent, that means there is a density operator, ρπψ ?>, and observable, πψ(A) ?>, such that a unitary operator, U, transforms ρπφ ?> into ρπψ ?> via ρπψ=UρπφU−1 ?> and πφ(A) ?> into πψ(A) ?> via πψ(A)=Uπφ(A)U−1. ?> The empirical equivalence of the expectation values follows from these relations and the cyclic property of the Tr operation (that is, Tr(ABC)=Tr(CAB)), and UU−1=U−1U=I ?>, where I is the identity operator (IA = AI = A): Tr (ρπψπψ(A))= ?> Tr (UρπφU−1Uπψ(A)U−1)= ?>Tr (UρπφIπψ(A)U−1)= ?> Tr (Uρπφπψ(A)U−1)= ?>Tr (U−1Uρπφπψ(A)) \\= ?>Tr (Iρπφπψ(A)= ?>Tr (ρπφπψ(A) ?>). Thus, for two unitarily equivalent representations and any expectation value in one representation, a state and observable exists in the other representation that gives the same expectation value. 9 Wallace does not argue that UIRs are mathematical surplus; he ([2011], p. 123) says that the algebraic approach is useful for tackling long-range divergences in QFT. 10 Good sources for more on the mathematical details of AQFT are (Emch [1972]; Halvorson [2007]). 11 A C*-algebra A ?> is a vector space over the field of complex numbers C ?> with an associative distributive product that has an involution * (which satisfies ( A*)*=A, (AB)*=B*A*, (λA+μB)*=λ¯A*+μ¯B* ?> for all A,B∈A ?> and λ,μ∈C ?> where λ¯ ?> and μ¯ ?> are the complex conjugates of λ and μ) and is complete in the norm ∥.∥ ?> obeying ∥AB∥≤∥A∥∥B∥ ?> and ∥A*A∥=∥A∥2 ?> for all A,B∈A ?>. 12 A linear functional ω is a mapping from A ?> to the complex numbers C ?> ( ω:A→C ?>) such that ω(αA+βB)=αω(A)+βω(B) ?>, where α,β∈C ?> and A,B∈A ?>. 13 A C*-algebra is called a W*-algebra if it is the dual space of a Banach space. A* ?> is a Banach space, so A** ?> is a W*-algebra. 14 To say that πω ?> preserves the algebraic relations between the elements of A ?> means that the following conditions are satisfied for any A,B∈A ?> and α,β∈C ?>: πω(αA+βB)=απω(A)+βπω(B),  πω(AB)=πω(A)πω(B) ?>, and πω(A*)=πω(A)*. ?> 15 Let R⊆B(H) ?> be a von Neumann algebra. R ?>’s commutant is defined as R′={B∈B(H)|AB=BA ?>, for all A∈A} ?>, and R ?>’s bicommutant is R″=(R′)′ ?>. 16 To say that a subset R ?> of B(H) ?> is ‘weakly closed’, that is, that it is ‘closed in the weak operator topology’ means that any sequence {Tn} ?> of elements of R ?> converges to another element T∈R ?> in the sense that ⟨Φ|Tn|Ψ⟩→⟨Φ|T|Ψ⟩ ?> for all Φ,Ψ∈H ?>. 17 A linear functional ρ∼ ?> on a W*-algebra A ?> is said to be ‘normal’ if and only if ρ∼(supαTα)=supαρ∼(Tα) ?> for every uniformly bounded increasing directed set {Tα} ?> of positive elements of A** ?>. 18 For more details about representations of A** ?>, see (Bing-Ren [1992], pp. 221-2). 19 Roughly, a W*-representation is a *-homomorphism from an abstract W*-algebra to B(Hπω) ?>, which is continuous in the topologies at the abstract and concrete levels. For the details, see (Bing-Ren [1992], p. 221). 20 Fell ([1960], Theorem 1.2) used a family of representations in his equivalence theorem, but that additional complexity is not needed. Two of the four equivalent conditions from Fell’s theorem are not listed in my presentation of Fell’s theorem and will not be needed in what follows. 21 It is assumed throughout this article that all representations are non-degenerate, which can intuitively be thought of as assuming that the representations are not the zero representation. 22 A representation is ‘faithful’ if and only if its kernel is trivial, that is, the kernel only contains the zero operator from A ?>: kerπ={0} ?>, where 0∈A ?>. 23 Alternatively, two representations, πϕ ?> and πψ ?>, of a C*-algebra A ?> are weakly equivalent if and only if every weak*-neighbourhood of any state belonging to folium Fϕ ?> has a non-empty intersection with the weak*-neighbourhood of some state in folium Fψ ?> and vice versa. 24 In Fell’s theorem, there is no subscript on the ε. In what follows, the value of ε in the discussion of the weak*-topology on A* ?> can be considered the maximum experimental error, that is, the largest of the εi. 25 Ruetsche ([2011], pp. 133-9) distinguishes two types of algebraic imperialist: a ‘bold algebraic imperialist’ and an ‘apologetic algebraic imperialist’. The algebraic imperialist that uses weak equivalence as an argument for algebraic imperialism is an ‘apologetic algebraic imperialist’, and that is the type of algebraic imperialist that I am arguing against here. 26 Summers ([2001], p. 145) also criticizes physical equivalence. 27 There are also arguments that different UIRs have different empirical content. Emch and Liu ([2002]) discuss ‘marker’ observables and Ruetsche ([2011]) discusses ‘phase’ observables for broken symmetries. These observables belong to the centre of a von Neumann algebra. In Section 6, I will discuss observables in the centre of A** ?> and how they can be used to distinguish different UIRs. 28 Two representations, πϕ ?> and πψ ?>, of a C*-algebra A ?> are ‘unitarily equivalent’ to each other if there exists an isomorphism U:Hπϕ→Hπψ ?> such that πψ(A)=Uπϕ(A)U−1 ?>. πϕ ?> and πψ ?> are ‘quasi-equivalent’ to each other if there is an isomorphism α:πϕ(A)″→πψ(A)″ ?> between the von Neumann algebras of πϕ ?> and πψ ?> such that α[πϕ(A)]=πψ(A) ?> for all A∈A ?>. 29 The equivalence between the quasi-equivalence of two C*-representations and the weak equivalence of their associated W*-representations was proved by Emch ([1972], pp. 122–4). 30 The centre ℨπω(A)″ ?> of a von Neumann algebra πω(A)″ ?> is the intersection of the von Neumann algebra and its commutant πω(A)′ ?>, that is, ℨπω(A)″=πω(A)″∩πω(A)′ ?>. A factor is a von Neumann algebra with a trivial centre, that is, ℨπω(A)″=πω(A)″∩πω(A)′={λIπω} ?>. In other words, a factor’s centre consists of scalar multiples of the Hilbert space Hπω ?> identity operator Iπω ?>. 31 For more technical details, see chapter six of Kadison and Ringrose ([1997]). 32 A number of results that indicate that local algebras of relativistic QFT are type III von Neumann algebras are discussed in (Halvorson and Mueger [2007], pp. 748-52). 33 For more information about KMS states and AQSM, see (Kronz and Lupher [2005]; Ruetsche [2003]). An abstract state ωβ ?> is a KMS state with respect to an automorphism group αt, inverse temperature β if and only if the following holds for all operators A, B in a dense subalgebra A ?>: ωβ[Aαiβ(B)]=ωβ(BA) ?>. This equation is the KMS condition and it uniquely determines with respect to αt with value β the Gibbs grand canonical equilibrium state. 34 A representation πω(A) ?> of a C*-algebra A ?> is called ‘irreducible’ if the only closed subspaces of Hπω ?> that are invariant under the action of the elements of πω(A) ?> are Hπω ?> and 0. Irreducibility will be discussed more below. 35 More precisely, quasi-equivalence implies unitary equivalence for KMS representations that have a non-zero finite temperature in the thermodynamic limit and that are not where a phase transition occurs. In order to be a bit more succinct, in this section I will just use the phrase ‘KMS representations’ instead of putting in those conditions every time. 36 The case of irreducible representations was proved by Kadison and Ringrose ([1997], p. 740). Dixmier ([1977], p. 128) proved that quasi-equivalence implies unitary equivalence for type III factors with separable Hilbert spaces. KMS representations that have a non-zero finite temperature in the thermodynamic limit and that are not where a phase transition occurs are type III factors, so Dixmier’s proof for type III factors can be used for these KMS representations. This does not cover special cases such as a KMS state with an infinite temperature, which is a type II1 ?> factor, and a KMS state at temperature zero, which is a type I factor. KMS representations that occur at a phase transition are a direct sum or direct integral of type III factors. For more on KMS representations and type III factors, see (Ruetsche [2011], p. 167). 37 According to Haag ([1996], p. 54), one of the advantages of irreducible representations of the equal-time canonical commutation relations for quantum fields is that all observables can be expressed in terms of the quantum field and its conjugate momentum. 38 The condition of disjointness is often used in discussing unitary inequivalence. It is usually contrasted with quasi-equivalence, but quasi-equivalence and disjointness are not mutually exclusive concepts unless both representations are factors. Two representations, πϕ ?> and πψ ?>, of a C*-algebra A ?> are disjoint when either (i) no sub-representation of πϕ ?> is unitarily equivalent to a sub-representation of πψ ?>, or (ii) the intersection of their folia is empty: Fϕ∩Fψ=∅ ?>. (ii) implies that no density operator in Hπϕ ?> can be expressed as a density operator in Hπψ ?>, and vice versa. 39 Kadison and Ringrose use the term ‘equivalent’ instead of ‘unitarily equivalent’, but ‘equivalent’ means ‘unitarily equivalent’. 40 Proof: If a KMS state has a finite non-zero temperature, then its representation is a type III factor. Under those conditions, Takesaki ([1970]) proved that πϕ ?> and πφ ?> are disjoint. Since πϕ ?> and πφ ?> are disjoint factor representations, they are not quasi-equivalent. From Table 1, since πϕ ?> and πφ ?> are not quasi-equivalent, πϕ∼ ?> and πψ∼ ?> are not weakly (physically) equivalent.□ 41 Proof: Under those conditions, Müller-Herold ([1980]) proved that ϕ ?> and ψ are disjoint. The rest of the proof is exactly the same as Theorem 1.□ 42 ℨ(A**) ?> is the set of elements of A** ?> that commute with every element of A** ?>, that is, ℨ(A**)={A∈A**|AB=BAfor allB∈A**} ?>. Proof of Theorem 3: Emch ([1972], p. 139) cited a result by Combes ([1967]) that two states ϕ ?> and ψ that generate factor representations πϕ ?> and πψ ?>, respectively, are quasi-equivalence if and only if their extensions ϕ∼ ?> and ψ∼ ?> coincide on ℨ ?>. Since πϕ ?> and πψ ?> are quasi-equivalent if and only if πϕ∼ ?> and πψ∼ ?> are weakly equivalent, this shows that πϕ∼ ?> and πψ∼ ?> are weakly equivalent if and only if ϕ∼(Z)=ψ∼(Z) ?> for every observable Z in the centre of the bidual, Z∈ℨ(A**) ?>.□ 43 Belonging to the centre of an abstract algebra is a necessary condition for an observable to be classical, but it is probably not a sufficient condition. Hence, my use of quotes around classical. For a more detailed discussion of classicality and quantum theory, see (Landsman [2007]). 44 This is not to say that the Z that does distinguish those UIRs is something that can be measured. ℨ(A**) ?> is a huge collection of mathematical operators and there is no reason to think that every one of those mathematical operators corresponds to a physical property that can be measured. Nonetheless, for the algebraic imperialist argument to succeed, it would have to offer an additional argument that every Z∈ℨ(A**) ?> has no physical significance. Given the examples given in the previous section, the algebraic imperialist would have to provide an argument for why temperature and chemical potential are not physical observables. 45 Hans Halvorson suggested the name when we were discussing these issues. 46 Ruetsche ([2011], p. 134) calls an observable ‘parochial’ when the observable is not an element of the abstract C*-algebra but is affiliated with a von Neumann algebra. 47 A more detailed presentation of bidualism will be given in a future paper. 48 For details, see (Arageorgis et al. [2002a], Section 2). 49 For simplicity, only the right Rindler wedge will be considered; however, the results apply to the left Rindler wedge and the algebra of observables defined on both the left and right wedges; see (Clifton and Halvorson [2001]) for details. 50 This C*-algebra is generated from the Weyl form of the canonical commutation relations. For details, see (Clifton and Halvorson [2001]). 51 Clifton and Halvorson ([2001]) actually prove that the Minkowski and Rindler representations are disjoint. Since both representations are factors, it follows that they are unitarily inequivalent. 52 Proof: Clifton and Halvorson ([2001], p. 463) proved that πωR⊲ ?> and πωM⊲ ?> are disjoint. Since both πωR⊲ ?> and πωM⊲ ?> are factors and disjoint, they are not quasi-equivalent, hence πω∼R⊲ ?> and πω∼M⊲ ?> are weakly inequivalent.□ 53 The automorphism group for the KMS states must be specified. See (Arageorgis et al. [2002a], p. 189) for more details. References Arageorgis A. [ 1995 ]: Fields, Particles, and Curvature: Foundations and Philosophical Aspects of Quantum Field Theory in Curved Spacetime , Ph.D. thesis, University of Pittsburgh . Arageorgis A. , Earman J. , Ruetsche L. [ 2002a ]: ‘Fulling Non-uniqueness and the Unruh Effect: A Primer on Some Aspects of Quantum Field Theory’ , Philosophy of Science , 70 , pp. 164 – 202 . Google Scholar CrossRef Search ADS Arageorgis A. , Earman J. , Ruetsche L. [ 2002b ]: ‘Weyling the Time Away: The Non-unitary Implementability of Quantum Fied Dynamics on Curved Spacetime’ , Studies in the History and Philosophy of Modern Physics , 33 , pp. 151 – 184 . Google Scholar CrossRef Search ADS Bing-Ren L. [ 1992 ]: Introduction to Operator Algebras , River Edge, NJ : World Scientific . Google Scholar CrossRef Search ADS Bratteli O. , Robinson D. [ 1997 ]: Operator Algebras and Quantum Statistical Mechanics 2 , Berlin : Springer . Google Scholar CrossRef Search ADS Clifton R. , Halvorson H. [ 2001 ]: ‘Are Rindler Quanta Real? Inequivalent Particle Concepts in Quantum Field Theory’ , British Journal for the Philosophy of Science , 52 , pp. 417 – 70 . Google Scholar CrossRef Search ADS Combes F. [ 1967 ]: ‘Sur les états factoriels d’une C*-algèbre’ , Comptes Rendus de l’Académie des Sciences , 265 , pp. 736 – 9 . Dixmier J. [ 1977 ]: C*-Algebras , New York : North-Holland . Earman J. [ 2004 ]: ‘Curie’s Principle and Spontaneous Symmetry Breaking’ , International Studies in the Philosophy of Science , 18 , pp. 173 – 98 . Google Scholar CrossRef Search ADS Earman J. , Fraser D. [ 2006 ]: ‘Haag’s Theorem and Its Implications for the Foundations of Quantum Field Theory’ , Erkenntnis , 64 , pp. 305 – 44 . Google Scholar CrossRef Search ADS Emch G. [ 1972 ]: Algebraic Methods in Statistical Mechanics and Quantum Field Theory , New York : John Wiley . Emch G. , Liu C. [ 2002 ]: The Logic of Thermostatistical Physics , Berlin : Springer . Google Scholar CrossRef Search ADS Fabri E. , Picasso L. , Strocchi F. [ 1967 ]: ‘Broken Symmetries in Quantum Field Theory’ , Nuovo Cimento A , 48 , pp. 376 – 85 . Google Scholar CrossRef Search ADS Fell J. M. G. [ 1960 ]: ‘The Dual Spaces of C*-Algebras’ , Transactions of the American Mathematical Society , 94 , pp. 365 – 403 . Haag R. [ 1996 ]: Local Quantum Physics , Berlin : Springer . Google Scholar CrossRef Search ADS Haag R. , Kastler D. [ 1964 ]: ‘An Algebraic Approach to Quantum Field Theory’ , Journal of Mathematical Physics , 5 , pp. 848 – 61 . Google Scholar CrossRef Search ADS Halvorson H. , Mueger M. [ 2007 ]: ‘Algebraic Quantum Field Theory’ , in Butterfield J. , Earman J. (eds), Philosophy of Physics , Amsterdam : Elsevier , pp. 731 – 922 . Google Scholar CrossRef Search ADS Huggett N. , Weingard R. [ 1996 ]: ‘Critical Review: Paul Teller’s Interpretive Introduction to Quantum Field Theory’ , Philosophy of Science , 63 , pp. 302 – 14 . Google Scholar CrossRef Search ADS Kadison R. , Ringrose J. [ 1997 ]: Fundamentals of the Theory of Operator Algebras , New York : American Mathematical Society . Kastler D. [ 1964 ]: ‘A C*-Algebra Approach to Field Theory’ , in Martin T. , Segal I. (eds), Proceedings of the Conference on Theory and Applications of Functional Analysis , Cambridge, MA : MIT Press , pp. 179 – 91 . Kronz F. , Lupher T. [ 2005 ]: ‘Unitarily Inequivalent Representations in Algebraic Quantum Theory’ , International Journal of Theoretical Physics , 44 , pp. 1239 – 58 . Google Scholar CrossRef Search ADS Landsman N. [ 2007 ]: ‘Between Classical and Quantum’ , in Butterfield J. , Earman J. (eds), Philosophy of Physics , Amsterdam : Elsevier , pp. 417 – 533 . Google Scholar CrossRef Search ADS Müller-Herold U. [ 1980 ]: ‘Disjointness of β-KMS States with Different Chemical Potential’ , Letters in Mathematical Physics , 4 , pp. 45 – 8 . Google Scholar CrossRef Search ADS Robinson D. [ 1966 ]: ‘Algebraic Aspects of Relativistic Quantum Field Theory’ , in Chrétien M. , Deser S. (eds), Axiomatic Field Theory , New York : Gordon and Breach , pp. 389 – 516 . Ruetsche L. [ 2003 ]: ‘A Matter of Degree: Putting Unitary Inequivalence to Work’ , Philosophy of Science , 70 , pp. 1329 – 42 . Google Scholar CrossRef Search ADS Ruetsche L. [ 2011 ]: Interpreting Quantum Theories , Oxford : Oxford University Press . Google Scholar CrossRef Search ADS Segal I. [ 1967 ]: ‘Representations of the Canonical Commutation Relations’ , in Lurçat F. (ed.), Cargèse Lectures in Theoretical Physics: Application of Mathematics to Problems in Theoretical Physics , New York : Gordon and Breach , pp. 107 – 70 . Sklar L. [ 2000 ]: Theory and Truth: Philosophical Critique within Foundational Science , Oxford : Oxford University Press . Summers S. [ 2001 ]: ‘On the Stone-von Neumann Uniqueness Theorem and Its Ramifications’ , in Rédei M. , Stöltzner M. (eds), John von Neumann and the Foundations of Quantum Physics , Dordrecht : Springer , pp. 135 – 52 . Google Scholar CrossRef Search ADS Takesaki M. [ 1970 ]: ‘Disjointness of the KMS-states of Different Temperatures’ , Communications of Mathematical Physics , 17 , pp. 33 – 41 . Google Scholar CrossRef Search ADS Thirring W. [ 1983 ]: Quantum Mechanics of Large Systems , New York : Springer . Wallace D. [ 2006 ]: ‘In Defense of Naiveté: The Conceptual Status of Lagrangian Quantum Field Theory’ , Synthese , 151 , pp. 33 – 80 . Google Scholar CrossRef Search ADS Wallace D. [ 2011 ]: ‘Taking Particle Physics Seriously: A Critique of the Algebraic Approach to Quantum Field Theory’ , Studies in History and Philosophy of Modern Physics , 42 , pp. 116 – 25 . Google Scholar CrossRef Search ADS © The Author 2016. Published by Oxford University Press on behalf of British Society for the Philosophy of Science. All rights reserved. For Permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The British Journal for the Philosophy of Science Oxford University Press

The Limits of Physical Equivalence in Algebraic Quantum Field Theory

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Abstract

Abstract Some physicists and philosophers argue that unitarily inequivalent representations (UIRs) in quantum field theory (QFT) are mathematical surplus structure. Support for that view, sometimes called ‘algebraic imperialism’, relies on Fell’s theorem and its deployment in the algebraic approach to QFT. The algebraic imperialist uses Fell’s theorem to argue that UIRs are ‘physically equivalent’ to each other. The mathematical, conceptual, and dynamical aspects of Fell’s theorem will be examined. Its use as a criterion for physical equivalence is examined in detail and it is proven that Fell’s theorem does not apply to the vast number of representations used in the algebraic approach. UIRs are not another case of theoretical underdetermination, because they make different predictions about ‘classical’ operators. These results are applied to the Unruh effect where there is a continuum of UIRs to which Fell’s theorem does not apply. 1 Introduction 2 Weak Equivalence and Physical Equivalence 3 Mathematical Overview of Algebraic Quantum Field Theory 4 Fell’s Theorem and Philosophical Responses to Weak Equivalence 5 Weak Equivalence in C*-Algebras and W*-Algebras 6 Classical Equivalence and Weak Equivalence 7 Interlude: Is Weak Equivalence Really Physical Equivalence? 8 The Unruh Effect 9 Time Evolution and Symmetries 10 Conclusions Appendix  1 Introduction The interpretive significance of unitarily inequivalent representations (UIRs) in quantum field theory (QFT) has generated intense philosophical discussion. What is the interpretive significance of UIRs in QFT? Does it make any difference which representation we use? Do UIRs make different predictions? Is the existence of UIRs just another case of theoretical underdetermination? Philosophers have radically different views on the significance of UIRs. At one end of the spectrum, UIRs are involved in questions about incommensurable particle ontologies (Arageorgis et al. [2002b], pp. 133–6), Haag’s theorem (Earman and Fraser [2006]), the Unruh effect (Arageorgis et al. [2002a]; Clifton and Halvorson [2001]), and algebraic quantum statistical mechanics (AQSM) (Ruetsche [2003]; Kronz and Lupher [2005]).1 Those authors treat UIRs as physically significant for foundational investigations of QFT. At the other end of the spectrum, Wallace ([2006]) has argued that the appearance of UIRs is not a problem for the foundations of QFT.2 Using Fell’s theorem, Wallace argues that different UIRs are empirically equivalent to each other.3 That argument was originally put forward by two physicists (Haag and Kastler [1964]) and has been influential in the development of algebraic quantum field theory (AQFT), particularly in the 1960s (for example, Robinson [1966], p. 488; Kastler [1964]). Fell’s theorem has been used to argue that different UIRs are, in some sense, ‘physically equivalent’ to each other, and that the choice of a particular representation is underdetermined by any finite amount of evidence we could ever obtain. Fell’s theorem has been used to argue for ‘algebraic imperialism’ (Ruetsche [2003]; Arageorgis [1995]). That position claims that the physical content of AQFT is contained in an abstract C*-algebra—not in UIRs. While some philosophers (Ruetsche [2003]; Arageorgis [1995]) have criticized the use of Fell’s theorem to classify UIRs as physically insignificant, no paper has provided an in-depth examination of the mathematical details surrounding Fell’s theorem in AQFT.4 The applicability of Fell’s theorem depends on the kind of algebraic structure under consideration. In particular, for UIRs of a C*-algebra, Fell’s theorem often cannot be extended to their W*-algebra extensions. If Fell’s theorem is the mathematical embodiment of physical equivalence, then the W*-algebra extensions of those UIRs are not physically equivalent. Another theorem will show that different UIRs cannot be considered empirically equivalent, or even approximately empirically equivalent, for all observables.5 For most UIRs, there exists an observable that will distinguish the UIRs. Thus, UIRs cannot be considered just another case of theoretical underdetermination. They will make different predictions, but the surprising twist is that the observable they will make different predictions about is a classical observable.6 While the results in this article show the limitations of a certain type of algebraic imperialism, they also point toward a new distinction in algebraic imperialism. In short, one can be an algebraic imperialist with respect to an abstract C*-algebra or an abstract W*-algebra. To put my cards on the table, I am an algebraic imperialist of the W*-algebra. For reasons that will become apparent later, this position is called ‘bidualism’. This interpretation will be particularly appealing to people who are sympathetic to algebraic imperialism but think that UIRs are physically significant and cannot be dismissed by appeal to Fell’s theorem. The plan for the article is as follows: Section 2 discusses Haag and Kastler’s interpretation of the mathematical condition of weak equivalence as a kind of physical equivalence. To understand the limitations of weak equivalence, it is crucial to understand how the different algebraic structures are related to each other. A brief overview of the mathematical details of AQFT is provided in Section 3. Fell’s theorem—or, more accurately, the mathematical conditions for the weak equivalence of two representations—the philosophical critique of physical equivalence, and the position of algebraic imperialism are discussed in Section 4. The connections among unitary equivalence, quasi-equivalence, and weak equivalence for representations of C*-algebras and W*-algebras are examined in Section 5. I will also show that most of the physically interesting UIRs are not weakly equivalent in that section. In Section 6, I argue that UIRs do make different empirical predictions for a ‘classical’ observable. The question of whether the mathematical condition of weak equivalence captures the idea of physical equivalence will be examined in Section 7. A brief application of these results is given for the Unruh effect in Section 8, where it is shown that there is a continuum of physically inequivalent representations. In Section 9, I will examine Fell’s theorem under symmetry transformations and time evolution. I will show how Fell's theorem fails in those cases and in the case of spontaneous symmetry breaking. Conclusions are provided in Section 10. 2 Weak Equivalence and Physical Equivalence The condition of unitary equivalence has been important in the history of quantum theory. For example, it follows from the Stone–von Neumann theorem that Schrödinger’s wave mechanics and Heisenberg’s matrix mechanics are unitarily equivalent representations of the canonical commutation relations.7 Unitarily equivalent representations are empirically equivalent to each other, in that they give the same predictions.8 Either formalism can be used since both will give the same result. Using wave mechanics or matrix mechanics is similar to using a particular coordinate system to solve a problem. The loss of unitary equivalence raises the worry of whether different UIRs make different empirical predictions, in which case it matters which representation is chosen. The problem of whether UIRs are physically significant or can be rejected as mathematical surplus structure is particularly acute in QFT because, in many cases, there is a continuum of different UIRs available. The vast number of UIRs might give aid to the intuition that there are far too many possibilities for different UIRs to count as physically significant. The argument that all UIRs are, in a sense, physically equivalent is based on an interpretation of a mathematical result in AQFT known as Fell’s theorem. That theorem introduces the mathematical idea of ‘weak equivalence’ and it is important to make a distinction between the notions of weak equivalence and physical equivalence. Weak equivalence is a mathematical concept introduced by Fell ([1960], p. 375), while physical equivalence, which was introduced by Haag and Kastler ([1964]), is a semantic extrapolation of weak equivalence. To avoid confusion, I will use ‘weak equivalence’ even though Haag and Kastler talk about physical equivalence, because I will argue in Section 7 that weak equivalence does not mathematically encode the concept of physical equivalence. I will explain weak equivalence in Section 4, after introducing some of the mathematical structures in AQFT. Weak equivalence was used by Haag and Kastler ([1964]), among others, in AQFT to elevate the abstract C*-algebra to a position of physical importance while denigrating the significance of representations, in particular that of UIRs, using the following operationalist justification. Since only a finite number of experiments can be carried out and each experiment has finite accuracy, given any state on a Hilbert space associated with one representation, there is a state on a Hilbert space associated with a different representation that is unitarily inequivalent to the first representation such that both states have roughly the same expectation values (within some small ε>0 ?>) for the same observables. Thus, both representations should be considered physically equivalent. UIRs would be mathematical surplus structure. This is how Wallace ([2006], p. 54) explains Fell’s theorem9: Fell’s theorem says that a finite number of measurements, each conducted with finite accuracy, cannot distinguish between representations. Hence we could reproduce any experimental results using whichever representation was most convenient. (Wallace [2006], p. 54) The problem of choosing a particular representation from a continuum of possible choices would evaporate since any choice would be physically equivalent to any other choice. However, as we will see in Section 5, the mathematical condition of weak equivalence, upon which Haag and Kastler’s argument rests, fails to be satisfied by most UIRs and, thus, they will fail to be physically equivalent. The failure of weak equivalence depends on the kind of algebra under consideration. The different algebraic options and their interconnections are explained in the next section. 3 Mathematical Overview of Algebraic Quantum Field Theory What follows is a brief partial overview of the conceptual landscape of AQFT.10 The diagram in Figure 1 is one of the keys for understanding the mathematical limitations of Haag and Kastler’s notion of physical equivalence. Figure 1. View largeDownload slide Figure 1. View largeDownload slide The first thing to notice is that there is an abstract and a concrete level in the algebraic approach on the left side of Figure 1. On a first pass, one can think of the concrete level as involving operators defined on Hilbert spaces and the abstract level as where operators are not defined on a Hilbert space. Beginning in the upper left-hand corner of Figure 1, there is an abstract C*-algebra A ?>.11 An algebra is essentially a set of elements that is closed under linear combinations and products. It is a mathematical structure that has more structure than the more familiar notion of a vector space. Moving to the right, the dual of A ?>, denoted by A* ?>, is the set of bounded linear functionals on A ?>.12 A state is a linear functional that is positive ( ω(A*A)>0 ?> for all A∈A ?>) and normed ( ω(I)=1 ?>, where I is the multiplicative identity in A ?>). The set of states in A* ?> is denoted as A1*+ ?> and A1*+⊂A* ?>. Taking the dual of A* ?> generates a W*-algebra, A** ?>, called the ‘bidual’.13 It is rather surprising that A** ?> has algebraic structure. A** ?> will play a key role in what follows. Moving to the concrete row at the bottom left of Figure 1, Hilbert spaces make their appearance. To get there, we will start at the abstract level in the upper left-hand corner of Figure 1 at A ?> and move downward. A representation, πω ?>, can be built by picking an abstract state, ω, where ω∈A1*+ ?>. πω(A) ?> is a ‘concrete’ C*-algebra and πω ?> is a C*-representation. A representation of a C*-algebra A ?> is a map πω ?> from A ?> into the set of bounded operators B(Hπω) ?> of an associated Hilbert space, Hπω ?>, that preserves the algebraic relations between the elements of A ?>.14 Given an abstract state ω, a Hilbert space Hπω ?> can be constructed from it. This should not be too surprising since A ?> is already a vector space. The standard procedure for generating Hilbert spaces from a C*-algebra A ?> is called the ‘GNS construction’ (named after Gelfand, Naimark, and Segal who showed how to do it). There is also a ‘reverse GNS process’, where states in a Hilbert space have abstract counterparts in A1*+ ?>. For each Hilbert space, Hπω ?>, associated with a representation, πω ?>, there will be a collection of vector states and density matrices defined on Hπω ?>. For each vector state, |Ψπω⟩ ?>, and density matrix, ρπω ?>, there is a corresponding abstract state in A1*+ ?> ( Ψω ?> and ρω ?>, respectively) defined by the following relations: Ψω(A)=⟨Ψπω|πω(A)|Ψπω⟩, ρω(A)=Tr(ρπωπω(A)). In other words, for each concrete state (vector state or density operator) in a Hilbert space, there corresponds an abstract counterpart in A1*+ ?>. The set of all abstract states that are density operators in Hπω ?> is a norm closed convex subset of A1*+ ?> called the folium Fω ?>. Staying at the concrete C*-level and moving to the right, there are two equivalent ways to construct a von Neumann algebra (also called a ‘concrete W*-algebra’) from a concrete C*-algebra, πω(A) ?>: (i) take the bicommutant of πω(A) ?>, that is πω(A)″ ?>,15 or (ii) close πω(A) ?> in the weak operator topology,16 that is, πω(A)− ?>. The most common way to construct a von Neumann algebra is to start with a C*-algebra A ?> and an abstract state ω∈A1*+ ?>, build a representation πω(A) ?> and a Hilbert space (Hπω,πω) ?> via the GNS construction, and then take the bicommutant, πω(A)″ ?>. However, there is an equivalent alternative way to build πω(A)″ ?> using the bidual A** ?>. Since C*-algebras and W*-algebras are both *-algebras, the GNS construction can be used to build Hilbert spaces for both algebras. For any state ω, it can be extended to be a normal state ω∼ ?> on A** ?>.17 If the GNS construction is done using ω∼ ?> and A** ?>, then a von Neumann algebra πω∼(A**) ?> is generated.18 A von Neumann algebra is a concrete W*-algebra; it is a W*-representation of A** ?>.19 All representations (Hπω,πω) ?> of A ?> can be uniquely extended (Hπω,πω∼) ?> to be a W*-representation of A** ?> and this extension is equal to the weak closure of πω(A) ?>, that is, πω(A)″=πω∼(A**)=πω(A)− ?>; see (Bing-Ren [1992], pp. 221-2, Theorem 4.2.7). 4 Fell’s Theorem and Philosophical Responses to Weak Equivalence There are a number of different formulations of Fell’s theorem. To disentangle them, I will begin with Fell’s ([1960]) original paper, which focused on the dual space A* ?> of C*-algebras A ?>. There are two crucial ideas: the topology of the state space and kernels. A topology may be defined on A* ?>, called the weak*-topology on A* ?>, which is sometimes referred to as the σ(A,A*) ?> topology, in which a basis of neighbourhoods N(ϕ,{Ai}i=1k,ε) ?> consists of all sets of the form {ω∈A*||ϕ(Ai)−ω(Ai)|<ε} ?>, where Ai∈A ?> and ε>0 ?>. The weak*-neighbourhoods of an abstract state, ϕ ?>, are indexed by a finite subset, Ai of A ?>, and a positive real number, ε. The second important concept is the kernel of a representation, π, which is defined as the set of elements a representation π maps to the zero operator in its Hilbert space: kerπ={A|π(A)=0,A∈A} ?>. The key theorem of Fell ([1960], p. 367) can now be expressed as follows20: Fell’s Theorem Let A ?> be a C*-algebra and πϕ ?> and πψ ?> be two representations of A ?>.21 The following conditions are equivalent: the kernel of πψ ?> is contained in πϕ ?>, that is, kerπψ⊆kerπϕ ?>, every positive functional η on A ?> associated with πϕ ?> is a weak*-limit of finite sums, γ, of positive functionals associated with πψ ?> for which ||η||≤||γ|| ?>. There are two important aspects of Fell’s theorem: when kerπψ⊆kerπϕ ?>, (i) a vector state associated with πϕ ?> is the weak*-limit of vector states associated with πψ ?> and (ii) the weak*-neighbourhood of an abstract state ρϕ ?> that is a density operator associated with πϕ ?>—that is, ρϕ ?> belongs to the folium Fϕ ?>—contains an abstract state ρψ ?> that is a density operator associated with πψ ?>, that is, ρψ ?> belongs to the folium Fψ ?>. If either (1) or (2) is satisfied, then πψ ?> is ‘weakly contained’ in πϕ ?>. Since the kernel of πϕ ?> is larger than πψ, πϕ ?> maps more elements of A ?> to the zero operator, that is, it is less faithful.22 Fell’s theorem is used to define weak equivalence: two representations πϕ ?> and πψ ?> of A ?> are weakly equivalent if and only if they are weakly contained in each other. The key feature doing work in weak equivalence is the notion of a weak*-neighbourhood at the abstract level of states—not at the concrete level of representations. This is where the limitation of restricting the number of observables, Ai, to a finite set comes from—not from any operationalist scruples about measuring an infinite number of observables. The definition of weak equivalence and Fell’s theorem permits the formulation of another version of Fell’s theorem, which is the basis of a number of presentations of weak equivalence (Haag and Kastler [1964]; Robinson [1966]). Fell’s Theorem 2 Two representations, πϕ ?> and πψ ?>, of a C*-algebra A ?> are weakly equivalent if and only if kerπϕ=kerπψ ?>.23 In Fell’s Theorem 2, the notion of weak equivalence depends on the kernels of the two representations being equal. This provides a simple test for whether two representations are weakly equivalent. If one considers all faithful representations—that is, representations that have trivial kernels—then those representations are all weakly equivalent to each other. This is how weak equivalence is often presented in the literature. Weak Equivalence: All faithful representations of a C*-algebra are weakly equivalent. If the C*-algebra is simple (that is, it has no closed two-sided ideals), then all non-zero representations (that is, π(A)≠0 ?> for some A∈A ?>) of it are faithful (Emch [1972], p. 80). Thus, each non-zero representation is weakly equivalent to every other representation of a simple C*-algebra. Two physically important simple algebras are the Weyl canonical commutation relations C*-algebra (Bratelli and Robinson [1997], pp. 19–22) and the Weyl canonical anti-commutation relations C*-algebra (Bratelli and Robinson [1997], pp. 15–6). Haag and Kastler ([1964]) interpreted the mathematical concept of weak equivalence as a kind of physical equivalence. In setting up an experiment, or a series of experiments, there are three important features: (i) the state, ρπϕ ?>, of the system prior to measurement, (ii) the observable(s), Ai, that are going to be measured, and (iii) the accuracy of each measurement, εi. Let ρπϕ ?> be the density operator in (Hπϕ,πϕ) ?> that is the state of the system prior to a measurement and ρϕ ?> its abstract state counterpart that is a member of the folium Fϕ ?>. Once (ii) and (iii) have been fixed, there will be a weak*-neighbourhood N(ρϕ,Ai,εi) ?>24 around ρϕ ?> in A1*+ ?>, where 1≤i≤k ?>. Let (Hπψ,πψ) ?> be a UIR with respect to (Hπϕ,πϕ) ?>. Will there be an abstract state, ρψ ?>, in the folium of Fψ ?> that is weakly equivalent in the above sense? If kerπϕ=kerπψ ?>, then weak equivalence guarantees that such a state can be found in Fψ ?>, that is, |ρϕ(Ai)−ρψ(Ai)|<εi ?>. As mentioned above, if the algebra is simple, then both representations are weakly equivalent and hence are interpreted as physically equivalent by Haag and Kastler. Various phrases such as ‘algebraic imperialism’ (Arageorgis [1995], p. 132), ‘apologetic imperialism’ (Ruetsche [2011], p. 133), and ‘algebraic chauvinism’ (Ruetsche [2003], p. 1334) have been used to describe the position of many algebraic quantum field theorists in the 1960s. Based on the notion of physical equivalence, it was claimed that Hilbert spaces and representations of a C*-algebra were dispensable and that the abstract C*-algebra contained all of the physical content of a theory.25 […] the important thing here is that the observables form some algebra, and not the representation Hilbert space on which they act. (Segal [1967], p. 128) It is in this new notion of equivalence in field theory that the algebraic approach has its greatest justification. All the physical content of the theory is contained in the algebra itself; nothing of fundamental significance is added to a theory by its expression in a particular representation. From this point of view it becomes clear that only faithful representations are worth consideration because the existence of a non-trivial kernel for a representation implies that there are redundant elements in the algebra. (Robinson [1966], p. 488) The relevant object is the abstract algebra and not the representation. The selection of a particular (faithful) representation is a matter of convenience without physical implications. It may provide a more or less handy analytical apparatus. (Haag and Kastler [1964], pp. 851–2) […] the specification of a special representation is physically irrelevant, all the physical information being contained in the algebraic structure of the abstract algebra A ?> alone. [Interpreting weak equivalence as physical equivalence] shows indeed that the physically relevant object is not a concrete realization of A ?> but the algebra A ?> itself, since any two different concrete realizations (i.e., faithful *-representations, or representations with zero kernel) will be physically equivalent. (Kastler [1964], p. 180–1) Some philosophers of physics have criticized weak equivalence as a sufficient reason to dismiss UIRs as mere mathematical possibilities.26 Explanations of representation dependent features such as symmetry breaking (Arageorgis [1995], pp. 159, 165-6) and phase transitions (Ruetsche [2003], p. 1138) use UIRs.27 Hence, it has been argued that these different UIRs cannot be considered physically equivalent. That argument is unlikely to convince Wallace that different UIRs must be used to explain phase transitions since ‘it is provably impossible for us to ever discover’ (Wallace [2006], p. 54) which representation is the correct representation for describing a particular phase. Since Fell’s theorem is taken to show that a finite number of measurements each carried out with finite accuracy cannot distinguish different representations, an infinite number of measurements is needed to find the correct representation. Rather than focus on semantic or methodological arguments against the algebraic imperialist’s use of Fell’s theorem, I am going to examine the mathematical heart of the matter: the notion of weak equivalence. Using weak equivalence to argue for the physical insignificance of UIRs would be particularly telling if the only type of representation used in algebraic quantum physics is a C*-representation. However, as I will explain below, the most commonly used concrete algebra in both AQFT and AQSM is a von Neumann algebra, which can be viewed as a W*-representation of the bidual A** ?>. If weak equivalence could be shown to hold for W*-representations as well, then a strong case could be made for the algebraic imperialist position on UIRs. However, the mere weak equivalence of C*-representations is not enough to guarantee the weak equivalence of their W*-representations. 5 Weak Equivalence in C*-Algebras and W*-Algebras To evaluate weak equivalence for von Neumann algebras, a new type of equivalence for C*-representations must be introduced. Quasi-equivalence is a weaker notion than unitary equivalence, but it is stronger than weak equivalence28: Unitary Equivalence ⇒ Quasi-equivalence ⇒ Weak Equivalence For C*-representations and W*-representations, the results are summarized below.29 Surprisingly, the weak equivalence of two C*-representations is not sufficient to guarantee the weak equivalence of their respective W*-representations. The stronger condition of quasi-equivalence for C*-representations must be satisfied for their W*-representations to be weakly equivalent. One response to Table 1 might be that C*-representations are more useful for modelling various physical situations and that W*-representations are not needed. However, it is the W*-representations, or von Neumann algebras, that are used more for modelling different systems. Every von Neumann algebra can be written as a direct sum, or direct integral, of types I, II, or III factors.30 There are three factors types: type I, type II, and type III. The rough idea behind the classification is the following31: Every von Neumann algebra is generated from its projections. A dimension function, d, can be defined to classify the range of the projections as follows: Type In ?>: d: {0, 1,…,n} where n is a natural number Physical example: Non-relativistic finite dimensional quantum mechanics (n × n complex matrices) Type I∞ ?>: d: {0, 1,…, ∞ ?>} Physical example: Free boson Fock space Type II1 ?>: d: [0, 1] Physical examples: Free fermion Fock space Infinite temperature maximally chaotic KMS state Type II∞ ?>: d: [0, ∞ ?>) Physical example: The tensor product of the free boson and fermion Fock spaces Type III: d: {0, ∞ ?>} (two-element set) Physical examples: Typically local algebras in AQFT32 KMS states with finite non-zero temperature in AQSM While the type III factor seems to be pathological, ‘[t]o the malicious delight of mathematicians’ (Thirring [1983], p. 41), type III factors are the most useful factor for modelling many physical systems. In particular, local algebras, which are defined on open regions of Minkowski spacetime, are usually type III factors (see Footnote 32). A KMS state (named after Kubo, Martin, and Schwinger) that corresponds to a pure phase of an infinite system at a non-zero finite temperature induces a representation that is also a type III factor.33 Table 1. C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇒ ?> Unitary Equivalence ⇓ ?> ⇓ ?> Quasi-Equivalence ⇔ ?> Weak Equivalence ⇓ ?> Weak (Physical) Equivalence C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇒ ?> Unitary Equivalence ⇓ ?> ⇓ ?> Quasi-Equivalence ⇔ ?> Weak Equivalence ⇓ ?> Weak (Physical) Equivalence Table 1. C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇒ ?> Unitary Equivalence ⇓ ?> ⇓ ?> Quasi-Equivalence ⇔ ?> Weak Equivalence ⇓ ?> Weak (Physical) Equivalence C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇒ ?> Unitary Equivalence ⇓ ?> ⇓ ?> Quasi-Equivalence ⇔ ?> Weak Equivalence ⇓ ?> Weak (Physical) Equivalence As the examples above show, W*-representations, and in particular type III factors, are necessary to model many of the important physical examples in AQFT and AQSM. One might still hope that quasi-equivalence could be used to banish UIRs to the realm of mathematical annoyances. If UIRs could be shown to be quasi-equivalent as C*-representations, then their W*-representations would still be weakly equivalent. However, the distinction between unitary equivalence and quasi-equivalence collapses for many physically important representations such as irreducible representations,34 KMS representations (that is, representations generated by abstract states that satisfy the KMS condition35), and type III factors. In those cases, mere unitary inequivalence is enough to guarantee the weak inequivalence of the W*-representations. For irreducible representations, KMS representations, and type III factors it can be proven that quasi-equivalence implies unitary equivalence, in which case the distinction between quasi-equivalence and unitary equivalence collapses (Table 2).36 Table 2. Irreducible Representations, KMS Representations, and Type III Factors Quasi-equivalence ⇒ Unitary Equivalence ?> Irreducible Representations, KMS Representations, and Type III Factors Quasi-equivalence ⇒ Unitary Equivalence ?> Table 2. Irreducible Representations, KMS Representations, and Type III Factors Quasi-equivalence ⇒ Unitary Equivalence ?> Irreducible Representations, KMS Representations, and Type III Factors Quasi-equivalence ⇒ Unitary Equivalence ?> Table 3. C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇔ ?> Unitary Equivalence ⇓ ?> Weak (Physical) Equivalence C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇔ ?> Unitary Equivalence ⇓ ?> Weak (Physical) Equivalence Table 3. C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇔ ?> Unitary Equivalence ⇓ ?> Weak (Physical) Equivalence C*-Representations: πϕ, πψ ?> Associated W*-Representations: πϕ∼ ?>, πψ∼ ?> Unitary Equivalence ⇔ ?> Unitary Equivalence ⇓ ?> Weak (Physical) Equivalence This modifies the relationships from Table 1 for irreducible representations, KMS representations, and type III factors. Irreducibility can intuitively be thought of as a representation that is as ‘small’ as possible. Some physicists view them as the most important kind of representation since they can serve as the building blocks for other representations. Whether one is looking for representations of the Lorentz group or of the algebra of the canonical commutation relations (Weyl algebra), irreducibility has been a desideratum.37 Irreducibility collapses the distinction between unitary equivalence and quasi-equivalence, as well as the distinction between disjointness and unitary inequivalence38; see (Kadison and Ringrose [1997], p. 740).39 For irreducible representations, KMS representations, and type III factors, two representations are either quasi-equivalent or disjoint. If they are disjoint, then they are not quasi-equivalent. That implies that they are not unitarily equivalent and that their associated W*-representations are not weakly equivalent. Thus, for those three cases mere unitary inequivalence is sufficient to show that their W*-representations are weakly inequivalent. In those cases, the situation is even worse since W*-representations are only going to be weakly equivalent when their C*-representations are unitarily equivalent! Here are some specific physical examples of UIRs that do not satisfy the condition of weak equivalence for W*-representations. Theorem 1 Let ϕ ?> and ψ be (αt,β) ?> and (αt,β′) ?>-KMS states with automorphism group αt, inverse temperatures β and β′ ?>, respectively, such that β≠β′ ?> and at least one temperature is non-zero and finite. Then πϕ∼ ?> and πψ∼ ?> are not weakly equivalent.40 Theorem 1 is a rich source of UIRs that are not weakly equivalent W*-representations. Given any two KMS states with different non-zero finite temperatures, the representations they generate will be unitarily inequivalent and their W*-representations will not be weakly equivalent no matter how small the difference is between their temperatures. Thus, there will be a continuum of UIRs whose W*-representations will not be weakly equivalent—one representation for each temperature! The next theorem proves that UIRs with different chemical potentials are weakly inequivalent. Theorem 2 Let ϕ ?> and ψ be KMS states with chemical potentials μ and μ′ ?>, respectively, such that μ≠μ′ ?>. Then πϕ∼ ?> and πψ∼ ?> are not weakly equivalent.41 As in the case of temperature, Theorem 2 provides a continuum of UIRs whose W*-representations are weakly inequivalent, regardless of how small the difference is between chemical potentials μ and μ′ ?>. These theorems show that KMS states are extremely sensitive to variations in temperature or chemical potential. Though these examples show how to generate UIRs whose W*-representations are weakly inequivalent in AQSM, there are similar examples in AQFT. Later in this article, it will be shown that there is a continuum of such cases for the Unruh effect. How does the failure of weak equivalence for W*-representations connect with the notion of faithful representations that is key to many presentations of Fell’s theorem? Recall that πϕ∼ ?> and πψ∼ ?> are weakly inequivalent if and only if kerπϕ∼≠kerπψ∼ ?>. First, notice that for von Neumann algebras, the faithfulness of a representation is no longer with respect to the original C*-algebra A ?>, but rather to the W*-algebra A** ?>. If kerπϕ∼≠kerπψ∼ ?>, then there are three possible cases: (i) πϕ∼ ?> is faithful and πψ∼ ?> is not faithful, (ii) πϕ∼ ?> is not faithful and πψ∼ ?> is faithful, and (iii) both πϕ∼ ?> and πψ∼ ?> are not faithful. In any of these cases, there is an A∈A** ?> such that A≠0 ?> and, for example, in (ii), πϕ∼(A)=0 ?>. It follows that for some density operator, ρπϕ∼ ?>, associated with πϕ∼(A**) ?>, there does not exist a density operator, ρπψ∼ ?>, associated with πψ∼(A**) ?> that can approximate ρπϕ∼ ?> for A within a given ε > 0, that is, for some ρπϕ∼ ?>, A, and ε that for every ρπψ∼ ?>. Thus, there will always be some observable, A in A** ?>, that will distinguish any pair of weakly inequivalent W*-representations. What is rather surprising is that this observable has classical properties. 6 Classical Equivalence and Weak Equivalence The mathematical basis for the algebraic imperialist argument against UIRs has now been shown to apply only in very exceptional circumstances. For most UIRs, it can be proven that they are not weakly equivalent to each other. Someone might hold onto the hope that different UIRs do not make different predictions. If that is the case, then the appearance of UIRs is just another case of theoretical underdetermination. It would then make no difference which representation is used since they would all make the same, or approximately the same, predictions. What is the difference, if any, between W*-representations that are weakly inequivalent? Does weak equivalence have any usefulness for W*-representations? To borrow a phrase from Ruetsche ([2003]), weak equivalence for W*-representations can be ‘put to work’. In fact, it shows us where to locate the physical differences between any two UIRs. But there is a surprising twist. Two W*-representations that are factors are weakly equivalent if and only if they assign the same expectation values to all classical—not quantum—properties. Theorem 3 Let ϕ ?> and ψ be states that generate factor representations πϕ ?> and πψ ?>, respectively. The W*-representations πϕ∼ ?> and πψ∼ ?> are weakly equivalent if and only if the expectation values are the same for the extensions of ϕ ?> and ψ to be states on A** ?>, that is, ϕ∼(Z)=ψ∼(Z) ?> for every observable Z in the centre of the bidual, Z∈ℨ(A**) ?>.42 Since Z commutes with every observable in A** ?>, it can be considered a ‘classical’ observable.43 The weak equivalence of two W*-representations is sufficient for their being classically equivalent to each other since their states will have exactly the same expectation values for every element in ℨ(A**) ?>. Since every element in ℨ(A**) ?> commutes with every element in A** ?> and A** ?> includes every element in the original C*-algebra A ?>—that is, A⊆A** ?>—this is the largest collection of classical observables. Thus, the weak equivalence of W*-representations is more appropriately a mathematical condition for classical equivalence than physical equivalence. The converse of Theorem 3, namely that πϕ∼ ?> and πψ∼ ?>, are weakly inequivalent if and only if there exists an observable Z∈ℨ(A**) ?> such that ϕ∼(Z)≠ψ∼(Z) ?>, shows that there exists an observable that will distinguish different UIRs, or more precisely, it will distinguish C*-representations that are not quasi-equivalent. It is the classical observables that are members of the much larger algebra A** ?> that serve to distinguish different UIRs. Thus, UIRs are not just another example of theoretical underdetermination, they make different predictions; but these predictions are with respect to classical observables in a W*-algebra A** ?>, not the original C*-algebra A ?>. In the cases discussed in Section 5, the classical properties that distinguished different UIRs were temperature and chemical potential. These results point to the limitations of the original C*-algebra A ?>, namely, that it is not large enough. UIRs can be distinguished by observables in a larger algebra: the bidual A** ?>.44 The argument that UIRs are not another case of undetermination can be strengthened. Weak equivalence says that for any abstract state, ϕ ?>, any finite collection of abstract observables, Ai, any corresponding finite collection of positive real numbers, εi, and any representation, there is a state ψ in a different representation for which the expectation value for each Ai lies within ±εi ?> from its expectation value in ϕ ?>. For a classical observable in a particular representation, all (concrete) states in that representation will have the same value, for example, in πϕ∼ ?> each vector state and density matrix will have the same expectation value for a particular Z∈ℨ(A**) ?>. Given any Z∈ℨ(A**) ?>, it can be assumed without loss of generality that Z either has the value 1 or 0. For any πϕ∼ ?> and πψ∼ ?> that are weakly inequivalent, there exists an observable Z∈ℨ(A**) ?> such that ϕ∼(Z)≠ψ∼(Z) ?>. The only way ϕ∼(Z)≠ψ∼(Z) ?> is that ϕ∼(Z)=1 ?> and ψ∼(Z)=0 ?>, or ϕ∼(Z)=0 ?> and ψ∼(Z)=1 ?>. If ψ∼(Z)=1 ?>, can I find any states in πψ∼ ?> such that ? No, precisely because every density matrix in πψ∼ ?> will have the same expectation value for Z, namely, 0! All of these results show that an algebraic imperialism based on weak equivalence has significant drawbacks. But the type of algebraic imperialism that is being criticized here is of a specific type. A better name for it would be ‘C*-algebraic imperialism’ since the physical content is supposed to be captured by A ?>. These results lend support to a new type of algebraic imperialism, which will be called ‘bidualism’.45 It’s only in the bidual A** ?> that the algebraic imperialist has access to enough observables to distinguish different states, for example, states of different thermodynamic phases (or KMS states of different temperatures). A** ?> contains the abstract counterparts to Ruetsche’s ‘parochial’ observables, as well as all of the observables in the original C*-algebra A ?>.46 For the bidualist, physical content is captured at the abstract algebra level by the observables in A** ?> and the normal states defined on A** ?>. It allows one to be an algebraic imperialist and endorse the physical significance of UIRs.47 7 Interlude: Is Weak Equivalence Really Physical Equivalence? These theorems also show the semantic failure of interpreting weak equivalence as physical equivalence. The interpretation of weak equivalence as physical equivalence was motivated by operationalist considerations, namely, that only a finite number of experiments can be carried out and that each experiment has some finite error associated with it. The problem is that weak equivalence is both too strong and too weak to capture the idea of physical equivalence. To see why it is too weak, consider two KMS states with different temperatures and let the difference in their temperatures be so small that they cannot be distinguished by any experiment. Those two KMS states should satisfy the operationalist’s notion of both systems being physically equivalent to each other since no measurement can distinguish them. However, as shown in the last section, those states are weakly inequivalent to each other, and hence are physically inequivalent to each other. Thus, while two KMS states with slightly different temperatures would satisfy an operationalist understanding of physical equivalence, the mathematical condition of weak equivalence would fail to classify those KMS states as being physically equivalent. The concept of weak equivalence is also too strong to capture the idea of physical equivalence. If the strategy of using weak equivalence to classify all UIRs as physically equivalent and empirically equivalent succeeded, the consequences would be unacceptable. One very rough way of describing Haag’s theorem is that the representations of a free and an interacting system are unitarily inequivalent to each other. If weak equivalence truly captured the concept of physical equivalence and the representations for all free and interacting systems are weakly equivalent to each other, then not only would a free and interacting system be physically equivalent, but they would make roughly the same predictions! It would also classify different phases (for example, a gas and a solid) as physically equivalent. These considerations should indeed make us wary of the semantic extrapolations from weak equivalence to physical equivalence (Emch [1972], p. 108). 8 The Unruh Effect While philosophers have focused on the fate of particles in the Unruh effect and the role of UIRs, there has been no discussion of weak equivalence in the Unruh effect. I will extend the results of the previous sections to show that the Minkowski and Rindler representations are not weakly equivalent as W*-representations. The classical observable that serves to distinguish those representations is temperature. What is even more surprising is that the Unruh effect has more than two UIRs that are not weakly equivalent as W*-representations. A continuum of W*-representations that are not weakly equivalent can be found in the Unruh effect, with one representation for each temperature or acceleration value. There are a number of excellent papers on the Unruh effect (Clifton and Halvorson [2001]; Arageorgis et al. [2002a]), so I will only give a brief overview. The Unruh effect is generated by using Rindler coordinates in Minkowski spacetime.48 There are two spacetime wedges that have two types of trajectories defined on them: inertial motion and uniformly accelerated motion. The Rindler observer in, say, the right wedge is uniformly accelerating while a Minkowski observer’s trajectory is inertial.49 Both observers will associate a special C*-algebra called the Weyl algebra W ?> with the right wedge and build a representation via the GNS construction50. For my purposes, the salient feature of the Unruh effect is that the Rindler representation πωR⊲ ?>, which is only defined on the right wedge, and the Minkowski representation πωM⊲ ?>, which is restricted to the right wedge, are unitarily inequivalent (Clifton and Halvorson [2001], p. 463).51 Given this result, it is straightforward to prove the following theorem: Theorem 4 Let πω∼R⊲ ?> and πω∼M⊲ ?> be the W*-representations of W** ?> associated with πωR⊲ ?> and πωM⊲ ?>, respectively. Then πω∼R⊲ ?> and πω∼M⊲ ?> are weakly inequivalent.52 Thus, the Unruh effect is an example in AQFT of two weakly inequivalent W*-representations. The explanation of their weak inequivalence can be deepened. Since both W*-representations are factors and they are weakly inequivalent, the following theorem immediately follows from Theorem 3 and Theorem 4: Theorem 5 There exists an observable Z∈ℨ(W⊲**) ?> in the centre of the bidual of the Weyl algebra W⊲** ?> on the right Rindler wedge, such that ω∼R⊲(Z)≠ω∼M⊲(Z) ?>. What physical quantity is Z? The Minkowski vacuum state, ωM⊲ ?>, when restricted to the right wedge, satisfies the KMS condition (Arageorgis et al. [2002a], pp. 188–9). Suppressing some technical details53 the temperature of ωM⊲ ?> is proportional to its acceleration in the right wedge. There is a continuum of these ωM⊲ ?> KMS states, one for each acceleration a∈(0,∞) ?> at temperature T=a2π ?>. Using Theorem 1, it then follows that if ωM,T⊲ ?> is a KMS state at temperature T=a2π ?> with acceleration a, ωM,T′⊲ ?> is a KMS state at temperature T′=a′2π ?> with acceleration a′ ?>, T≠T′ ?>, then πω∼M,T⊲ ?> and πω∼M,T′⊲ ?> are weakly inequivalent. Thus, there exists a continuum of UIRs in the Unruh effect (one UIR for each temperature, or acceleration, value), and they are each weakly inequivalent to each other and to πω∼R⊲ ?>! Further, the classical observable that distinguishes them is temperature. 9 Time Evolution and Symmetries One area that has not been explored is whether weak equivalence is preserved under time evolution or symmetry transformations. Symmetries and time evolution are represented in AQFT as automorphisms of the C*-algebra A ?>. An automorphism can be inner or outer depending on whether it can be implemented by a unitary operator. If it can be unitarily implemented, then the automorphism is called an inner automorphism, otherwise it is an outer automorphism. If the automorphism is outer, then there is a spontaneously broken symmetry or non-unitary dynamics. For two states, ω1 and ω2, to be weakly equivalent there must be a finite collection of observables, Ai∈A ?>, where 1≤i≤n ?>, such that |ω1(Ai)−ω2(Ai) |<εi ?>. Let α be an automorphism of A ?>. What does weak equivalence imply about |ω1(α(Ai))−ω2(α(Ai)) | ?>? A result from (Fabri et al. [1967], p. 383) states that for all A∈A, 0≤ |ω(A)−ω(α(A)) |≤2‖A‖ ?> for any pure state ω. Suppose that ω1 and ω2 are both pure and weakly equivalent ( |ω1(Ai)−ω2(Ai) |<εi ?>). Does this imply that ω1 and ω2 are still weakly equivalent under an automorphism α, that is, |ω1(α(Ai))−ω2(α(Ai)) |<εi ?>? No. The inequality becomes |ω1(α(Ai))−ω2(α(Ai)) |<4‖Ai‖+εi ?> (see Appendix for the proof). The only way for the upper bound to be preserved is for ‖Ai‖=0 ?> for all i. As long as ‖Ai‖>0 ?> for some i, weak equivalence will not necessarily hold. As Earman ([2004], p. 185) notes (using a theorem from (Fabri et al. [1967], p. 383), if ω is a pure state and α is an automorphism that is not unitarily implementable with respect to ω, then there exists an X∈A ?> with ‖X‖=1 ?> such that |ω(X)−ω(α(X)) |=2 ?>. Now suppose that α is not unitarily implementable with respect to ω1. If we pick that X as one of the C*-observables in our collection for weak equivalence, then |ω1(α(X))−ω2(α(X))|≤4+ε ?>. The weak equivalence between the representations is blown out of the water by a spontaneously broken symmetry. Even if the algebraic imperialist bashes the detectors with a baseball bat to force the representations to be physically equivalent, detectors that are only 25% efficient could still distinguish the representations. However, two representations, πφ ?> and πψ ?>, are weakly equivalent just in case they have the same kernel. If πφ ?> and πψ ?> have the same kernel, then πα*φ ?> and πα*ψ ?> have the same kernel. Thus, πα*φ ?> and πα*ψ ?> are weakly equivalent. How do we resolve these two results? At the level of representations, two weakly equivalent representations will continue to be weakly equivalent under automorphisms. However, the price is that the state needed to keep the expectation values within εi may change! If the automorphism represents the time evolution of two weakly equivalent representations, a new state may be required at each moment of time to satisfy the inequality! These results also show that weak equivalence is not a strong enough mathematical condition to be interpreted as physical equivalence. Intuitively, if we start with two states that are physically equivalent to each other and rotate the laboratory by a tiny angle, then we should still have two physically equivalent states. However, the weak equivalence between two states is not necessarily preserved even if the rotation is unitarily implementable in both representations. Suppose there are two weakly equivalent representations πφ ?> and πψ ?>. Once we fix the state we want to use in the folium of πφ ?>, the finite set of observables, Ai, and the εi, there will be some state in the folium of πψ ?> that will be weakly equivalent to it. However, once the symmetry transformation is applied, a second state in the folium of πψ ?> may well be needed to maintain weak equivalence. Thus, while only one state is needed from the folium of πφ ?>, two states from the folium of πψ ?> would be needed to maintain weak equivalence. Using two states from the folium of one representation to maintain weak equivalence, does not capture the idea of physical equivalence. 10 Conclusions The algebraic imperialist uses weak equivalence to argue that UIRs are mathematical surplus structure. If it is assumed that Fell’s theorem shows that any two representations of a C*-algebra are weakly equivalent to each other, and hence physically equivalent according to the algebraic imperialist, it does not follow that their extensions to W*-representations are weakly equivalent. However, weak equivalence can be put to use as a criterion for when two W*-representations are classically equivalent. The theorems in this article show that the difference between UIRs that are weakly inequivalent is that they differ in their expectation values for a classical observable such as temperature or chemical potential. These observables are not part of the original C*-algebra A ?>, but a larger W*-algebra: the bidual A** ?>. Thus, UIRs are not just another case of theoretical underdetermination. Nor is the interpretation of weak equivalence as physical equivalence supported. In the case of a unitarily implementable symmetry, different states would be needed before and after the symmetry is applied in order to save the phenomena. The choice of a particular UIR is enforced by the dynamics or macroscopic constraints such as temperature. The lesson of UIRs is that the C*-algebra A ?> was too small to be able to capture all of the possible physical content in either AQFT or AQSM. It also shows the path to a new kind of algebraic imperialism that locates physical content in the bidual A** ?>. Appendix Automorphisms and Weak Equivalence If ω1 and ω2 are both pure and weakly equivalent ( |ω1(Ai)−ω2(Ai) |<εi ?>), then |ω1(α(Ai))−ω2(α(Ai)) |<4‖Ai‖+εi ?> To prove this we need two results: (1) (Fabri et al. [1967], p. 383) for all A∈A,0≤ |ω(A)−ω(α(A)) |≤2‖A‖ ?> and (2) the triangle inequality |φ(A)+ψ(A) |≤ |φ(A) |+ |ψ(A) | ?>. |ω1(α(Ai))−ω2(α(Ai)) |= |ω1(α(Ai))+ω1(Ai)−ω1(Ai)+ω2(Ai)−ω2(Ai)−ω2(α(Ai)) |= |ω1(α(Ai))−ω1(Ai)+ω2(Ai)−ω2(α(Ai))+ω1(Ai)−ω2(Ai) |≤ |ω1(α(Ai))−ω1(Ai) |+ |ω2(Ai)−ω2(α(Ai)) |+ |ω1(Ai)−ω2(Ai) |(A.1)≤ |ω1(Ai)−ω1(α(Ai)) |+ |ω2(Ai)−ω2(α(Ai)) |+ |ω1(Ai)−ω2(Ai) |≤2‖Ai‖+2‖Ai‖+ |ω1(Ai)−ω2(Ai) |(A.2)≤4‖Ai‖+ |ω1(Ai)−ω2(Ai) |<4‖Ai‖+εi.(A.3) The triangle inequality is used in (A.1), the Fabri et al. ([1967]) result is used in (A.2), and the assumption of weak equivalence in (A.3).□ Acknowledgements I would like to thank Aristidis Arageorgis, David Baker, Benjamin Feintzeig, Hans Halvorson, and Fred Kronz for stimulating conversations and comments on earlier drafts of this paper. I would also like to thank the University of Western Ontario, the European Philosophy of Science Association 2013 conference, and the University of Oxford for opportunities to present this material. Footnotes 1Huggett and Weingard ([1996]) and Arageorgis ([1995]) also argue for the significance of UIRs in QFT. 2 After discussing Fell’s theorem, Wallace offers two arguments against the significance of UIRs for the foundations of QFT. Those arguments focus on UIRs that arise from short distance and global features. My discussion of UIRs will focus on showing the limitations of Fell’s theorem. 3Sklar ([2000], p. 20) argues that UIRs are observationally equivalent. Though Sklar does not make specific reference to Fell’s theorem, it seems to be what he has in mind. 4 Though Kronz and Lupher ([2005]) do provide an explicit discussion of how Fell’s theorem fails to apply to certain representations. 5 I will follow the standard terminology of referring to elements of a C*-algebra or W*-algebra as ‘observables’, though many elements of the algebra may not be something we can measure in an experiment. The observables that will distinguish different UIRs belong to an abstract W*-algebra that is the bidual of a C*-algebra (see Section 6 for details). 6 The observable can be considered ‘classical’ because it commutes with all of the other observables in the algebra; see Section 5 for more details. 7 For more information on the Stone–von Neumann theorem, see (Summers [2001]). 8 To see roughly why unitary equivalence of two representations of the canonical commutation relations, or of a C*-algebra or W*-algebra, are empirically equivalent, suppose the representations πφ ?> and πψ ?> are unitarily equivalent. (Precise definitions of what representations are and the unitary equivalence of two representations will be given shortly.) Let ρπφ ?> be a density operator acting on the Hilbert space Hπφ ?> and πφ(A) ?> be a representation of an observable, A, acting on Hπφ ?>. The expectation value of the state ρπφ ?> for the observable πφ(A) ?> is given by Tr (ρπψπψ(A)) ?>. Since πφ ?> and πψ ?> are unitarily equivalent, that means there is a density operator, ρπψ ?>, and observable, πψ(A) ?>, such that a unitary operator, U, transforms ρπφ ?> into ρπψ ?> via ρπψ=UρπφU−1 ?> and πφ(A) ?> into πψ(A) ?> via πψ(A)=Uπφ(A)U−1. ?> The empirical equivalence of the expectation values follows from these relations and the cyclic property of the Tr operation (that is, Tr(ABC)=Tr(CAB)), and UU−1=U−1U=I ?>, where I is the identity operator (IA = AI = A): Tr (ρπψπψ(A))= ?> Tr (UρπφU−1Uπψ(A)U−1)= ?>Tr (UρπφIπψ(A)U−1)= ?> Tr (Uρπφπψ(A)U−1)= ?>Tr (U−1Uρπφπψ(A)) \\= ?>Tr (Iρπφπψ(A)= ?>Tr (ρπφπψ(A) ?>). Thus, for two unitarily equivalent representations and any expectation value in one representation, a state and observable exists in the other representation that gives the same expectation value. 9 Wallace does not argue that UIRs are mathematical surplus; he ([2011], p. 123) says that the algebraic approach is useful for tackling long-range divergences in QFT. 10 Good sources for more on the mathematical details of AQFT are (Emch [1972]; Halvorson [2007]). 11 A C*-algebra A ?> is a vector space over the field of complex numbers C ?> with an associative distributive product that has an involution * (which satisfies ( A*)*=A, (AB)*=B*A*, (λA+μB)*=λ¯A*+μ¯B* ?> for all A,B∈A ?> and λ,μ∈C ?> where λ¯ ?> and μ¯ ?> are the complex conjugates of λ and μ) and is complete in the norm ∥.∥ ?> obeying ∥AB∥≤∥A∥∥B∥ ?> and ∥A*A∥=∥A∥2 ?> for all A,B∈A ?>. 12 A linear functional ω is a mapping from A ?> to the complex numbers C ?> ( ω:A→C ?>) such that ω(αA+βB)=αω(A)+βω(B) ?>, where α,β∈C ?> and A,B∈A ?>. 13 A C*-algebra is called a W*-algebra if it is the dual space of a Banach space. A* ?> is a Banach space, so A** ?> is a W*-algebra. 14 To say that πω ?> preserves the algebraic relations between the elements of A ?> means that the following conditions are satisfied for any A,B∈A ?> and α,β∈C ?>: πω(αA+βB)=απω(A)+βπω(B),  πω(AB)=πω(A)πω(B) ?>, and πω(A*)=πω(A)*. ?> 15 Let R⊆B(H) ?> be a von Neumann algebra. R ?>’s commutant is defined as R′={B∈B(H)|AB=BA ?>, for all A∈A} ?>, and R ?>’s bicommutant is R″=(R′)′ ?>. 16 To say that a subset R ?> of B(H) ?> is ‘weakly closed’, that is, that it is ‘closed in the weak operator topology’ means that any sequence {Tn} ?> of elements of R ?> converges to another element T∈R ?> in the sense that ⟨Φ|Tn|Ψ⟩→⟨Φ|T|Ψ⟩ ?> for all Φ,Ψ∈H ?>. 17 A linear functional ρ∼ ?> on a W*-algebra A ?> is said to be ‘normal’ if and only if ρ∼(supαTα)=supαρ∼(Tα) ?> for every uniformly bounded increasing directed set {Tα} ?> of positive elements of A** ?>. 18 For more details about representations of A** ?>, see (Bing-Ren [1992], pp. 221-2). 19 Roughly, a W*-representation is a *-homomorphism from an abstract W*-algebra to B(Hπω) ?>, which is continuous in the topologies at the abstract and concrete levels. For the details, see (Bing-Ren [1992], p. 221). 20 Fell ([1960], Theorem 1.2) used a family of representations in his equivalence theorem, but that additional complexity is not needed. Two of the four equivalent conditions from Fell’s theorem are not listed in my presentation of Fell’s theorem and will not be needed in what follows. 21 It is assumed throughout this article that all representations are non-degenerate, which can intuitively be thought of as assuming that the representations are not the zero representation. 22 A representation is ‘faithful’ if and only if its kernel is trivial, that is, the kernel only contains the zero operator from A ?>: kerπ={0} ?>, where 0∈A ?>. 23 Alternatively, two representations, πϕ ?> and πψ ?>, of a C*-algebra A ?> are weakly equivalent if and only if every weak*-neighbourhood of any state belonging to folium Fϕ ?> has a non-empty intersection with the weak*-neighbourhood of some state in folium Fψ ?> and vice versa. 24 In Fell’s theorem, there is no subscript on the ε. In what follows, the value of ε in the discussion of the weak*-topology on A* ?> can be considered the maximum experimental error, that is, the largest of the εi. 25 Ruetsche ([2011], pp. 133-9) distinguishes two types of algebraic imperialist: a ‘bold algebraic imperialist’ and an ‘apologetic algebraic imperialist’. The algebraic imperialist that uses weak equivalence as an argument for algebraic imperialism is an ‘apologetic algebraic imperialist’, and that is the type of algebraic imperialist that I am arguing against here. 26 Summers ([2001], p. 145) also criticizes physical equivalence. 27 There are also arguments that different UIRs have different empirical content. Emch and Liu ([2002]) discuss ‘marker’ observables and Ruetsche ([2011]) discusses ‘phase’ observables for broken symmetries. These observables belong to the centre of a von Neumann algebra. In Section 6, I will discuss observables in the centre of A** ?> and how they can be used to distinguish different UIRs. 28 Two representations, πϕ ?> and πψ ?>, of a C*-algebra A ?> are ‘unitarily equivalent’ to each other if there exists an isomorphism U:Hπϕ→Hπψ ?> such that πψ(A)=Uπϕ(A)U−1 ?>. πϕ ?> and πψ ?> are ‘quasi-equivalent’ to each other if there is an isomorphism α:πϕ(A)″→πψ(A)″ ?> between the von Neumann algebras of πϕ ?> and πψ ?> such that α[πϕ(A)]=πψ(A) ?> for all A∈A ?>. 29 The equivalence between the quasi-equivalence of two C*-representations and the weak equivalence of their associated W*-representations was proved by Emch ([1972], pp. 122–4). 30 The centre ℨπω(A)″ ?> of a von Neumann algebra πω(A)″ ?> is the intersection of the von Neumann algebra and its commutant πω(A)′ ?>, that is, ℨπω(A)″=πω(A)″∩πω(A)′ ?>. A factor is a von Neumann algebra with a trivial centre, that is, ℨπω(A)″=πω(A)″∩πω(A)′={λIπω} ?>. In other words, a factor’s centre consists of scalar multiples of the Hilbert space Hπω ?> identity operator Iπω ?>. 31 For more technical details, see chapter six of Kadison and Ringrose ([1997]). 32 A number of results that indicate that local algebras of relativistic QFT are type III von Neumann algebras are discussed in (Halvorson and Mueger [2007], pp. 748-52). 33 For more information about KMS states and AQSM, see (Kronz and Lupher [2005]; Ruetsche [2003]). An abstract state ωβ ?> is a KMS state with respect to an automorphism group αt, inverse temperature β if and only if the following holds for all operators A, B in a dense subalgebra A ?>: ωβ[Aαiβ(B)]=ωβ(BA) ?>. This equation is the KMS condition and it uniquely determines with respect to αt with value β the Gibbs grand canonical equilibrium state. 34 A representation πω(A) ?> of a C*-algebra A ?> is called ‘irreducible’ if the only closed subspaces of Hπω ?> that are invariant under the action of the elements of πω(A) ?> are Hπω ?> and 0. Irreducibility will be discussed more below. 35 More precisely, quasi-equivalence implies unitary equivalence for KMS representations that have a non-zero finite temperature in the thermodynamic limit and that are not where a phase transition occurs. In order to be a bit more succinct, in this section I will just use the phrase ‘KMS representations’ instead of putting in those conditions every time. 36 The case of irreducible representations was proved by Kadison and Ringrose ([1997], p. 740). Dixmier ([1977], p. 128) proved that quasi-equivalence implies unitary equivalence for type III factors with separable Hilbert spaces. KMS representations that have a non-zero finite temperature in the thermodynamic limit and that are not where a phase transition occurs are type III factors, so Dixmier’s proof for type III factors can be used for these KMS representations. This does not cover special cases such as a KMS state with an infinite temperature, which is a type II1 ?> factor, and a KMS state at temperature zero, which is a type I factor. KMS representations that occur at a phase transition are a direct sum or direct integral of type III factors. For more on KMS representations and type III factors, see (Ruetsche [2011], p. 167). 37 According to Haag ([1996], p. 54), one of the advantages of irreducible representations of the equal-time canonical commutation relations for quantum fields is that all observables can be expressed in terms of the quantum field and its conjugate momentum. 38 The condition of disjointness is often used in discussing unitary inequivalence. It is usually contrasted with quasi-equivalence, but quasi-equivalence and disjointness are not mutually exclusive concepts unless both representations are factors. Two representations, πϕ ?> and πψ ?>, of a C*-algebra A ?> are disjoint when either (i) no sub-representation of πϕ ?> is unitarily equivalent to a sub-representation of πψ ?>, or (ii) the intersection of their folia is empty: Fϕ∩Fψ=∅ ?>. (ii) implies that no density operator in Hπϕ ?> can be expressed as a density operator in Hπψ ?>, and vice versa. 39 Kadison and Ringrose use the term ‘equivalent’ instead of ‘unitarily equivalent’, but ‘equivalent’ means ‘unitarily equivalent’. 40 Proof: If a KMS state has a finite non-zero temperature, then its representation is a type III factor. Under those conditions, Takesaki ([1970]) proved that πϕ ?> and πφ ?> are disjoint. Since πϕ ?> and πφ ?> are disjoint factor representations, they are not quasi-equivalent. From Table 1, since πϕ ?> and πφ ?> are not quasi-equivalent, πϕ∼ ?> and πψ∼ ?> are not weakly (physically) equivalent.□ 41 Proof: Under those conditions, Müller-Herold ([1980]) proved that ϕ ?> and ψ are disjoint. The rest of the proof is exactly the same as Theorem 1.□ 42 ℨ(A**) ?> is the set of elements of A** ?> that commute with every element of A** ?>, that is, ℨ(A**)={A∈A**|AB=BAfor allB∈A**} ?>. Proof of Theorem 3: Emch ([1972], p. 139) cited a result by Combes ([1967]) that two states ϕ ?> and ψ that generate factor representations πϕ ?> and πψ ?>, respectively, are quasi-equivalence if and only if their extensions ϕ∼ ?> and ψ∼ ?> coincide on ℨ ?>. Since πϕ ?> and πψ ?> are quasi-equivalent if and only if πϕ∼ ?> and πψ∼ ?> are weakly equivalent, this shows that πϕ∼ ?> and πψ∼ ?> are weakly equivalent if and only if ϕ∼(Z)=ψ∼(Z) ?> for every observable Z in the centre of the bidual, Z∈ℨ(A**) ?>.□ 43 Belonging to the centre of an abstract algebra is a necessary condition for an observable to be classical, but it is probably not a sufficient condition. Hence, my use of quotes around classical. For a more detailed discussion of classicality and quantum theory, see (Landsman [2007]). 44 This is not to say that the Z that does distinguish those UIRs is something that can be measured. ℨ(A**) ?> is a huge collection of mathematical operators and there is no reason to think that every one of those mathematical operators corresponds to a physical property that can be measured. Nonetheless, for the algebraic imperialist argument to succeed, it would have to offer an additional argument that every Z∈ℨ(A**) ?> has no physical significance. Given the examples given in the previous section, the algebraic imperialist would have to provide an argument for why temperature and chemical potential are not physical observables. 45 Hans Halvorson suggested the name when we were discussing these issues. 46 Ruetsche ([2011], p. 134) calls an observable ‘parochial’ when the observable is not an element of the abstract C*-algebra but is affiliated with a von Neumann algebra. 47 A more detailed presentation of bidualism will be given in a future paper. 48 For details, see (Arageorgis et al. [2002a], Section 2). 49 For simplicity, only the right Rindler wedge will be considered; however, the results apply to the left Rindler wedge and the algebra of observables defined on both the left and right wedges; see (Clifton and Halvorson [2001]) for details. 50 This C*-algebra is generated from the Weyl form of the canonical commutation relations. For details, see (Clifton and Halvorson [2001]). 51 Clifton and Halvorson ([2001]) actually prove that the Minkowski and Rindler representations are disjoint. Since both representations are factors, it follows that they are unitarily inequivalent. 52 Proof: Clifton and Halvorson ([2001], p. 463) proved that πωR⊲ ?> and πωM⊲ ?> are disjoint. Since both πωR⊲ ?> and πωM⊲ ?> are factors and disjoint, they are not quasi-equivalent, hence πω∼R⊲ ?> and πω∼M⊲ ?> are weakly inequivalent.□ 53 The automorphism group for the KMS states must be specified. See (Arageorgis et al. [2002a], p. 189) for more details. References Arageorgis A. [ 1995 ]: Fields, Particles, and Curvature: Foundations and Philosophical Aspects of Quantum Field Theory in Curved Spacetime , Ph.D. thesis, University of Pittsburgh . Arageorgis A. , Earman J. , Ruetsche L. [ 2002a ]: ‘Fulling Non-uniqueness and the Unruh Effect: A Primer on Some Aspects of Quantum Field Theory’ , Philosophy of Science , 70 , pp. 164 – 202 . 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