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The British Journal for the Philosophy of Science
, Volume 69 (1) – Mar 1, 2018

31 pages

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- Publisher
- Oxford University Press
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- © 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
- ISSN
- 0007-0882
- eISSN
- 1464-3537
- D.O.I.
- 10.1093/bjps/axw010
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- See Article on Publisher Site

ABSTRACT Ruetsche ([2011]) claims that an abstract C*-algebra of observables will not contain all of the physically significant observables for a quantum system with infinitely many degrees of freedom. This would signal that in addition to the abstract algebra, one must use Hilbert space representations for some purposes. I argue to the contrary that there is a way to recover all of the physically significant observables by purely algebraic methods. 1 Introduction 2 Preliminaries 3 Three Extremist Interpretations 3.1 Algebraic imperialism 3.2 Hilbert space conservatism 3.3 Universalism 4 Parochial Observables 4.1 Parochial observables for the imperialist 4.2 Parochial observables for the universalist 5 Conclusion 1 Introduction It is by now well known that in quantum theories with infinitely many degrees of freedom, like quantum field theory and quantum statistical mechanics, the presence of unitarily inequivalent representations stymies the extension of our interpretive practices from the ordinary quantum mechanics of finite systems. Ruetsche ([2011]) lays out the interpretive options in the wake of this problem of unitarily inequivalent representations. One can be a ‘Hilbert space conservative’ and maintain that possible worlds correspond to density operators on a particular privileged Hilbert space containing a concrete irreducible representation of the algebra of observables. Or one can be an ‘algebraic imperialist’ and hold that possible worlds are represented by the states on the abstract C*-algebra of observables, which captures the structure all representations have in common. Finally, one can be a ‘universalist’ aligning with the Conservative in viewing states as density operators on a Hilbert space, but picking the (reducible) universal representation of the abstract C*-algebra as the collection of physically measurable quantities.1 Ruetsche argues that none of the interpretations just mentioned is adequate. This article will focus on Ruetsche’s argument against imperialism and universalism,2 which I call ‘the problem of parochial observables’. Ruetsche argues that there are certain observables, such as particle number and net magnetization, that the Hilbert space conservative acquires and employs in physically significant explanations, and which the imperialist and universalist do not have access to. Ruetsche calls these the ‘parochial observables’. Since the imperialist and universalist do not have access to these observables, they cannot recover physically significant explanations of (for example) particle content, phase transitions, and symmetry breaking. Since an interpretation is adequate only insofar as it can recover physically significant explanations, Ruetsche argues that imperialism and universalism are unacceptable interpretations. I will argue that the problem of parochial observables is only an apparent problem, which disappears once one recognizes that the imperialist and universalist have additional resources for representing parochial observables. Parochial observables arise for the Conservative as approximations to, or idealizations from, observables in a particular Hilbert space representation (once we have chosen a physically relevant notion of idealization). I will show that the imperialist and the universalist can use the same sorts of tools in their respective settings to acquire the parochial observables. Namely, the imperialist can define a physically relevant notion of approximation or idealization on the abstract algebra, and the universalist can similarly define a physically relevant notion of approximation or idealization on the bounded operators on the universal Hilbert space. According to both of these notions, parochial observables arise as idealizations from the observables we began with in the abstract algebra or the universal representation, respectively. When the imperialist and universalist are allowed the same mathematical tools that the Hilbert space conservative uses for representing approximations and idealizations, the problem of parochial observables does not arise. The imperialist and universalist each have access to all of the observables they need. Furthermore, when we allow the imperialist and the universalist access to these idealized observables, one can show a precise sense in which they are equivalent interpretations. There is a sense in which the imperialist and the universalist subscribe to the same physical possibilities; they simply use different mathematical tools to describe these possibilities. Thus, imperialism and universalism amount to the same interpretation, and it is one which avoids Ruetsche’s problem of parochial observables. 2 Preliminaries A standard presentation of the non-relativistic quantum theory of a single particle consists in a Hilbert space, H, containing operators P and Q satisfying the canonical commutation relations. The states of the theory are the vectors or density operators on H, and the observables are self-adjoint operators on H. A result known as the Stone–von Neumann theorem (described below) tells us that this formulation is equivalent to a more abstract algebraic approach.3 The algebraic approach, which provides a useful way of generalizing to quantum field theory and quantum statistical mechanics, proceeds as follows: One begins with a C*-algebra A generated by canonical (anti-)commutation relations.4 The observables of the theory are represented by the self-adjoint elements of A. A ‘state’ is a positive, normalized linear functional ω:A→C with the following interpretation: for each self-adjoint A∈A, ω(A) represents the expectation value of A in the state ω. Importantly, this algebraic formalism can be translated back into the familiar Hilbert space theory through a representation. A ‘representation’ of A is a pair (π,H), where π:A→B(H) is a *-homomorphism into the bounded linear operators on some Hilbert space, H. A representation is called ‘irreducible’ if the only if subspaces of H invariant under π(A) are H and {0}. One may find representations of A on different Hilbert spaces, and in this case one wants to know when these can be understood as ‘the same representation’. This notion of ‘sameness’ is given by the concept of unitary equivalence5: two representations (π1,H1) and (π2,H2) are ‘unitarily equivalent’ if there is a unitary mapping U:H1→H2 which intertwines the representations, that is, for each A∈A, Uπ1(A)=π2(A)U. The specified unitary mapping, U, sets up a way of translating between density operator states on H1 and density operator states on H2, and between observables in B(H1) and observables in B(H2). The translated states reproduce the same expectation values on the translated observables that the original states assigned to the original observables. The Stone–von Neumann theorem tells us that for systems with finitely many degrees of freedom, the usual Schrödinger representation is the unique irreducible representation of the algebra of observables up to unitary equivalence.6 This means that for ordinary, non-relativistic quantum theories, it does not make any difference to physics whether we work in the abstract algebra or one of its representations because there is only one suitable representation available. But it is well known that the assumptions of the Stone–von Neumann theorem fail for the algebras used in quantum field theory and quantum statistical mechanics in the thermodynamic limit because these theories describe systems with infinitely many degrees of freedom. In this case, there exist unitarily inequivalent irreducible representations of the relevant algebra of observables, which appear to provide us with inequivalent formulations of the theory. These unitarily inequivalent Hilbert spaces representations appear to be distinct quantum theories because the set of states that can be represented as density operators on a given irreducible Hilbert space representation is disjoint from the set of states that can be represented as density operators on a unitarily inequivalent irreducible Hilbert space representation (Kadison and Ringrose [1997], p. 741). Different Hilbert spaces seem to specify completely different collections of ways the world might be. Luckily, even for systems with infinitely many degrees of freedom, we always know that we can construct a Hilbert space theory if we desire. One of the most fundamental results in the theory of C*-algebras, known as the GNS theorem (Kadison and Ringrose [1997], p. 278, Theorem 4.5.2), asserts that for each state ω on A, there exists a representation (πω,Hω) of A, known as the ‘GNS representation for ω’, and a (cyclic) vector Ωω∈Hω, such that for all A∈A, ω(A)=⟨Ωω,πω(A)Ωω⟩. The GNS representation for state ω is unique in the sense that any other representation (π,H) of A containing a cyclic vector corresponding to ω is unitarily equivalent to (πω,Hω) (Kadison and Ringrose [1997], p. 279, Proposition 4.5.3). This means that one can always start with an algebraic theory and get back to an ordinary Hilbert space theory by using the GNS construction; for example, the Minkowski quanta Fock space representation of a quantum field system can be constructed as the GNS representation for the Minkowski vacuum state (Clifton and Halvorson [2001], p. 436). However, it is important to note that for systems with infinitely many degrees of freedom, even the GNS representation does not give a unique way of generating a representation for algebra A. Even though the GNS representation of A for each state is unique, by cycling through the GNS representations of different states on the abstract algebra, one can generate unitarily inequivalent representations. For example, by taking the GNS representation for the Rindler vacuum state, one can construct a Fock space representation that is unitarily inequivalent to the Minkowski quanta Fock space representation (Clifton and Halvorson [2001], p. 440) How, then, are we supposed to interpret infinite quantum theories like quantum field theory and quantum statistical mechanics? According to Ruetsche, to interpret a physical theory is to specify the collection of worlds that the theory deems possible (Ruetsche [2011], p. 6). Although I do not endorse this view of interpretation,7 I will accept it provisionally in this article. For our purposes, then, an interpretation will consist in what Ruetsche calls a kinematic pair (Ruetsche [2011], p. 35) specifying the physically measurable quantities and the physically possible states of a system. The presence of unitarily inequivalent representations in quantum theories with infinitely many degrees of freedom gives rise to the three interpretations discussed in the next section.8 3 Three Extremist Interpretations 3.1 Algebraic imperialism First, one can be an algebraic imperialist by asserting that a quantum theory is given in full by the abstract algebra of observables and the states on that algebra rather than its Hilbert space representations. The abstract algebra captures a structure that all Hilbert space representations have in common, so the algebraic imperialist chooses to focus only on this structure. To do so is to proclaim that all the work that has been done on interpreting the Hilbert space formalism for ordinary quantum mechanics with finitely many degrees of freedom cannot yield a complete and adequate interpretation for the case of infinitely many degrees of freedom. According to the algebraic imperialist, ‘the extra structure one obtains along with a concrete representation of [ A] is extraneous’ (Ruetsche [2011], p. 132). All that matters is the abstract algebraic structure. The physically measurable quantities are given by the observables (self-adjoint elements) in A, and the physically possible states are given by the states on A. 3.2 Hilbert space conservatism On the other hand, if one wants to be a Hilbert space conservative and maintain an interpretation via the Hilbert space formalism like those usually discussed for ordinary quantum mechanics, then one must pick a particular Hilbert space representation to interpret. When one is working in the context of a particular Hilbert space, H, one can define the weak operator topology on B(H) by the following criterion for convergence (Reed and Simon [1980], p. 183): a net {Ai} converges in the weak operator topology to A just in case for all φ,ψ∈H, ⟨φ,Aiψ⟩→⟨φ,Aψ⟩ in C. Recall that the GNS theorem allows us to take any C*-algebra of observables A and, having chosen some state ω, represent it via the representation πω as a sub-algebra πω(A)⊆B(Hω) for some Hilbert space, Hω. Using the weak operator topology on B(Hω) as a physically relevant notion of approximation,9 one can include in any algebra of observables the operators that are physically indistinguishable from or arbitrarily well approximated by the observables already picked out in our algebra. To do so, we take the weak operator closure (written πω(A)¯) of our original algebra of observables πω(A)—this adds to our original algebra all limit points of weak operator converging nets. Any weak operator closed sub-algebra of a Hilbert space is called a von Neumann algebra, so πω(A)¯ will be called the von Neumann algebra affiliated with the representation πω. If the state ω that we used to take our GNS representation is pure,10 then because the representation πω is irreducible (Kadison and Ringrose [1997], p. 728, Theorem 10.2.3), it follows that πω(A)¯=B(Hω) (see Sakai [1971], Proposition 1.21.9, p. 52). So taking the GNS representation for a pure state and closing in the weak operator topology brings us back to the familiar situation where our observables are all of the bounded self-adjoint operators on a Hilbert space. For the Hilbert space conservative, one such particular irreducible Hilbert space representation of the abstract algebra specifies the physical possibilities. Having chosen a pure state, ω on A, and obtained its GNS representation (πω,Hω), the Hilbert space conservative follows the standard practice in ordinary quantum mechanics (for finitely many degrees of freedom). For the Hilbert space conservative, the physically measurable quantities are the self-adjoint elements of πω(A)¯=B(Hω), and the physically possible states are density operators on Hω. Ruetsche argues that Hilbert Space Conservatism is inadequate because it does not give us access to enough states. In theories with infinitely many degrees of freedom, the existence of unitarily inequivalent representations entails that for any privileged irreducible representation (π,H), there is some algebraic state that cannot be implemented as a density operator on H. This would be fine if we only ever needed the density operator states on a single Hilbert space to accomplish the goals of physics, but there are instances in which we need to appeal to two states that cannot be represented as density operators on the same irreducible Hilbert space representation in the course of giving a single physically significant explanation (Ruetsche [2003], [2006]). For example, states of different pure thermodynamic phases cannot be represented as density operators on the same irreducible Hilbert space representation. But certainly we need to be able to account for states of different pure thermodynamic phases as simultaneously physically possible in order to explain phase transitions in quantum statistical mechanics. So the Hilbert space conservative lacks the resources to recover such physically significant explanations.11 This may push the Hilbert space conservative to try to find a Hilbert space on which all states can be represented as density operators. The universalist seeks to do precisely this. 3.3 Universalism The universalist agrees with the Hilbert space conservative that we need a Hilbert space representation of the abstract algebra A to interpret our quantum theory, but disagrees that we need an irreducible representation. The universalist holds that the ‘universal representation’ is the privileged representation of the abstract algebra. Letting SA denote the set of states on A and letting (πω,Hω) denote the GNS representation of any ω∈SA, the universal representation is given by (πU,HU), where HU=⊕ω∈SAHω is the universal Hilbert space and for each A∈A, πU(A)=⊕ω∈SAπω(A). The universal representation is guaranteed to be faithful (Kadison and Ringrose [1997], p. 281, Remark 4.5.8), so every element A∈A has a unique counterpart in πU(A). The universalist, like the Hilbert space conservative, has access to more observables than the algebraic imperialist. The universal Hilbert space carries its own weak operator topology defined precisely as above, which allows us to take the weak operator closure, πU(A)¯, of our original algebra, thereby obtaining ‘the universal enveloping von Neumann algebra’ of A. Since πU is reducible, it follows that πU(A)⊆πU(A)¯⊊B(HU). For the universalist, the physically measurable quantities are the self-adjoint elements of πU(A)¯, and the physically possible states are density operators on HU. Furthermore, every state on the abstract algebra A can be represented by a density operator (in fact, a vector and hence a finite rank density operator; see Kadison and Ringrose [1997], p. 281, Remark 4.5.8) on the universal Hilbert space HU, so the universalist has access to as many states as there are on the abstract algebra A and avoids one of the pitfalls of the Hilbert space conservative. It is worth noting that although universalism uses a Hilbert space representation to think about states and observables, it differs crucially from the ways we are accustomed to thinking about these mathematical tools in ordinary quantum mechanics, and thus it may not give the Hilbert space conservative everything she desired. First, in ordinary quantum mechanics we are accustomed to working in an irreducible representation of the algebra of observables. We have seen that the universal representation is manifestly reducible. Second, in ordinary quantum mechanics we are accustomed to thinking of vector states as pure states (because we work in an irreducible representation). Vector states on the universal representation, or universal enveloping von Neumann algebra, are not necessarily pure states. In fact, every mixed state on A can be thought of as a vector state as well. Third, in ordinary quantum mechanics we are accustomed to working in a separable Hilbert space. Because the universal Hilbert space is the direct sum of an uncountable number of non-trivial Hilbert spaces, it follows that the universal Hilbert space is non-separable. These are certainly important differences from the mathematical tools that we use in ordinary quantum mechanics, and I save further discussion of their significance for future work. 4 Parochial Observables Ruetsche’s ‘problem of parochial observables’ begins as an argument against algebraic imperialism and at times is extended to an argument against universalism. The basic claim of the argument is that the algebraic imperialist does not have the resources to represent all of the physically possible observables (Ruetsche [2002], p. 367, [2003], p. 1330). We saw that the Hilbert space conservative, having privileged some pure state, ω, and its GNS representation (πω,Hω), acquires all of the observables in πω(A)¯=B(Hω). The new observables are the ones that Ruetsche calls ‘parochial observables’; these are limit points in the weak operator topology of nets of observables from the original algebra and so they can be thought of as approximations to, or idealizations from, the observables we already recognized.12 Many of these observables have real physical import but have no analogue in the abstract algebra. So the Hilbert space conservative gains access to more observables than the algebraic imperialist. These observables are physically significant for giving explanations of thermodynamic phase transitions, as just one example. So, according to Ruetsche, the algebraic imperialist runs into a problem because she cannot recognize these operators as physically possible observables, and so cannot vindicate such explanations. It may be helpful to see the problem of parochial observables in some more detail in the context of an example: the infinite one-dimensional spin chain. To construct the spin chain, one assigns Pauli spin operators σxk,σyk,σzk to each point, k, in the lattice, Z, of the integers. These operators are required to satisfy the usual canonical anti-commutation relations. The relevant algebra, A, for the total system is the inductive limit (quasilocal algebra) of algebras of spin operators assigned to finite regions of the chain (Ruetsche [2006], p. 477). In order to describe and explain the phenomenon of broken symmetry of the spin chain—the fact that nature appears to select a preferred direction in space despite the invariance of the laws governing the system under rotations—one appeals to the global magnetization observable (which specifies the aforementioned preferred direction in space). We can define the global magnetization in the z-direction by starting with the average magnetization in the z-direction over finite regions mn=12n+1∑k=−nk=+nσzk. Taking the limit as n goes to infinity yields the desired global observable. The algebraic imperialist hits a roadblock here because this sequence does not converge in the norm topology on A, so there is no global magnetization in the abstract algebra. On the other hand, the global magnetization does converge in the weak operator topology of some representations. For example, consider the GNS representation of the state representing determinate spins aligned in the positive z-direction; the global magnetization does converge in the weak operator topology of this representation and so resides in B(H), where H is the representing Hilbert space. Since only the Hilbert space conservative has access to the weak operator topology on a representation, it appears that we need to use the resources of the Hilbert space conservative in order to understand the global magnetization as a limit of observables we already recognized. I will argue, contra Ruetsche, that there is a way for both the imperialist and the universalist to account for parochial observables. The problem of parochial observables only appears because we have given the Conservative more tools than the imperialist—specifically, tools for representing idealizations and approximations. Once we give the imperialist the analogous tools on the abstract algebra, she has no trouble representing the parochial observables. We will see that when the imperialist is allowed access to a topology that is analogous to the weak operator topology induced by a representation, she too gains access to all of the parochial observables by using the relevant limiting procedures. 4.1 Parochial observables for the imperialist When one thinks of an abstract C*-algebra, one usually thinks of it as coming equipped with the topology induced by its norm. This, of course, corresponds to the uniform topology of a concrete C*-algebra of operators acting on a Hilbert space. But just as one can consider alternative topologies on concrete algebras of operators, one can consider alternative topologies on the abstract algebra prior to taking a representation. One of the alternative topologies on the abstract algebra corresponds, in a certain sense, to the algebraic translation of the weak operator topology. To motivate this, we must first think about the significance of the weak operator topology. I stated in the previous section that the weak operator topology gives us a criterion of convergence based on expectation values and transition probabilities, and so gives us a notion of approximation relevant to the empirical content of the theory. But this can be made more precise. The following proposition shows that the weak operator topology of a representation is the topology for convergence of expectation values with respect to a privileged collection of states—namely, the finite rank density operators on a Hilbert space representation.13 Proposition 1 Let (π,H) be a representation of a C*-algebra, A. Let {Ai}⊆π(A) be a net of operators. Then the following are equivalent: The net {Ai} converges to A∈B(H) in the weak operator topology on B(H). For all states φ on A implementable by a finite rank density operator Φ on H, φ(Ai)=Tr(ΦAi) converges to Tr(ΦA) in C. One remark before we proceed: in clause (2), the expectation values converge as complex numbers even though there is, in general, no element of the abstract algebra to which the net of observables converges. This is because the parochial observable (the limit point of the net) will not, in general, have an analogue in the abstract algebra. Proposition 1 shows that the weak operator topology on a representation of A gives us a notion of approximation relevant to only those states that can be implemented as finite rank density operators in this representation. A Hilbert space conservative might have reason to restrict attention to these states, but an algebraic imperialist does not. The algebraic imperialist sees all states on the abstract algebra as on a par and believes that the empirical content of the theory comes from all expectation values. However, the algebraic imperialist has access to a different topology that defines a notion of convergence using the expectation values of all of the states she deems physically possible. The ‘weak ( σ(A,A*)) topology’ on an abstract C*-algebra A is defined as follows14: Let A* denote the dual space of bounded linear functionals on the Banach space, A. A net, {Ai}∈A, converges in the weak ( σ(A,A*)) topology to A∈A just in case, for all φ∈A*, φ(Ai)→φ(A) in C. The notation σ(A,A*) signifies that the weak topology is the weakest topology on A that makes all of the bounded linear functionals in A* continuous. Now we see from Proposition 1 that the notion of convergence given by the weak operator topology on a representation is simply the notion of convergence that we get by using the weak ( σ(A,A*)) topology with attention restricted to a particular set of states—those that can be represented by finite rank density operators. Likewise, the weak ( σ(A,A*)) topology is the analogue of the weak operator topology for the abstract algebra, in the sense that it is an appropriate generalization to a notion of convergence with respect to the expectation values of all states on the abstract algebra. Insofar as the Hilbert space conservative is justified in using the weak operator topology as a physically relevant standard of approximation or idealization, the algebraic imperialist is justified in using the weak ( σ(A,A*)) topology as a physically relevant standard of approximation or idealization too. The imperialist is concerned with approximation and idealization with respect to the expectation values of all states on the abstract algebra.15 There is a sense in which the abstract algebra A and its representations are not complete with respect to the weak and weak operator topologies, respectively (even though they are both complete with respect to the norm topology). There are nets of observables whose relevant expectation values converge in C, but which have no limit point in the abstract algebra. To find those limit points, we must think of the algebra of observables as living in some kind of ambient space of observables; for the Hilbert space conservative, this is just the collection B(H) of operators on some Hilbert space that the imperialist eschews. But the imperialist also has access to an ambient space of observables of her own, and we will see that this ambient space allows us to find the limit points of nets of observables in the weak ( σ(A,A*)) topology, just as B(H) allows the Hilbert space conservative to find the limit points of nets of observables in the weak operator topology. Recall that the ‘dual’ of any Banach space X, written X*, is the set of continuous linear functionals on X (in the norm topology), and the ‘bidual’ of X, written X**, is the set of continuous linear functionals on X*. The original space, X, can be embedded in X** via the canonical evaluation map J:X→X** given by x∈X↦x^∈X**, where we define x^ by x^(l)=l(x) for all l∈X*. When X is finite-dimensional, J always provides an isomorphism; but, in general, when X is infinite-dimensional, J may not be an isomorphism because it may not be onto. The bidual A** of the abstract algebra A provides the ambient space of observables in which to look for limit points or idealizations of observables from the original algebra. The elements of A** can be thought of as observables because for each state ω on A, we can think of the expectation value of any A∈A** as the value A(ω). Notice that this makes sense because for any element of our original algebra A∈A, letting A^=J(A)∈A**, we see that A^(ω)=ω(A) is the expectation value of A in the state ω. Thinking of observables and expectation values in the bidual like this suggests a way of extending the weak topology of A to the bidual A**: focus on convergence of expectation values with respect to the same collection of states now considered as states on an enlarged algebra of observables. This brings us to the ‘weak* ( σ(A**,A*)) topology’ on the bidual A**. A net {Ai}∈A** converges in the weak* ( σ(A**,A*)) topology to A∈A** just in case for all φ∈A* , Ai(φ)→A(φ) in C. The notation σ(A**,A*) signifies that the weak* topology on the bidual is the weakest topology that makes all of the bounded linear functionals in A* continuous when considered as linear functionals on A** (that is, when the elements of A* are considered as elements of A*** by the canonical evaluation map). The weak* topology on A** is the natural extension of the weak topology on A because it makes precisely the same linear functionals, and more specifically states, continuous—namely, the linear functionals and states in A*. Furthermore, each state on A has a unique continuous extension to a state on A** in the weak* topology. The following proposition shows that every element of the bidual A** can be understood as a limit point of a net of observables in the abstract algebra A in the weak* ( σ(A**,A*)) topology on the bidual A**. This shows that elements of the bidual can be thought of as approximations to, or idealizations from, our original observables in a topology that the algebraic imperialist deems physically relevant. Proposition 2 Let A be a C*-algebra, A** be its bidual, and J:A→A** be the canonical evaluation map. Then J(A) is dense in A** in the weak* ( σ(A**,A*)) topology. Just as the Hilbert space conservative, upon adding limit points in the weak operator topology of some irreducible representation, arrives at the algebra B(H) of bounded operators on the Hilbert space, H, the algebraic imperialist, upon adding limit points in the weak (or really weak*) topology, arrives at the collection of observables, A**. When allowed the same methods for constructing idealizations, the algebraic imperialist gains access to more observables than reside in abstract algebra A. Now that we have given the algebraic imperialist access to more observables than reside in abstract algebra A, we must ask: does the algebraic imperialist gain access to the observables that Ruetsche argued are physically significant for giving explanations? Recall that the reason the problem of parochial observables was supposed to be a problem was that the algebraic imperialist appears to not allow us to reconstruct the physics of, say, phase transitions and symmetry breaking. The idealizations the Hilbert space conservative constructs are, in a certain sense, the ‘right’ ones because they allow us to give these physically significant explanations. We need to check that the observables the algebraic imperialist constructs by moving to the bidual have enough structure to be able to recover those physically significant explanations, too. First, notice that although A** is initially only a Banach space, it can be made into a C*-algebra by defining multiplication and involution operations. We get these operations by Proposition 2 as the unique extensions of the multiplication and involution operations on the algebra J(A) (with algebraic structure inherited from A) such that multiplication is separately continuous in its individual arguments in the weak* topology and involution is continuous in its only argument in the weak* topology.16 When we refer to C*-algebraic structure on A** in what follows, these are the operations we will have in mind. The following proposition shows that the algebraic imperialist, using this structure on the bidual, has access to every idealized parochial observable to which the Hilbert space conservative has access (for each possible distinct privileged representation). Proposition 3 If π is a representation of a C*-algebra A, then there is a central projection P∈A** and a *-isomorphism α from A**P onto π(A)¯ such that π(A)=α(J(A)P) for all A∈A.17 So the von Neumann algebra affiliated with any representation of A is canonically isomorphic to a sub-algebra of A**. This shows that every parochial observable can be thought of as an element of the bidual A** and so can be thought of as an idealization from observables in the abstract algebra that we arrive at by a notion of approximation or idealization that is physically relevant by the imperialist’s lights. The algebraic imperialist, when allowed the same tools for constructing idealizations that we allowed the Hilbert space conservative, has access to all of the parochial observables—such as net magnetization and particle number—that we need to give the kinds of physically significant explanations Ruetsche is worried about.18,19 Having presented the central technical results that provide the algebraic imperialist with the parochial observables, it’s worth pausing for a moment to remark on the kind of imperialism we’ve ended up with. One might want to distinguish between two different kinds of algebraic imperialists specifying different interpretations via different kinematic pairs: one algebraic imperialist who believes the physically significant observables reside in A, and another who believes the physically significant observables reside in A**. This article provides an argument that the latter algebraic imperialist, who chooses A** as her algebra of observables,20 overcomes the problem of parochial observables. In this respect, my preferred interpretation differs from that of many of the original algebraic imperialists in the physics community who explicitly disavowed parochial observables (for example, Haag and Kastler [1964], p. 853). For example, rather than seeing all representations as specifying the same physical possibilities as did Haag and Kastler ([1964], p. 851),21 on my view a representation focuses our gaze on a particular collection of states and observables constituting only a small subset of the physically possible states and observables of the theory.22 My interpretation also differs from that of Segal ([1959], pp. 348–9), who presents a version of algebraic imperialism in which the norm topology on A is centrally important because of its operational significance. I have shown that even if one takes the norm topology to be physically significant, one has reason to take the weak Banach space topology on A as physically (or even operationally) significant as well. Because of this, the algebraic imperialist need not restrict attention to the algebra A, which is only complete in the norm topology but not the weak topology. I believe Proposition 2 justifies the algebraic imperialist in moving to the larger algebra A** for exactly the same reasons the Hilbert space conservative is justified in moving from π(A) to the larger algebra π(A)¯.23 And we have seen that the algebraic imperialist acquires all parochial observables as elements of A**. 4.2 Parochial observables for the universalist When we think about universalism, we do not need to go through the rigamarole of defining a new kind of topology as we did for imperialism in the previous section. The universal representation already comes with a topology that is precisely analogous to the weak operator topology used by the Hilbert space conservative, namely, the weak operator topology on the collection B(HU) of bounded operators on the universal Hilbert space, HU. But how does the weak operator topology of the universalist compare to the weak operator topology of the Hilbert space conservative? While the Hilbert space conservative restricts attention to a privileged collection of states from the larger collection of states on the abstract algebra, the universalist considers all states on the abstract algebra to be physically possible because they can all be implemented as finite rank density operators on the universal Hilbert space. In this sense, the universalist is just like the imperialist, so we expect their notions of approximation with respect to the empirical content of the theory to match up. As the following proposition shows, the notion of convergence of the weak operator topology on the universal representation corresponds exactly to the notion of convergence of the weak (or weak*) topology on the abstract algebra. Proposition 4 Let (πU,HU) be the universal representation of A. Let {Ai}⊆πU(A) be a net of operators. Then the following are equivalent: The net {Ai} converges to A∈B(HU) in the weak operator topology on B(HU). For all states φ on A, φ(Ai)=Tr(ΦAi) converges to Tr(ΦA) in C, where Φ is any finite rank density operator (and there is always at least one) implementing the state φ on πU(A). Just as for Proposition 1, in clause (2) the expectation values converge as complex numbers even though there is no element of the abstract algebra to which the net is converging. The collection of bounded operators on the universal Hilbert space provides the ambient space of observables in which to think about these limit points or idealizations. This allows us to construct the universal enveloping von Neumann algebra, which contains the idealizations of our original observables with respect to a notion of idealization that is physically relevant by the lights of the universalist. We also note that Proposition 4 shows a sense in which the universal representation is privileged if one wants to think algebraically—its weak operator topology reproduces the precise condition of convergence of the weak (or weak*) topology on abstract algebra A. Restricting attention to finite rank density operator states on the universal representation amounts to no restriction at all because every state is implementable as a finite rank density operator on the universal representation. We already knew that the universalist could acquire more observables by using the universal enveloping von Neumann algebra, but as in the previous section, we still need to ask whether the universalist acquires the right ones. Does the universalist have access to parochial observables like net magnetization and particle number? The following proposition shows that the universalist has access to every parochial observable that every possible (for each distinct privileged representation) Hilbert space conservative has access to (compare with Proposition 3). Proposition 5 If π is a representation of a C*-algebra A and πU is the universal representation of A, then there is a central projection, P, in πU(A)¯ and a *-isomorphism, α, from the von Neumann algebra πU(A)P¯ onto π(A)¯ such that π(A)=α(πU(A)P) for all A∈A.24 The von Neumann algebra affiliated with any representation is canonicaly isomorphic to a sub-algebra of the universal enveloping von Neumann algebra. This shows that every parochial observable can be thought of as an element of the universal enveloping von Neumann algebra, and so can also be thought of as an idealization from observables in the universal representation with respect to a notion of idealization that is physically relevant by the lights of the universalist. Just as in the previous section, the universalist gains access to all of the observables, including net magnetization and particle number, that we need to recover the physically significant explanations of concern to Ruetsche. Ruetsche, however, presents a number of objections to the claim that the universal representation contains all parochial observables. I will consider what I take to be two of the most prominent objections here and argue that they fail. Seeing why they fail illustrates how the universal representation gives us access to all parochial observables. First, Ruetsche ([2011], p. 145, Footnote 10) asserts that in the universal representation, we would expect an observable πω(A) from the GNS representation for ω to be implemented in the universal representation by a portmanteau operator of the form πω(A)⊕φ≠ω0 and, similarly, for the parochial observables from the representation πω, which will be weak operator limits of these portmanteau operators. But Ruetsche correctly argues that these operators may not be contained in the universal enveloping von Neumann algebra because it follows from (Kadison and Ringrose [1997], p. 738, Theorem 10.3.5) that πU(A)¯≠⊕ω∈SAπω(A)¯. This objection, however, fades in light of Proposition 5. The portmanteau operators are obviously intended to capture the observables from the GNS representation for ω in the universal representation. Proposition 5 explicitly asserts that we can think of any observable πω(A) in the GNS representation for ω as the observable πU(A)P in the universal representation (and similarly for parochial observables), where P is a central projection in the universal enveloping von Neumann algebra. Because of the presence of projection P, such an observable acts non-trivially only on some subspace of universal Hilbert space HU and acts like the zero operator everywhere else. This means that operator πU(A)P has some properties analogous to operator πω(A)⊕φ≠ω0. But since operators of the latter form are not required to belong to the universal enveloping von Neumann algebra, it follows that P may not be the projection on to the GNS representation for ω. I will show explicitly that P is not the projection onto the GNS representation for ω and that we should not be surprised by this fact.25 Intuitively, observable πU(A)P ought to act like πω(A) on all of the vector states that πω(A) acts on. But, in general, there will be many vectors in the universal Hilbert space, HU, that implement any vector state, ψ, corresponding to the vector Ψ∈Hω, which is the sort of vector that πω(A) acts on. For example, the vector Ψ⊕φ≠ω0 will implement state ψ in the universal representation. But so will the vector Ωψ⊕φ≠ψ0, where Ωψ is the cyclic vector implementing ψ in the GNS representation for ψ. In fact, there will be a vector implementing ψ in each direct summand of the universal representation that corresponds to a GNS representation unitarily equivalent to πω. And furthermore, if φ is any mixture having non-zero component on state ω, then the GNS representation of φ will be a direct sum containing the GNS representation of ω as one of its summands. It follows by similar considerations that there will be some vector in the GNS representation of φ (and hence in another summand of the universal representation) that implements the state ψ. The following proposition shows that the range of projection P must contain all of these vectors implementing state ψ. Proposition 6 Let (π,H) be a representation of a C*-algebra A, let (πU,HU) be the universal representation of A, and let P be the corresponding central projection in Proposition 5. Choose an arbitrary unit vector, Ψ∈H. Then for any vector, Φ∈HU, implementing vector state Ψ in the sense that ⟨Φ,πU(A)Φ⟩=⟨Ψ,π(A)Ψ⟩ for all A∈A, it follows that PΦ=Φ. This shows that the operators in the universal representation that correspond to operators from some particular GNS representation must take a more complicated form than the portmanteau operators Ruetsche suggests. In the universal representation, the correct operator must act on a whole host of vectors implementing the states from the GNS representation that we started with, and many of these vectors will lie elsewhere in the universal representation, outside of the GNS representation we began with. However, it is important to recognize that even though the elements of the universal representation do not take the simple form we might have expected, Proposition 5 guarantees us that there is always some operator in the universal enveloping von Neumann algebra corresponding to the observable in which we are interested. Ruetsche’s ([2011], p. 283) second objection claims that parochial observables (in particular, phase observables like the net magnetization of a ferromagnet (Ruetsche [2006], p. 478)) may have a domain that is a proper subset of SA and that no observable qua bounded operator on the entire universal Hilbert space can have this property. But we have already seen that parochial observables, by virtue of Proposition 5, can be thought of as acting on the subspace of HU that is the range of projection P given in that proposition. Parochial observables act like the zero operator everywhere else in HU and so we can think of the orthogonal complement of the range of P as the subspace generated by the collection of states that are outside the domain of that parochial observable. Hence, we have a way of making sense, in the universal representation, of Ruetsche’s claim that the domain of parochial observables may be smaller than SA. It may be helpful to consider a particular example. Ruetsche ([2011], p. 226) considers and rejects the universal representation as a way of accounting for inequivalent particle notions in quantum field theory.26,27 The universal representation gives rise to a total number operator representing the number of particles of any variety as an observable in the universal enveloping von Neumann algebra.28 Ruetsche argues that this total number operator does not suffice for doing physics because it identifies states with the same total number of quanta spread among the different varieties and, of course, a state with n quanta of one variety is different from a state with n quanta of a different variety. For example, the total number operator cannot distinguish the state with one Minkowski quantum from the state with one Rindler quantum because it tells us that both states simply have one total quantum.29 But Proposition 5 tells us that the universal representation will contain (in addition to the total number operator for all varieties) the Minkowski quanta total number operator and the Rindler quanta total number operator, which are parochial observables to particular GNS representations of the abstract algebra. The Minkowski quanta total number operator and the Rindler quanta total number operator will have different expectation values in the state with one Minkwoski quantum and the state with one Rindler quantum. Hence, they will distinguish between these two distinct states. The universal representation, by virtue of containing all of the parochial observables, gives us the ability to make as many distinctions between states as we might like. I hope that these remarks concerning Ruetsche’s objections suffice to show that all parochial observables really are contained in the universal representation or, more specifically, the universal enveloping von Neumann algebra. Now I want to remark upon the fact that the technical results I have presented for the imperialist's and universalist's solutions to the problem of parochial observables appear so similar—this is no coincidence. We have already mentioned that the universal representation (or, really, universal enveloping von Neumann algebra) and the abstract algebra (or, really, its bidual) share the same topological structure in the sense that the notion of convergence provided by the weak topology on A (or weak* topology on A**) reproduces precisely the notion of convergence provided by the weak operator topology on B(HU). But these objects share much more structure than that. The following proposition shows that the bidual of a C*-algebra in a certain sense carries the same algebraic structure as the universal enveloping von Neumann algebra. Proposition 7 There is a *-isomorphism, α, from bidual A** of C*-algebra A to its universal enveloping von Neumann algebra πU(A)¯ such that πU(A)=α(J(A)) for all A∈A.30,31 Since the imperialist and the universalist invoke the same algebraic and topological structure to represent quantum systems, they end up believing in the same physically significant observables and the same physically possible states, while using the same notion of approximation or idealization. This shows a sense in which algebraic imperialism and universalism amount to the same position. Of course, I do not claim that imperialism and universalism are equivalent with respect to every purpose to which we put quantum theory, but at least they are equivalent with respect to the interpretive uses just outlined. One may have pragmatic reasons for choosing one or the other—for example, one might want to use the universal representation if one is familiar with interpreting Hilbert spaces from ordinary quantum mechanics. Nevertheless, whatever quantum states and observables the imperialist can represent, the universalist can represent too (and vice versa). So the imperialist and the universalist, when allowed the same tools as the Hilbert space conservative for representing idealizations, come up with the same solution to the problem of parochial observables. 5 Conclusion I have argued that the algebraic imperialist and the universalist both have solutions to Ruetsche’s problem of parochial observables,32 and that these are, in a certain sense, equivalent interpretations of quantum theories. Proposition 3 shows that every parochial observable from every possible representation can be thought of as an element of bidual A** and Proposition 2 shows that each of these elements of A can be acquired as a limit (in the weak* topology) of some net of observables that the algebraic imperialist already recognized. Similarly, Proposition 5 shows that the universal enveloping von Neumann algebra, πU(A)¯, contains every parochial observable from every possible representation, and since all elements of πU(A)¯ can be constructed as the limit of some net of observables in πU(A), the universalist also acquires all parochial observables. Furthermore, Proposition 7 shows that these extended spaces of observables— A** for the imperialist and πU(A)¯ for the universalist—are isomorphic by the relevant notion of isomorphism, and so they specify the same physically measurable quantities for the system. This shows that algebraic imperialism and universalism are equivalent interpretations, and both of them give us the tools to represent all of the parochial observables. Since the problem of parochial observables is Ruetsche’s main reason for rejecting algebraic imperialism and universalism, and since we have seen that both of these positions can adequately solve this problem, this leaves the path open for imperialism and universalism as viable interpretations of quantum theories with infinitely many degrees of freedom. Appendix A: Proofs of Propositions This Appendix contains the proofs of propositions from the body of the article. Propositions 1 and 4 follow immediately from the following lemma, which is a restatement of facts described in (Reed and Simon [1980], p. 213). Lemma 1 Let H be a Hilbert space and let {Ai}⊆B(H) be a net of operators. Then the following are equivalent: The net {Ai} converges to A∈B(H) in the weak operator topology on B(H). For all finite rank density operators, ρ on H, Tr(ρAi) converges to Tr(ρA) in C. Proposition 1 Let (π,H) be a representation of a C*-algebra A. Let {Ai}⊆π(A) be a net of operators. Then the following are equivalent: The net {Ai} converges to A∈B(H) in the weak operator topology on B(H). For all states φ on A implementable by a finite rank density operator Φ on H, φ(Ai)=Tr(ΦAi) converges to Tr(ΦA) in C. Proof (1⇒2) This follows immediately by Lemma 1. (2⇒1) Suppose that for all states φ on A implementable by a finite rank density operator Φ on H, φ(Ai)=Tr(ΦAi) converges to Tr(ΦA) in C. Every finite rank density operator, ρ on H, defines a state on A implementable by a finite rank density operator. So it follows from the assumption that Tr(ρAi) converges to Tr(ρA) for all finite rank density operators and hence Ai converges to A by Lemma 1.□ Proposition 4 Let (πU,HU) be the universal representation of A. Let {Ai}⊆πU(A) be a net of operators. Then the following are equivalent: The net {Ai} converges to A∈B(HU) in the weak operator topology on B(HU). For all states φ on A, φ(Ai)=Tr(ΦAi) converges to Tr(ΦA) in C, where Φ is any finite rank density operator (and there is always at least one) implementing the state φ on πU(A). Proof ( 1⇒2) Suppose Ai→A in the weak operator topology on B(HU). Let Φ be any finite rank density operator (there is always at least one) implementing state φ on πU(A). Then by Proposition 1, φ(Ai)=Tr(ΦAi) converges to Tr(ΦA) in C. ( 2⇒1) Suppose that for all states φ on A, Tr(ΦAi) converges to Tr(ΦA) in C, where Φ is a finite rank density operator implementing φ. Let Φ be any finite rank density operator on H. Then X↦Tr(ΦX), for all X∈πU(A), defines a state on πU(A) and hence Tr(ΦAi) converges to Tr(ΦA) in C, which shows, by Proposition 1, that Ai converges in the weak operator topology to A. □ Proposition 2 is an immediate corollary of the following elementary lemma about Banach spaces. Lemma 2 Let X be a Banach space, X** its bidual, and J:X→X** the canonical evaluation map. Then J(X) is dense in X** in the weak* ( σ(X**,X*)) topology. Proof Let x∼∈X** and let U be an open neighborhood of x∼. We will show that U contains some element y=J(x)∈J(X). By the definition of the σ(X**,X*)-topology, there are linear functionals l1,...,ln∈X* and ε1,...,εn>0 such that ∩i=1n(x∼+N(li,εi))⊆U, where x∼+N(li,εi)={y∼∈X**:|y∼(li)−x∼(li)|<εi}. Now let N′(li,εi)={x∈X: |li(x)−x∼(li)|<εi}. We will show that ∩i=1nN′(li,εi)⊆∩i=1n(x∼+N(li,εi))∩J(X) is non-empty or, in other words, we will show that there is a y=J(x)∈J(X) such that, for all 1≤i≤n, |y(li)−x∼(li)|=|li(x)−x∼(li)|<εi. (1) It suffices to consider only a linearly independent subset of the linear functionals l1,...,ln∈X* forming a basis for the subspace of X* spanned by these functionals: if x∈X satisfies the above inequality for this linearly independent subset of l1,...,ln∈X*, then it must also satisfy the inequalities for the rest of the linear functionals, or else the inequalities would contradict each other and then ∩i=1n(x∼+N(li,εi)) would have to be empty, which it is not because it contains x∼. Choose this linearly independent set of functionals lk1,...,lkm. We know (Schechter [2001], p. 93, Lemma 4.14) that there exists a dual basis, e1,...,em∈X, for a subspace of X such that lki(ej)=δij for all 1≤i,j≤m. Consider the vector x=∑i=1mx∼(lki)ei. We have for all 1≤j≤m, lj(x)=∑i=1mx∼(lki)lj(ei)=x∼(lkj). Hence, x satisfies the above inequalities in Equation (1) and it follows that y=J(x)∈∩i=1nN′(li,εi)⊆∩i=1n(x∼+N(li,εi))⊆U. Therefore, since U∩J(X)≠Ø, J(X) is dense in X**. □ Proposition 2 Let A be a C*-algebra, A** be its bidual, and J:A→A** be the canonical evaluation map. Then J(A) is dense in A** in the weak* ( σ(A**,A*)) topology. Proof Since every C*-algebra A is a Banach space, this follows immediately from Lemma 2.□ Proposition 3 is an immediate corollary of Proposition 5 and Proposition 7, whose proofs are contained in (Kadison and Ringrose [1997], p. 719, Theorem 10.1.12) and ([1997], p. 726, Proposition 10.1.21), respectively. Proposition 3 If π is a representation of C*-algebra A, then there is a central projection P∈A** and a *-isomorphism α from A**P onto π(A)¯ such that π(A)=α(J(A)P) for all A∈A. Proof By Proposition 5, there is a projection, P∼, in the centre of πU(A)¯ and a *-isomorphism, α1, from πU(A)¯P∼ to π(A) such that π(A)=α1(πU(A)P∼) for all A∈A. By Proposition 7, there is a *-isomorphism, α2, from the bidual A** to πU(A)¯ such that πU(A)=α2(J(A)) for all A∈A. Projection P=α2−1(P∼) and *-isomorphism α=α1°(α2)|A**P serve as a witness to the current theorem, because for any A∈A, α(J(A)P)=α1°α2(J(A)P)=α1[α2(J(A))·α2(P)]=α1(πU(A)P∼)=π(A). □ Proposition 6 Let (π,H) be a representation of C*-algebra A and let (πU,HU) be the universal representation of A. Let P be the central projection in Proposition 5, and choose some unit vector, Ψ∈H. Then for any vector Φ∈HU, implementing vector state Ψ in the sense that ⟨Φ,πU(A)Φ⟩=⟨Ψ,π(A)Ψ⟩ for all A∈A, it follows that PΦ=Φ. Proof By construction (see Kadison and Ringrose [1997], p. 443, Theorem 6.8.8, [1997], p. 719, Theorem 10.1.12), projection P takes the form P=I−E, E=⋁T∈Ker(β¯)R(T*), where β¯ is the ultra-weakly continuous extension of the map β=π°πU−1 and R(T*) is the projection on to the range of T*. With Φ and Ψ as above, PΦ=Φ−EΦ. It suffices to show that ⟨Φ,EΦ⟩=0, which shows that the second term above is zero. We know that E=⋁T∈Ker(β¯)R(T*)∈Ker(β¯), so it follows that ⟨Ψ,β¯(E)Ψ⟩=0. Choose a net, Ai∈A, such that πU(Ai) converges in the weak operator topology on B(HU) to E. It follows immediately that π(Ai)=β¯(πU(Ai)) converges in the weak operator topology on B(H) to β¯(E), so ⟨Φ,EΦ⟩=⟨Φ,w−lim(πU(Ai))Φ⟩ =lim⟨Φ,πU(Ai)Φ⟩ =lim⟨Ψ,π(Ai)Ψ⟩ =⟨Ψ,w−lim(π(Ai))Ψ⟩ =⟨Ψ,β¯(E)Ψ⟩=0, where w−lim denotes the weak operator limit in the relevant Hilbert space. □ Appendix B: An Illustration in Classical Systems Suppose that the system under consideration is classical so that A is abelian. Examining the weak topology and weak operator topologies of representations of this algebra illustrates the concepts of Section 4 in a somewhat more familiar and concrete setting (although admittedly not the most familiar or concrete!).33 This appendix aims at an audience who is familiar with the different notions of convergence on spaces of functions; I show that the topologies on C*-algebras and their representations are simply extensions of these familiar notions of convergence to the non-commutative setting. Recall that when A is abelian, it is *-isomorphic to C(P(A)), the continuous functions on the compact Hausdorff space, P(A), of pure states of A with the weak* ( σ(P(A), A)) topology (Kadison and Ringrose [1997], p. 270, Theorem 4.4.3). As such, each observable A∈A corresponds to a function A^∈C(P(A)) defined by A^(ω)=ω(A) for each pure state ω∈P(A). Taking a representation of A amounts to choosing a measure on the space P(A) (see Kadison and Ringrose [1997], p. 744; Landsman [1998], p. 55), which defines an (L2) inner product, hence constructing a Hilbert space as follows: By the Riesz-Markov theorem (Reed and Simon [1980], p. 107, Theorem IV.14), each state ω on A corresponds to unique regular Borel measure μω on P(A) such that for all A∈A, ω(A)=∫P(A)A^dμω The GNS representation of A for the state ω is unitarily equivalent to the representation (πω,Hω) on Hilbert space Hω=L2(P(A),dμω),34 with πω defined by πω:A↦MA^, where the operator MA^ is defined as multiplication by the function A^, that is, for any ψ∈Hω, MA^ψ=A^·ψ. Now, we can pull the discussion of topologies on A back to the more familiar topologies on ordinary functions on P(A). We recall these topologies now before proceeding. Let fn be a net of functions on P(A). We say that fn converges to the function f ‘uniformly’ if supψ∈P(A)|fn(ψ)−f(ψ)| converges to zero in C. We say that fn converges to the function f ‘pointwise’ if fn(ψ) converges to f(ψ) in C for each ψ∈P(A). And finally, we say that fn converges to the function f ‘pointwise almost everywhere’ with respect to measure μ on P(A) if fn(ψ) converges to f(ψ) in C for all ψ∈P(A), except possibly on a set of measure zero with respect to μ. Now it is easy to show that for the GNS representation, (πω,Hω), the weak operator topology on B(Hω) is the topology of pointwise convergence almost everywhere with respect to measure μω. This implies that while πω(A) is the collection of multiplication operators by continuous functions (which is uniformly closed), its weak operator closure πω(A) will be the collection of multiplication operators by essentially bounded measurable functions with respect to measure μω (that is, L∞(P(A),dμω)). In some cases, taking the weak operator closure (and hence, moving to the essentially bounded measurable functions) does not give rise to any new parochial observables. When ω is pure, ω(A)=A^(ω)=∫P(A)A^δ(ω) for all A∈A, where δ(ω) is the point mass or delta function centred on ω. It follows by the uniqueness clause of the Riesz–Markov theorem that dμω=δ(ω). So every vector ψ∈Hω will be defined by a single complex number—the value of ψ on ω∈P(A) and Hω will be one-dimensional. Hence, B(Hω) will be one-dimensional and since πω(A) contains the identity and is closed under scalar multiplication, it follows that πω(A)¯=B(Hω)=πω(A). This means that there are no parochial observables in this representation. Furthermore, even the GNS representations for many mixed states do not give rise to new parochial observables. Consider arbitrary state ω on A such that μω has support on only a countable subset of P(A). In such a special state, since the measure focuses our attention on only a countable subset of P(A), the continuous functions coincide with the essentially bounded measurable functions—every discontinuous but essentially bounded measurable function is equivalent to a continuous function when we ignore differences on sets of measure zero. In other words, focusing only on a countable subset of P(A) does not allow one to distinguish between continuous and merely bounded functions. So it similarly follows that πω(A)¯=πω(A) and there are no parochial observables in this representation. However, we can also have an arbitrary mixed state, ω on A, such that μω has support on an uncountable subset of P(A). In this case, we may acquire new parochial observables. The weak operator closure will include even discontinuous functions like characteristic functions (projection operators), where the original algebra did not. Since each one of these essentially bounded measurable functions is the weak operator limit point of a collection of continuous functions, we can understand them as idealizations from, or approximations to, collections of our original observables. So the essentially bounded (but discontinuous) measurable functions with respect to some measure are the parochial observables for an algebra of continuous functions in the representation defined by that measure. But, comes the obvious retort, in a similar sense every bounded (Borel) function (without considering any measure) can be considered as an idealization from, or approximation to, a collection of our original observables without appeal to any measure, and hence without appeal to any representation. The sense in which this is true uses the weak topology on A. In particular, every bounded function is the pointwise limit of a collection of continuous functions. The topology of pointwise convergence for functions is just the weak topology on A (really, extended to the weak* topology on A**), so A** is just the collection of bounded (Borel) functions on P(A). Since every parochial observable qua essentially bounded function is equivalent to a bounded (Borel) function (ignoring differences on sets of measure zero), it follows that every parochial observable can be thought of as the weak (pointwise) limit of observables in the abstract algebra. This provides our algebraic route to all of the parochial observables at once, without reference to any representation. On the other hand, we can gather all the parochial observables in a single representation, as in the previous appendix, by taking the universal representation (see Kadison and Ringrose [1997], p. 746). The universal representation acts on Hilbert space HU=⊕ω∈S(A)L2(P(A),dμω) by the representation πU(A)=⊕ω∈S(A)MA^|supp(dμω), where MA^|supp(dμω) is the multiplication operator by the restriction of A^ to the support of dμω because that is equivalent to MA^ on Hω. Operator πU(A), in a certain sense, amounts to the multiplication operator by A^ everywhere because we have taken the direct sum over spaces with all possible regular normalized Borel measures, and so for each point ω∈P(A) there is some summand in which {ω} gets assigned non-zero measure. This is why πU gives a faithful representation, whereas each individual GNS representation is not faithful (because functions that disagree only on a set of measure zero are mapped to the same multiplication operator by that GNS representation). Weak operator convergence in the universal representation is just pointwise convergence everywhere again because each point gets assigned non-zero measure by at least one of the regular normalized Borel measures defining the Hilbert space summands. Thus the weak operator closure in the universal representation gives us back all of the bounded functions as direct sums of essentially bounded measurable functions considered over all possible regular normalized Borel measures. The upshot is that the abstract algebra and its universal representation give us two routes to acquiring all of the parochial observables, which in this case are the bounded functions. One can stay with the abstract algebra and use the weak topology, which defines the topology of pointwise convergence. Or one can use the universal representation and its weak operator topology, which similarly defines the topology of pointwise convergence. Either way, we start with the continuous functions and construct certain discontinuous idealizations from them. I hope that this special example may take some of the mystery out of the parochial observables and where they come from. Funding This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under NSF grant no. DGE-1321846. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. Acknowledgments I would like to thank Tracy Lupher, Laura Ruetsche, an anonymous referee, and especially Jim Weatherall for extremely helpful comments and discussions concerning this article that lead to many improvements. I would also like to thank Sam Fletcher for motivating me to think more deeply about the significance of the topologies used in quantum theories. Footnotes 1 Of course, these are not the only possible interpretations of algebraic quantum theories (see Ruetsche [2011], Chapter 6), but we will restrict our attention to these positions for the purpose of this article. 2 In what follows, Hilbert Space Conservatism will mainly play the role of a foil to help us understand imperialism and universalism. We will mention Ruetsche’s argument against Conservatism only briefly. 3 For mathematical reviews of the Stone–von Neumann theorem, see (Petz [1990]; Summers [1999]). For philosophical discussions, see (Ruetsche [2011], p. 41; Clifton and Halvorson [2001], p. 427). 4 For more on the theory of C*-algebras and their representations, see (Kadison and Ringrose [1997]). For more on algebraic quantum field theory, see (Ruetsche [2011]; Halvorson [2006]). 5 See (Ruetsche [2011], Chapter 2.2; Clifton and Halvorson [2001], Section 2.2–2.3) for more on unitary equivalence as a notion of ‘sameness of representations’. 6 The Stone–von Neumann theorem carries other substantive assumptions as well. It assumes that the representation is continuous in an appropriate sense and that the phase space of the classical theory is symplectic (see Ruetsche [2011], Chapters 2–3). 7 In fact, I believe one can use the considerations of this article to demonstrate why this is a misleading framework for talking about interpretation, but I save this for future work. 8 For more on these positions and their advantages and disadvantages, see (Arageorgis [unpublished]; Ruetsche [2002], [2003], [2006], [2011], Chapter 6). Of course, as Ruetsche describes, there are many more subtle interpretive options, but we deal here only with three of the simplest cases. 9 The motivation for this standard practice is that in the weak operator topology, a net of observables well approximates (that is, converges to) another observable just in case it approximates it with respect to all possible expectation values and transition probabilities, and hence with respect to the empirical predictions of the theory. We will discuss the significance of this in the next section. 10 A state ω is ‘pure’ if whenever ω=a1ω1+a2ω2 for states ω1,ω2, it follows that ω1=ω2=ω. 11 This argument is elaborated in (Ruetsche [2002], [2003], [2006], [2011]) and discussed in (Feintzeig [forthcoming]). 12 By saying this, I do not mean to take a stance on whether the limit points are bona fide observables or ‘merely’ idealizations, and I also do not mean to take a stance on whether such idealizations here are indispensable (see, for example, Callender [2001]; Batterman [2005], [2009]). I hope only to invoke the notion that a topology captures a notion of similarity or resemblance and that limit points can be thought of as relevantly resembling elements of the original algebra to arbitrarily high accuracy so that we are licensed to use them for some scientific purposes. For more on limiting relations capturing a notion of similarity, resemblance, and approximation, see (Butterfield [2011a], [2011b]; Fletcher [forthcoming]). If one is bothered by my use of the terms ‘idealization’ and ‘approximation’, which are highly contested in the philosophical literature, one can view this discussion as merely being about the mathematical tools for taking certain limits, which end up being physically significant. 13 See the Appendix for proofs of Propositions 1, 2, 3, 4, and 6. The proofs of Propositions 5 and 7 are included in the references and are not reproduced here. 14 The nomenclature here is a bit unfortunate. The weak topology (sometimes called the weak Banach space topology) on an abstract C*-algebra is to be distinguished from the weak operator topology on its particular representations. We will see the difference in what follows. 15 I do not claim that the weak topology is the ‘right’ one for the algebraic imperialist or that there even is a ‘right’ topology to use. Just as the Hilbert space conservative may have access to multiple topologies on B(H), the algebraic imperialist may have access to multiple topologies on A. My claim is simply that the weak topology on A is analogous to the weak operator topology on B(H), in the sense that they derive from the same physical motivations for approximation and idealization, and they have analogous conditions for convergence. 16 In general, multiplication is not jointly continuous in the weak* topology. 17 The centre of a C*-algebra A consists in those elements A∈A such that for all B∈A, AB = BA. 18 See (Kronz and Lupher [2005]; Lupher [unpublished]), which also assert that the bidual contains all of the parochial observables. Here, I have added a precise characterization of how the parochial observables arise from the original algebra via limiting relations. 19 One might worry that the observables the algebraic imperialist gets in Proposition 3 (or the analogous observables for the universalist in the next section) are somehow not the ‘real’ or ‘fundamental’ net magnetization and particle number. I do not claim that the observables the imperialist acquires are fundamental—all I claim is that these observables suffice to give the kinds of explanations that Ruetsche worries about. 20 This is the same variant on algebraic imperialism that Lupher ([unpublished]) dubs ‘bidualism’. 21 See also (Robinson [1966], pp. 487–9; Ruetsche [2011], p. 135). 22 For more on this interpretation of states for the algebraic imperialist, see (Feintzeig [forthcoming]). 23 I also believe that one can make precise a sense in which this move from A to A** does not add any information or content to the theory—I save this for future work. 24 See (Kadison and Ringrose [1997], p. 719, Theorem 10.1.12) for a proof. 25 See (Kadison and Ringrose [1997], p. 443, Theorem 6.8.8, [1997], p. 719, Theorem 10.1.12) for the general construction of P. It is easy to show that P is the projection whose range is the class of all sub-representations of πU quasi-equivalent to πω, but this is beyond the scope of this article. 26 I am taking Ruetsche’s claims out of context here. Really, she rejects the universalized particle notion as a ‘fundamental’ particle notion (see Ruetsche [2011], Chapter 9 for more detail). However, considering her remarks as an objection to the view outlined here is illustrative. 27 See also (Clifton and Halvorson [2001]) for more on inequivalent particle notions. 28 Because the number operator is unbounded, it cannot strictly speaking belong to the universal enveloping von Neumann algebra. Instead, the number operator is affiliated with the algebra in the sense that any bounded function of it belongs to the algebra. We ignore this complication in what follows. 29 Really, Ruetsche makes this claim in the context of the reduced atomic representation (see Kadison and Ringrose [1997], pp. 740–1). But her objection may be carried over to the universal representation and my response may be carried back to the reduced atomic representation as well because (Kadison and Ringrose [1997], p. 741, Theorem 10.3.10) shows that the von Neumann algebra affiliated with the reduced atomic representation contains all of the parochial observables for all irreducible representations, just as the universal enveloping von Neumann algebra contains the parochial observables associated with all (not just irreducible) representations. 30 Proposition 3 shows that this *-isomorphism is a W*-isomorphism in the sense of Sakai ([1971], p. 40), that is, it is also a homeomorphism in the weak* and weak operator topologies, respectively. This is a relevant notion of isomorphism because both A** and πU(A)¯ are W*-algebras (the abstract version of von Neumann algebras). 31 Adapted from (Kadison and Ringrose [1997], p. 726, Proposition 10.1.21; Emch [1972], pp. 121–2, Theorem 11). See those sources for a proof and see (Lupher [unpublished], p. 95) for more discussion. 32 Note that although algebraic imperialism and universalism are what Ruetsche ([2011]) calls ‘pristine interpretations’, the arguments of this article do not vindicate the ideal of pristine interpretation, which is Ruetsche’s main target. The reason is that none of my manoeuvres for saving imperialism and universalism depend on that ideal. While my arguments here make imperialism and universalism viable interpretations, I believe the considerations of approximation and idealization in this article show that thinking of these as pristine interpretations is misleading at best. However, I must save this discussion for future work. 33 Similarly, Feintzeig ([forthcoming]) uses the classical case to gain insight about interpreting the algebraic formalism. This appendix can be understood as adding to that project. 34 Here, the relevant cyclic vector, Ωω, is the constant unit function. References Arageorgis A. [ unpublished]: Fields, Particles, and Curvature: Foundations and Philosophical Aspects of Quantum Field Theory in Curved Spacetime , PhD Thesis, University of Pittsburgh. Batterman R. [ 2005]: ‘Critical Phenomena and Breaking Drops: Infinite Idealizations in Physics’, Studies in the History and Philosophy of Modern Physics , 36, pp. 225– 44. Google Scholar CrossRef Search ADS Batterman R. [ 2009]: ‘Idealization and Modeling’, Synthese , 169, pp. 427– 46. Google Scholar CrossRef Search ADS Butterfield J. [ 2011a]: ‘Emergence, Reduction, and Supervenience: A Varied Landscape’, Foundations of Physics , 41, pp. 920– 59. Google Scholar CrossRef Search ADS Butterfield J. [ 2011b]: ‘Less Is Different: Emergence and Reduction Reconciled’, Foundations of Physics , 41, pp. 1065– 135. Google Scholar CrossRef Search ADS Callender C. [ 2001]: ‘Taking Thermodynamics Too Seriously’, Studies in the History and Philosophy of Modern Physics , 32, pp. 539– 53. 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 Emch G. [ 1972]: Algebraic Methods in Statistical Mechanics and Quantum Field Theory , New York: Wiley. Feintzeig B. [ forthcoming]: ‘Unitary Inequivalence in Classical Systems’, Synthese , doi: 10.1007/s11229-015-0875-1. Fletcher S. [ forthcoming]: ‘Similarity, Topology, and Physical Significance in Relativity Theory’, British Journal for the Philosophy of Science , doi: 10.1093/bjps/axu044. 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. [ 2006]: ‘Algebraic Quantum Field Theory’, in Butterfield J., Earman J. (eds), Handbook of the Philosophy of Physics , New York: North Holland, pp. 731– 864. Google Scholar CrossRef Search ADS Kadison R., Ringrose J. [ 1997]: Fundamentals of the Theory of Operator Algebras , Providence, RI: American Mathematical Society. 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. P. [ 1998]: Mathematical Topics between Classical and Quantum Mechanics , New York: Springer. Google Scholar CrossRef Search ADS Lupher T. [ unpublished]: The Philosophical Significance of Unitarily Inequivalent Representations in Quantum Field Theory , PhD Thesis, University of Texas. Petz D. [ 1990]: An Invitation to the Algebra of Canonical Commutation Relations , Leuven: Leuven University Press. Reed M., Simon B. [ 1980]: Functional Analysis , New York: Academic Press. Robinson D. [ 1966]: ‘Algebraic Aspects of Relativistic Quantum Field Theory’, in Chretien M., Deser S. (eds), Axiomatic Field Theory , New York: Gordon and Breach, pp. 389– 509. Ruetsche L. [ 2002]: ‘Interpreting Quantum Field Theory’, Philosophy of Science , 69, pp. 348– 78. Google Scholar CrossRef Search ADS 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. [ 2006]: ‘Johnny’s So Long at the Ferromagnet’, Philosophy of Science , 73, pp. 473– 86. Google Scholar CrossRef Search ADS Ruetsche L. [ 2011]: Interpreting Quantum Theories , New York: Oxford University Press. Google Scholar CrossRef Search ADS Sakai S. [ 1971]: C*-Algebras and W*-Algebras , New York: Springer. Schechter M. [ 2001]: Principles of Functional Analysis , Providence, RI: American Mathematical Society. Google Scholar CrossRef Search ADS Segal I. [ 1959]: ‘The Mathematical Meaning of Operationalism in Quantum Mechanics’, in Henkin L., Suppes P., Tarski A. (eds), The Axiomatic Method , Amsterdam: North Holland, pp. 341– 52. Google Scholar CrossRef Search ADS Summers S. [ 1999]: ‘On the Stone–von Neumann Uniqueness Theorem and its Ramifications’, in Rédei M., Stoeltzner M. (eds), John von Neumann and the Foundations of Quantum Physics , Dordrecht: Kluwer, pp. 135– 52. 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

The British Journal for the Philosophy of Science – Oxford University Press

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