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“Above the Slough of Despond”: Weylean invariantism and quantum physics
Studies in History and Philosophy of Modern Physics ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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“Above the Slough of Despond”: Weylean invariantism and quantum physics Iulian D. Toader The Research Institute, University of Bucharest; Descartes Centre for the History and Philosophy of the Sciences and the Humanities, Utrecht University, the Netherlands
art ic l e i nf o Article history: Received 18 February 2016 Received in revised form 20 April 2017 Accepted 21 April 2017
When citing this paper, please use the full journal title Studies in History and Philosophy of Modern Physics
1. Introduction The pursuit of scientific objectivity turned physical theories into systems of symbols or, as Weyl also put it sometimes, into symbolic constructions. What characterizes such constructions, at least in part, is a certain type of Begriffsbildung, according to which scientific concepts are freely created by the mind, i.e., implicitly defined via fundamental theoretical postulates (Toader, 2013). This idea, inspired by Hilbert, together with an approach to understanding influenced by Husserl, led Weyl to a form of skepticism about science, according to which if objectivity could be attained, understanding would thereby be sacrificed; and if understanding were to be pursued, this would render objectivity unattainable (Toader, 2011). Scientific objectivity can be attained, Weyl maintained, only if the relations described by a physical theory are invariant under the appropriate symmetry group, where symmetry is defined as a structure preserving mapping s: m → m′, for any m, m′ in the set M of all interpretations (or models in the formal semantics sense) of the theory. Weyl further believed that objectivity demands that s be the identity mapping, but this requirement should be rendered more general, as we will see, by letting s be a bijective mapping. This means that objectivity requires that all interpretations of a physical theory belong to the same isomorphism class, i.e., that objectivity requires categoricity. E-mail address: [email protected]
Under proper valuations of s and M , this more general view, which I will call here Weylean invariantism, seems adequate for quantum mechanics, or more exactly for quantum theories of systems with a finite number of degrees of freedom. For such systems, obeying canonical commutation relations, the Stone-von Neumann theorem suggests that the requirement above can be satisfied. This is because the theorem affirms the unitary equivalence of the irreducible Hilbert space representations of the Weyl algebra associated with the theory, without which the invariance that defines categoricity cannot be obtained. To be sure, Weyl acknowledged Stone's formulation of the theorem and saluted von Neumann's proof. However, as one soon realized, the theorem is false in quantum field theory (QFT), or more exactly for quantum theories of systems with an infinite number of degrees of freedom. This was, probably, the reason why, towards the end of his life, Weyl wrote: “Who could seriously pretend that the symbolic construct is the true real world? Objective Being, reality, becomes elusive; and science no longer claims to erect a sublime, truly objective world above the Slough of Despond in which our daily life moves…. the objective Being that we hoped to construct as one big piece of cloth each time tears off; what is left in our hands are – rags.” (Weyl, 1954 627) The unitary inequivalence of the Hilbert space representations of the Weyl algebra associated with QFT signals a failure of categoricity, which makes objectivity unattainable if Weylean invariantism is true. Confronting such dispiriting remarks, one may be strongly inclined to reject Weylean invariantism and argue that objectivity is attainable in the absence
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Please cite this article as: Toader, I. D. “Above the Slough of Despond”: Weylean invariantism and quantum physics. Studies in History and Philosophy of Modern Physics (2017), http://dx.doi.org/10.1016/j.shpsb.2017.04.004i
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of categoricity. But before one pursues that inclination, one should evaluate the claim that the invalidation of the Stone-von Neumann theorem does not constitute a threat to Weylean invariantism because one can, in fact, establish QFT's categoricity despite unitary inequivalence. I begin by articulating the type of invariantism that I am concerned with in more detail, and by describing what I take to be the general philosophical significance that Weyl attributed to categoricity. Briefly put, he took this to be a form of theoretical completeness, which led him to think that relations, rather than objects, constitute the physical content of a theory. I then turn to quantum mechanics and explain why this is complete in Weyl's sense. The explanation assumes a reading of the Stone-von Neumann theorem as a categoricity result, which also explains why QFT is considered incomplete, for the invalidation of the Stone-von Neumann theorem entails, on this reading, a failure of categoricity. Then, I discuss the claim that, despite unitary inequivalence, there may be ways of restoring QFT's categoricity. Some facts about other relations that may obtain between Hilbert space representations, like quasi-equivalence and weak equivalence, are analyzed to see if they are enough for establishing categoricity. I argue that, unless one is justified on physical grounds to discard physically significant representations, the claim is false: there is no categoricity without unitary equivalence. This points to a general problem for structuralist views of objectivity, like Weylean invariantism, a problem that had in fact been anticipated by Weyl. In contrast to such views, I believe that non-categoricity should be considered a resource, rather than a problem, but this line of thought will be pursued elsewhere.
2. Weylean invariantism Let us start by examining a requirement that Weyl thought ought to be imposed on scientific theorizing in general. This requirement is what he dubbed “concordance” (Einstimmigkeit) and defined in the following terms: “The definite value, which is assigned, in a certain individual case, to a quantity occurring in the theory, is determined on the basis of the theoretically posited connections and the contact with the perceptually given. Every such determination must lead to the same result.” (Weyl, 1949, 121.) This expresses the requirement that all measurements of a physical quantity should reach identical numerical results, if they are relative to the same theory and are based on conceptual and quantitative relations stipulated by this theory. Let's say that a scientific theory is concordant, or satisfies concordance, when such measurement results are approximately identical, i.e., within some measurement error margins. If concordant, a theory is “wellfounded” (op. cit., 185), or as recent commentators put it, “empirically grounded” (van Fraassen, 2009) or “confirmed” (Psillos, 2014). A similar requirement on scientific theorizing had been proposed by Reichenbach, and illustrated with respect to general relativity: “If, based on Einstein's theory, one calculates a light deflection of 1.7” by the sun, but finds instead 10”, then this is a contradiction, and such contradictions decide always upon the validity of a physical theory. The number 1,7” is obtained on the basis of equations and experiences on other material. The number 10” is, in principle, not obtained in a different manner, for it is by no means directly read off, but constructed from reading data with the help of quite complicated theories of the measuring instruments. One can thus say that one chain of calculations and experiences assigns to the real event the number 1,7, the other the number 10, and this is the contradiction.” (Reichenbach, 1920, 41) Scientific theorizing demands that such contradictions be discarded, that is, that all measurements of a physical quantity should
reach approximately identical results. Furthermore, Reichenbach thought that the satisfaction of this requirement was necessary for obtaining what he called “univocality” (Eindeutigkeit), i.e., a univocal correlation between the fundamental concepts of a scientific theory and their physical meaning or interpretation: “Univocality means…that a physical quantity, determined from different observational data, is expressed by [approximately] the same measure number.” (op. cit., 43). On Reichenbach's view, as I see it, this condition on measurement, i.e., precisely what Weyl called concordance, is necessary for a univocal relation between theory and physical reality: no Einstimmigkeit, no Eindeutigkeit. However, Weyl considered univocality an ideal that cannot be realized. In the winter semester of 1931, after having accepted a position as Hilbert's successor in Göttingen, Weyl taught a course on axiomatics. Speaking about completeness, he made the following comment: “We only require that [any] two contentual interpretations of a complete system of axioms be isomorphic to each other…. One designates this conception of completeness also as categoricity of the system.” (ETH manuscript Hs91a) To be sure, the idea of completeness as categoricity was already present in his 1927 monograph, Philosophie der Mathematik und Naturwissenschaften, although Weyl would use the term “categoricity” in print only in its 1949 English translation: “One might have thought of calling a system of axioms complete if the meaning of the basic concepts present in them were univocally fixed through the requirement that the axioms be valid. But this ideal cannot be realized, for the isomorphic mapping of a contentual interpretation is surely just another contentual interpretation. The final formulation is therefore this: a system of axioms is complete, or categorical, if any two contentual interpretations of it are necessarily isomorphic.” (Weyl, 1949 25) The ideal of completeness conceived of in Reichenbachian terms of univocality cannot be realized, according to Weyl, due to the fact that an axiomatized theory can be isomorphically interpreted over different domains of objects. This fact raises what has become known more recently as the problem of indeterminacy of reference (Putnam, 1980). Weyl's solution to this problem was to reconceptualize completeness in terms of categoricity. What is the significance of this solution? The existence of isomorphic interpretations of a scientific theory led Weyl to the view that relations, rather than objects, constitute the proper physical content of the theory: “A science can determine its domain only up to an isomorphic mapping. In particular it remains entirely indifferent as to the ‘essence’ of its objects. That which distinguishes the real points in space from number triples or other interpretation of geometry one can only know by immediate, living intuition. But intuition is not in itself some blessed tranquility, which it would never be able to leave behind. Rather, intuition presses on toward the chasm and adventure of cognition. However, it would be a chimera to expect cognition to reveal to intuition an essence deeper than that openly available to intuition. The idea of isomorphism designates the natural insurmountable boundary of scientific cognition.” (Weyl, 1949, 26) This view engenders a certain epistemological limitation: no scientific theory can be extended by adding a statement that expresses the difference between any isomorphic interpretations of the theory. Thus, no theory can offer scientific knowledge (i.e., what Weyl called cognition) about what makes this difference (i.e., what Weyl called the essence of objects) since this is particular to each domain of interpretation and is not invariant under isomorphic transformations. Therefore, a theory can offer scientific knowledge only if this is about relations, rather than objects, since only relations can be invariant under such transformations. On this view, all interpretations of a categorical theory are physically equivalent. Furthermore, as Weyl noted, this view has “enlightening value” for the problem of our knowledge of objective reality: such knowledge
Please cite this article as: Toader, I. D. “Above the Slough of Despond”: Weylean invariantism and quantum physics. Studies in History and Philosophy of Modern Physics (2017), http://dx.doi.org/10.1016/j.shpsb.2017.04.004i
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can be only about relations that remain invariant under isomorphic transformations. This takes us to Weyl's famous dictum: “objectivity means invariance with respect to the group of automorphisms”. As I see it, this expresses the view that objectivity can be attained only if the relations described by a physical theory are invariant under the group of automorphisms, i.e., only if the set M of all interpretations of the theory is such that for any m, m′ in M there is a structure preserving mapping s: m → m′, which is the identity mapping. However, making this type of invariance a requirement for objectivity presupposes either that there is only one domain of objects as a possible interpretation for the theory or that a unique domain among the possible ones can be identified as the domain with respect to which invariance is to be measured. That there is only one domain of objects as a possible interpretation may seem adequate for some theories, but it is not generally adequate since a physical theory can typically be interpreted over different domains of objects. If a unique domain among the possible ones can be identified as the domain with respect to which invariance is to be measured, then the unique domain either is not the domain of objects in the world, and so invariance would be insufficient for objectivity, or it is, and so invariance is redundant. Thus, I think that Weyl's invariance requirement should be rendered more general, by letting s be a bijective mapping. What this generalization says is that objectivity requires that all interpretations of the theory belong to the same isomorphism class, i.e., that objectivity requires categoricity. This claim is what I take to characterize Weylean invariantism. Attributing this view to Weyl seems justified not only in light of his reconceptualization of theoretical completeness in terms of categoricity. There is also indirect evidence that justifies this attribution, based on his skeptical remarks to the effect that objectivity cannot be attained because categoricity does not obtain. Weyl believed that a theory can be concordant, and thus wellfounded, without being objective: “The positing of the real external world does not guarantee that this constitutes itself from phenomena through the cognitive work of reason establishing concordance. For this much more is needed — that the world be ruled by simple elementary laws. […] the question about its reality mingles inseparably with the question about the reason for the lawful-mathematical harmony of the world.” (Weyl, 1949, 125) Objectivity would require, beside concordance, that the world be ruled by simple elementary laws, which as Weyl seems to indicate, entails that the world be lawful-mathematically harmonious. But he thought that this further requirement failed to be satisfied. First, he noted: “We still share [Kepler's] belief in a mathematical harmony of the universe. It has withstood the test of ever widening experience. But we no longer seek this harmony in static forms like the regular solids, but in dynamic laws.” (Weyl, 1952, 76) Mathematical harmony is determined by the dynamical laws that rule the world — its lawfulness, as it were. However, Weyl then remarked, such lawfulness does not characterize our world: “If nature were all lawfulness then every phenomenon would share the full symmetry of the universal laws of nature as formulated by the theory of relativity.” (ibid.) But, of course, not all physical quantities share the symmetry of the Poincaré group in special relativity, or the symmetry of the group of diffeomorphisms in general relativity. So nature is not all lawfulness. Thus, our belief in the mathematical harmony of the universe is unjustified. Weyl drew this conclusion from more general considerations as well: “The truth as we see it today is this: The laws of nature do not determine uniquely the one world that actually exists, not even if one concedes that two worlds arising from each other by an automorphic transformation, i.e., by a transformation which preserves the universal laws of nature, are to be considered the same world.” (Weyl, 1952, 27) The truth as he saw it is that the full symmetry of the universal laws does not physically obtain. This
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underlies Weyl's general point that concordance, although necessary, is not sufficient for objectivity. But it also emphasizes, I believe, that what was further required – invariance under automorphisms or, as I argued above, more generally, invariance under isomorphisms – does not obtain. This fact raises a problem to which Weyl did not seem to have any solution.
3. Quantum mechanics and categoricity On Weyl's conception of completeness as categoricity, quantum mechanics turns out to be a complete theory. The argument that quantum mechanics satisfies the Weylean invariantist requirement for objectivity is based on a reading of the Stone-von Neumann theorem as a categoricity result. In order to see why this reading is justified, consider a classical system with a finite number of degrees of freedom. The observables of the system, like momentum and position coordinates, are associated (in Heisenberg's formalism) with matrix operators pm and qn . Quantizing the system has these operators obey canonical commutation relations (CCRs):
pm pn − pn pm = 0 = qmqn − qnqm, pm qn − qnpm = − iℏδmn. The dynamics of the system relative to a canonical operator A is then given by
i(HAt − At H ) ∂A = ∂t
(1)
where H is the Hamiltonian operator, and ℏ is Planck's constant. In Schrödinger's formalism, the dynamics is represented by Schrödinger's equation:
i
∂ψt ∂t
(x1, x2 , …, x n) = Hψt (x1, x2 , …, x n).
(2)
where ψ is a wavefunction in the space L2(n) associated to the system's state space. With von Neumann's introduction of a new formalism, of course, the state space is associated with a Hilbert space = L2(n), which is a metrically complete, normed, vector space over the complex numbers, with a Hermitian inner product. The observables of the system are associated with self-adjoint, linear operators on , which obey corresponding CCRs:
^ [p^m , p^n ] = 0 = [q^m, q^n], [p^m , q^n] = − iℏδmnI . The expectation value of an observable O, described by the self^ ^ adjoint operator O , in a given state |ϕ〉 of the system, is 〈ϕ|O|ϕ〉. And the equivalence between representations (1) and (2) of the dynamics of the system is given by (Bratteli & Robinson, 1987, 5):
^ ^ 〈ϕt |At |ϕt 〉 = 〈ϕt |At0 |ϕt 〉. 0
0
Since is an infinite dimensional space, one normally associates to canonical operators p^m and q^n bounded unitary op^ ^ ^ ^ erators on the Hilbert space , that is Um(t ) = eipmt , and Vn(t ) = eiqnt . These are the so-called Weyl operators, which obey the Weyl form of the CCRs:
^ ^ ^ ^ ^ ^ ^ ^ Um(t )Un(s ) − Un(s )Um(t ) = 0 = Vm(t )Vn(s ) − Vn(s )Vm(t ) ^ ^ ^ ^ Um(t )Vn(s ) = eiℏstδmn Vn(s )Um(t ). These relations generate a certain type of C *-algebra, i.e., what is typically called a Weyl algebra. Now, if π : A → L( ) is a *-isomorphism from a Weyl algebra A to the set L( ) of bounded linear operators on , a faithful representation of A is a pair ( , π ). This is a norm preserving representation, where every state is a
Please cite this article as: Toader, I. D. “Above the Slough of Despond”: Weylean invariantism and quantum physics. Studies in History and Philosophy of Modern Physics (2017), http://dx.doi.org/10.1016/j.shpsb.2017.04.004i
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positive normalized linear functional on the Weyl algebra, and it is an irreducible representation if no (nontrivial) subspace of is invariant under the operators in π(A ). For any two such representations (1, π1) and (2, π2) of a Weyl algebra A , one can define the unitary equivalence relation in the following way: (1, π1) and (2, π2) are unitarily equivalent if and only if there is a ^ ^ ^ unitary operator U: 1 → 2 such that U π1(A) = π2(A)U for all A ∈ A. The Stone-von Neumann theorem, conjectured by Marshall Stone in 1930 and proved by von Neumann a year later, states that any irreducible representation ( , π ) of a Weyl algebra A is uniquely determined up to a unitary transformation (von Neumann, 1931, 577.) As Stone correctly noted, this result and its significance were already known to Weyl, who had famously claimed, in his 1928 book Gruppentheorie und Quantenmechanik, that the various formulations of the quantization of a physical system are essentially equivalent: “it can justly be maintained that the essence of the new Heisenberg-Schrödinger-Dirac quantum mechanics is to be found in the fact that there is associated with each physical system a set of quantities, constituting a non-commutative algebra in the technical mathematical sense.” (Weyl, 1928, 201) What is of interest to us here is that the theorem can be naturally read as a categoricity result, and also that Weyl could read it as such. This reading is justified by the fact that a unitary transformation is a mapping that preserves the relevant algebraic structure, i.e., the structure of the C *-algebra π(A) generated on a Hilbert space by the *-isomorphism π. More precisely, for any two irreducible Hilbert space representations (1, π1) and (2, π2), a unitary transformation is a *-isomorphism s: π1(A) → π2(A) implemented ^ by the unitary operator U: 1 → 2 such that s(π1(A)) = π2(A) fol^ lows from U π1(A) = π2(A)U^ for all A ∈ A . It is worth emphasizing that in quantum mechanics, s is always implemented by a unitary operator. A class of unitary equivalence of all irreducible Hilbert space representations of the Weyl algebra associated to the theory can thus be taken as a class of physically equivalent representations of the same algebra. This means that the states expressed as density matrices in an irreducible Hilbert space representation of the Weyl algebra are invariant. This reading of the Stone-von Neumann theorem justifies the claim that quantum mechanics realizes Weyl's ideal of completeness as categoricity and thus satisfies his requirement for objectivity. Quantum mechanics verifies Weylean invariantism.1 There are several assumptions behind this claim that should be mentioned. First, we should note that instrumental for the validity of the theorem was, of course, Weyl's replacement of unbounded operators with bounded unitary operators on Hilbert space. The canonical operators in the Heisenberg form of the CCRs can be realized only as unbounded operators, since there are no bounded linear operators on a Hilbert space, the commutator of which is a nonzero multiple of the identity operator. A mathematically consistent use of the Heisenberg from would thus require the existence of a common dense domain. The Weyl form of the CCRs sidesteps this problem. Furthermore, the algebraic structure generated by the Heisenberg form of the CCRs on a Hilbert space is not 1 Weyl's talk of “the one world that actually exists”, or an equivalence class of “worlds” determinable by universal laws, which I quoted in the previous section, may suggest that in quantum mechanics the proper valuation of M – the set of all interpretations of a theory – is the set of states expressed as density matrices in each of the Hilbert space representations of the Weyl algebra associated with the theory, rather than the set of representations, and that s is accordingly a mapping between states, rather than one between representations. Reading the Stone-von Neumann theorem as a categoricity result is in this case essential for verifying Weylean invariantism in quantum mechanics insofar as the unitary equivalence of representations is a necessary condition for the invariance of states across representations. (Thanks to one anonymous reviewer for pressing me on this point.)
preserved across Hilbert space representations, for there are such representations in which they have no solutions. Also, there exist Hilbert space representations in which the Heisenberg CCRs have solutions, but for which the relation of unitary equivalence fails (Schmüdgen, 1983). By contrast, there are no Hilbert space representations in which the Weyl CCRs lack solutions, and there are no unitarily inequivalent representations in which the Weyl CCRs have solutions. But this is true, of course, only for quantum mechanical systems, i.e., physical systems with a finite number of degrees of freedom.2 Secondly, the reading of the Stone-von Neumann theorem as a categoricity result, as a result that renders quantum mechanics a complete theory, in Weyl's sense, assumes that an irreducible Hilbert space representation of the Weyl algebra can be considered as a contentual interpretation of the theory. In other words, this reading of the theorem assumes that all (or at least part of the) physical content of the theory is located in the algebra π(A) on the Hilbert space, rather than in the Weyl algebra itself. While this raises a problem if one takes the latter as the exclusive repository of physical content, the assumption seems rather unproblematic if one restricts consideration of unitarily inequivalent representations to the faithful ones (as one surely does in the case of the Stone-von Neumann theorem).
4. No categoricity without unitary equivalence As already mentioned, the Stone-von Neumann theorem holds for a physical system with finitely many degrees of freedom, and guarantees unique quantization up to unitary equivalence. However, in QFT, just like in any quantum theory of systems with an infinite number of degrees of freedom, the theorem fails: there exists an infinity of unitarily inequivalent Hilbert space representations of the Weyl algebra associated to the theory. This means that, in general, there is no unitary transformation that preserves the relevant algebraic structure, i.e., the structure of the algebra π(A) generated on a Hilbert space by a *-isomorphism π. That is, there are irreducible Hilbert space representations, (1, π1) and (2, π2), such that there is no *-isomorphism s: π1(A) → π2(A) ^ implemented by a unitary operator U: 1 → 2. This entails that there is no class of unitary equivalence of all irreducible Hilbert space representations of the Weyl algebra associated to QFT that can be taken as a class of physical equivalence. In light of what has been discussed above, this fact suggests that QFT is non-categorical. Therefore, according to Weylean invariantism, QFT falls short of objectivity. As we have seen above, this led some, like Weyl, to believe that physical reality is “elusive”. To resist this conclusion, one needs a strategy for salvaging QFT's objectivity in the face of unitary inequivalence. One such strategy considers the quasilocal algebra arising as the C *-inductive limit of a net of local C *-algebras with common unit indexed by open bounded spacetime regions and satisfying isotony (Haag & Kastler, 1964). Objectivity is thought to be established since all representations of this algebra are taken to form a class of physical equivalence on the ground of their being weakly equivalent (see below). However, in addition to whether this is a valid ground for physical equivalence, the strategy discards as physically insignificant the so-called global observables (i.e., observables pertaining to infinitely extended spacetime regions such as total energy or total charge), since the quasilocal algebra cannot contain even bounded functions of such observables (Ruetsche, 2 As stated at the outset, only systems obeying CCRs (canonical commutation relations) are considered, but a similar discussion would consider systems that obey CARs (canonical anticommutation relations). For the latter, it is the JordanWigner theorem that guarantees unitary equivalence (Ruetsche, 2011, 62).
Please cite this article as: Toader, I. D. “Above the Slough of Despond”: Weylean invariantism and quantum physics. Studies in History and Philosophy of Modern Physics (2017), http://dx.doi.org/10.1016/j.shpsb.2017.04.004i
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2011, 134sq). Another strategy invokes certain restrictions on the range of validity of QFT and considers as physical only the Hilbert space representations within that range: unitarily inequivalent representations that pertain to long-distance, low energy, behavior (IR) are physical, but not “pathological” and may therefore be ignored; unitarily inequivalent representations that pertain to shortdistance, high energy, behavior (UV) are not physical, but mere artefacts of the mathematical formalism (Wallace, 2006). However, as has been pointed out in the literature, these latter unitarily inequivalent representations do have physical significance, insofar as they are necessary to account for a range of natural phenomena. For example, in order to explain phase transitions in quantum statistical mechanics, one typically needs a partition function displaying singularities, which require the thermodynamic limit and, thus, unitarily inequivalent representations (Ruetsche, 2003). Also, in quantum field theory, spontaneous symmetry breaking is needed to account for the mass of massive elementary particles, but this is usually thought to require unitarily inequivalent representations (Earman, 2004). Therefore, it seems fair to say that such a strategy should be avoided, if possible. Two other types of equivalence, which are often invoked in the literature, are quasi-equivalence and weak equivalence. These are defined as follows.3 Recall first that a von Neumann algebra is algebraically defined as a *-subalgebra N of the set L( ) of bounded linear operators on , such that N = N″, where N′ = {B ∈ L( ) : [B , A] = 0, for all A ∈ N} is the commutant of N , and N″ its double commutant. Two Hilbert space representations (1, π1) and (2, π2) are quasi-equivalent if and only if their associated von Neumann algebras π1(A)″ and π2(A)″ are *-isomorphic, i.e., if there is a *-isomorphism α : π1(A)″ → π2(A)″ such that α(π1(A)) = π2(A) for all A ∈ A . Note that α cannot be implemented by a unitary operator, unless one discards all reducible representations. Alternatively, (1, π1) and (2, π2) are weakly equivalent if and only if where Ker (π1) = Ker (π2), Ker (π ) = {A ∈ A π (A) ¼0}. It is a fact that all unitarily equivalent representations are quasi-equivalent, and all quasi-equivalent ones are weakly equivalent, but that not all weakly equivalent representations are quasi-equivalent, and not all quasi-equivalent ones are unitarily equivalent. The question is: can one establish QFT's categoricity by means of quasi-equivalence or weak equivalence? In other words, can any of these two relations help identify a mapping that preserves the relevant algebraic structure, and thus justifies one's taking an appropriate class of equivalence as a class of physical equivalence? The answer is that, even if one is flexible with respect to the relevant algebraic structure to be preserved, neither of these relations is sufficient for establishing categoricity. Quasi-equivalence seems to be sufficient for establishing categoricity, provided that one is flexible enough to take the relevant algebraic structure of any two Hilbert space representations (1, π1) and (2, π2) to be the structure of their von Neumann algebras π1(A)″ and π2(A)″, rather than the structure of the C *-algebras π1(A) and π2(A). For then we can take the *-isomorphism α : π1(A)″ → π2(A)″ to be implemented by a unitary operator ^ ^ ^ U: 1 → 2 such that α(π1(A)) = π2(A) follows from U π1(A) = π2(A)U for all A ∈ A . But it is a fact that we can do this only for quasiequivalent irreducible representations. A class of such representations is a class of unitary equivalence after all, since irreducibility ensures that quasi-equivalence collapses to unitary equivalence. Moreover, taking quasi-equivalent irreducible representations as a class of physical equivalence discards representations the quasi-equivalence of which cannot be unitarily 3
All subsequent definitions and results are from Bratteli and Robinson (1987).
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implemented. Thus, quasi-equivalence fails to squeeze all unitarily inequivalent representations of the Weyl algebra into one class of physical equivalence. Can weak equivalence be taken instead as a criterion of physical equivalence? Fell's theorem is usually thought to entail that all Hilbert space representations of the Weyl algebra of QFT are weakly equivalent. It follows from this that, for any two such representations, given any state ϕ expressed as a density matrix in the first representation, any finite set A of observables and any measurement error margin ϵ, there is a state ψ expressed as a density matrix in the second representation such that the expectation values of all observables in A are approximately identical (within ϵ) in the states ϕ and ψ. In other words, it is not possible to distinguish such representations by any given finite set of measurements. But although finite indistinguishability may be a necessary condition, it cannot be a sufficient condition for taking all weakly equivalent representations as one class of physical equivalence. The reason is known already from Weyl: finite indistinguishability guarantees Einstimmigkeit, i.e., that any given measurement yields approximately identical (within ϵ) expectation values to corresponding observables in weakly equivalent representations. But this, by itself, is not a guarantee of physical equivalence. Conversely, a careful analysis shows that many unitarily inequivalent representations are not weakly equivalent (Lupher, 2016), which enforces the point that weak equivalence fails to squeeze all representations into one class of physical equivalence. One recent strategy for salvaging QFT's objectivity argues that unitary equivalence is not required for physical equivalence (Baker & Halvorson, 2013). The argument is based on the fact that a ^ unitary operator, W , can be constructed that preserves transition probabilities between pure states, but fails to implement unitary equivalence. In other words, for any irreducible representations (1, π1) and (2, π2) of a Weyl algebra A , one can construct a ^ ^ ^ unitary operator W : 1 → 2 such that W π1(A) = π2(A)W , for most A ∈ A . For most, that is, because there are observables for which this relation does not hold, which is precisely the reason why unitary equivalence fails to obtain. A unitary operator that allows for unitary inequivalence meets the condition for implementing spontaneous symmetry breaking. But is it sufficient for physical equivalence? Physical equivalence, it is argued, requires that transition probabilities between pure states in any two irreducible representations be approximately identical (within ϵ). The unitary ^ operator W helps satisfy this requirement. But, as I see it, this merely restates Weyl's claim that Einstimmigkeit is a necessary condition for objectivity. His general point, however, was that this is not also a sufficient condition. In the case of QFT, what is further required is that a *-isomorphism s: π1(A) → π2(A) can be im^ plemented by a unitary operator U: 1 → 2 such that ^ ^ ^ U π1(A) = π2(A)U for all A ∈ A . But, clearly, W fails to implement s. Preservation of transition probabilities, in the absence of unitary equivalence, guarantees that QFT is a well-founded theory, but not that it is an objective one. To sum up, Weylean invariantism is adequate for quantum mechanics, as we have seen, for the Stone-von Neumann theorem can be read, in the context of this theory, as a categoricity result. But it seems inadequate for QFT, since here the theorem does not hold. In order to avoid the conclusion that this theory falls short of objectivity, while endorsing Weylean invariantism, one could appeal to quasi-equivalence or weak equivalence, rather than unitary equivalence, to restore categoricity. Such appeals are unsuccessful, however, as the above analysis briefly tried to show. Should one then share Weyl's dispiriting remarks, should one believe that physical reality is “elusive”? Or should one rather drop Weylean invariantism and articulate alternative conceptions of objectivity that do not require categoricity?
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5. Non-categoricity and modality Dropping Weylean invariantism may look like a radical move, but non-categoricity has not always been regarded as a theoretical defect. In the context of mathematical theories, for example, Zermelo argued that the non-categoricity of his system of axioms for set theory should be rather considered an advantage: the settheoretic antinomies could be dissolved, he claimed, by postulating an unlimited series of non-isomorphic interpretations (Zermelo, 1930). Restoring categoricity, as Fraenkel and von Neumann attempted to do, decreases the system's generality, reduces its range of applicability, and hinders the further development and enrichment of the theory. As a classic example supporting a view like Zermelo's, one should think of non-standard analysis, which could not have been developed had the non-categoricity of firstorder real analysis been considered a defect (Robinson, 1974). In the context of mathematical physics, one suggestion regarding the non-categoricity of QFT is to take unitarily inequivalent representations as complementary, in the sense that choosing one of them as physically significant does not necessarily entail that the others are physically insignificant (Clifton & Halvorson, 2001). More recently, Laura Ruetsche defended the view according to which the non-categoricity of a physical theory is an advantage, rather than a defect: “the manifold of interpretations functions as a theoretical resource. […] A theory that underdetermines its own interpretation is like a healthy breeding population: it has a shot at enough diversity to (under some interpretation or another) meet the variety of demands its scientific environment places on it.” (Ruetsche, 2011, 355) On her view, physical significance is partially determined by the applications of a theory and, in the context of QFT in particular, applications requiring the consideration of one unitarily inequivalent representation do not entail that others are unphysical, for it is possible that the latter are physically significant for other applications. On this approach, all unitarily inequivalent representations are thus placed “on a modal par,” i.e., just like the blades in a Swiss knife, each of which has its relevance partially determined by the demand its practical environment places on it (Ruetsche, 2003). There is a sense in which accommodating this view of unitary inequivalence can be used in explaining the nature of objective modality. For example, Steven French argued that “The fundamental structure [of the Weyl algebra, which admits of unitarily inequivalent representations] can be understood as inherently modal in the sense of encoding the full range of allowable physical possibilities. […] Thinking of Ruetsche's metaphor, actuality then emerges, depending on the representational blade that is pulled out of this structural knife as it were. Thinking in modal terms, the abstract [Weyl] algebra would have to be viewed as a structure that effectively encodes these modal features.” (French, 2014, 319) The idea of inherent modality is naturally directed against a Humean modal ontology, which ‘outsources’ modality to possible worlds and, as is well known, faces a whole battery of standard problems: quidditism & humility, quantum nonlocality, nomic indeterminacy, etc. (Berenstain & Ladyman, 2012) The idea of inherent modality is also conceived of as a plausible alternative to dispositionalism, as the latter is thought to be unable to properly accommodate the modal character of physical laws and symmetries (French, 2014). Nevertheless, the idea of inherent modality seems to be only one option, for French also makes the following suggestion: “If one wished one could adopt the possible worlds analysis of modality, insist that the actual world ‘contains’ no inherent modality and understand the structure [of the Weyl algebra] as spanning physically possible worlds, rather than being confined to the actual one.” (French, 2014, 319) Albeit pluralism seems always laudable, one should perhaps ask whether there is a
good way to decide between these two options. On the one hand, accommodating QFT's non-categoricity in a way that supports the idea of inherent modality suggests a view according to which the quantum states expressed as density matrices in a representation that is considered in a certain application context obtain in the actual world, and all states expressed as density matrices in the representations that are not considered in that context do not obtain, but are nevertheless contained, in the actual world. This seems to lead one to think of quantum states as abstract states that can be instantiated by a physical system. On the other hand, accommodating non-categoricity in a way that supports the possible worlds analysis of modality suggests a view according to which the states expressed as density matrices in a representation that is considered in a certain application context obtain in the actual world, and all states expressed as density matrices in the representations that are not considered in that context are not contained in the actual world, but in physically possible worlds. This allows one to think of quantum states as concrete states of a physical system. Assuming that this is a plausible reading of the two views of objective modality presented by French, and assuming that this reading can sufficiently be made sense of, then there is a good way to decide between them: it requires that one determine whether quantum states are concrete or abstract. In any case, it seems to me that one who defends the idea of inherent modality and, at the same time, endorses the “ontic” view that quantum states are concrete would be caught in a tight spot. But this discussion requires another paper.
6. Conclusion In the present paper, I argued that Weylean invariantism, that is the view that scientific objectivity requires categoricity, may correctly be attributed to Weyl, who took this condition to express a type of completeness of a scientific theory. This condition is satisfied by quantum mechanics, for the Stone-von Neumann theorem can be naturally interpreted as a categoricity result. However, QFT invalidates the theorem due to unitary inequivalence, which entails that either Weylean invariantism is false and should be rejected, or that categoricity can be established despite unitary inequivalence. Since the latter does not seem to be a viable option, I pointed out that one should take seriously QFT's non-categoricity, and that doing so may improve our understanding of the nature of objective modality. I think that both points deserve further attention.
Acknowledgements Thanks to audiences in Athens, Belgrade, Bristol, Budapest, Helsinki, Munich, and Paris, where my work on this paper has been presented, and especially to Aristidis Arageorgis, for very helpful comments. The Research Institute at the University of Bucharest, where the paper has been finalized, is also gratefully acknowledged for financial and institutional support.
References Baker, D. J., & Halvorson, H. (2013). How is spontaneous symmetry breaking possible? Understanding Wigner’s theorem in light of unitary inequivalence. In Studies in History and Philosophy of Modern Physics 44, (pp. 464–469). Berenstain, N. & Ladyman, J. (2012). Ontic Structural Realism and Modality. In Structural Realism: Structure, Object, and Causality, ed. by E. Landry and D. Rickles, Springer, (pp. 149–168). Bratteli, O. & Robinson, D. W. (1987). Operator Algebras and Quantum Statistical Mechanics, vol. I, second edition, Springer.
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I.D. Toader / Studies in History and Philosophy of Modern Physics ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Clifton, R. & Halvorson, H. (2001). Are Rindler quanta real? Inequivalent particle concepts in quantum field theory. In British Journal for the Philosophy of Science, 52, (pp. 417–470). Earman, J. (2004). Curie’s Principle and spontaneous symmetry breaking. In International Studies in the Philosophy of Science, 18, (pp. 173–198). French, S. (2014). The structure of the world: metaphysics and representation. Oxford University Press. Haag, R. & Kastler, D. (1964). An Algebraic Approach to Quantum Field Theory. In Journal of Mathematical Physics, 5, (pp. 848–861). Lupher, T. (2016). The Limits of Physical Equivalence in Algebraic Quantum Field Theory. In British Journal for the Philosophy of Science, http://dx.doi.org/10.1093/ bjps/axw017. Psillos, S. (2014). The View from Within and the View from Above: Looking at van Fraassen’s Perrin. In Bas van Fraassen's Approach to Representation and Models in Science, ed. by Gonzalez, W. J. Springer, Synthese Library, 368, (pp. 143–166). Putnam, H. (1980). Models and Reality. In Journal of Symbolic Logic, 45, (pp. 464– 482). Reichenbach, H. (1920). Relativitätstheorie und Erkenntnis apriori. Springer. Ruetsche, L. (2003). A Matter of Degree: Putting Unitary Inequivalence to Work. In Philosophy of Science, 70, (pp. 1329–1342). Ruetsche, L. (2011). Interpreting quantum theories. The art of the possible. Oxford University Press. Schmüdgen, K. (1983). On the Heisenberg commutation relation. In Journal of Functional Analysis, 50, (pp. 8–49).
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Toader, I. D. (2011) Objectivity Sans Intelligibility: Hermann Weyl’s Symbolic Constructivism, PhD dissertation, University of Notre Dame. Published by ProQuest, University of Michigan, 2012. Toader, I. D. (2013). Concept Formation and Scientific Objectivity: Weyl’s Turn Against Husserl. In HOPOS: The Journal of the International Society for the History of Philosophy of Science, 3, (pp. 281–305). van Fraassen, B. (2009). The Perils of Perrin, in the Hands of Philosophers. In Philosophical Studies, 143, (pp. 5–24). von Neumann, J. (1931). Die Eindeutigkeit der Schrödingerschen Operatoren. In Mathematische Annalen, 104, (pp. 570–578). Wallace, D. (2006). In Defence of naiveté: the Conceptual Status of Lagrangian Quantum Field Theory. In Synthese, 151, (pp. 33–80). Weyl, H. (1928) Gruppentheorie und Quantenmechanik, Leipzig. Hirzel. Weyl, H. (1949). Philosophy of mathematics and natural science. Princeton University Press. Weyl, H. (1952). Symmetry. Princeton University Press. Weyl, H. (1954).Erkenntnis und Besinnung (Ein Lebensrückblick). In Studia Philosophica, reprinted in Gesammelte Abhandlungen, vol. IV, ed. by K. Chandrasekharan, Springer, (pp. 631–649). Zermelo, E. (1930). Über Grenzzahlen und Mengenbereiche. Neue Untersuchungen über die Grundlagen der Mengenlehre. In Fundamenta Mathematicae, 16, 29–47, reprinted with an English translation in Collected Works, vol. I, ed. by Ebbinghaus, H. D. & Kanamori, A., Springer, (pp. 400–430).
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“Above the Slough of Despond”: Weylean invariantism and quantum physics Iulian D. Toader The Research Institute, University of Bucharest; Descartes Centre for the History and Philosophy of the Sciences and the Humanities, Utrecht University, the Netherlands
art ic l e i nf o Article history: Received 18 February 2016 Received in revised form 20 April 2017 Accepted 21 April 2017
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1. Introduction The pursuit of scientific objectivity turned physical theories into systems of symbols or, as Weyl also put it sometimes, into symbolic constructions. What characterizes such constructions, at least in part, is a certain type of Begriffsbildung, according to which scientific concepts are freely created by the mind, i.e., implicitly defined via fundamental theoretical postulates (Toader, 2013). This idea, inspired by Hilbert, together with an approach to understanding influenced by Husserl, led Weyl to a form of skepticism about science, according to which if objectivity could be attained, understanding would thereby be sacrificed; and if understanding were to be pursued, this would render objectivity unattainable (Toader, 2011). Scientific objectivity can be attained, Weyl maintained, only if the relations described by a physical theory are invariant under the appropriate symmetry group, where symmetry is defined as a structure preserving mapping s: m → m′, for any m, m′ in the set M of all interpretations (or models in the formal semantics sense) of the theory. Weyl further believed that objectivity demands that s be the identity mapping, but this requirement should be rendered more general, as we will see, by letting s be a bijective mapping. This means that objectivity requires that all interpretations of a physical theory belong to the same isomorphism class, i.e., that objectivity requires categoricity. E-mail address: [email protected]
Under proper valuations of s and M , this more general view, which I will call here Weylean invariantism, seems adequate for quantum mechanics, or more exactly for quantum theories of systems with a finite number of degrees of freedom. For such systems, obeying canonical commutation relations, the Stone-von Neumann theorem suggests that the requirement above can be satisfied. This is because the theorem affirms the unitary equivalence of the irreducible Hilbert space representations of the Weyl algebra associated with the theory, without which the invariance that defines categoricity cannot be obtained. To be sure, Weyl acknowledged Stone's formulation of the theorem and saluted von Neumann's proof. However, as one soon realized, the theorem is false in quantum field theory (QFT), or more exactly for quantum theories of systems with an infinite number of degrees of freedom. This was, probably, the reason why, towards the end of his life, Weyl wrote: “Who could seriously pretend that the symbolic construct is the true real world? Objective Being, reality, becomes elusive; and science no longer claims to erect a sublime, truly objective world above the Slough of Despond in which our daily life moves…. the objective Being that we hoped to construct as one big piece of cloth each time tears off; what is left in our hands are – rags.” (Weyl, 1954 627) The unitary inequivalence of the Hilbert space representations of the Weyl algebra associated with QFT signals a failure of categoricity, which makes objectivity unattainable if Weylean invariantism is true. Confronting such dispiriting remarks, one may be strongly inclined to reject Weylean invariantism and argue that objectivity is attainable in the absence
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of categoricity. But before one pursues that inclination, one should evaluate the claim that the invalidation of the Stone-von Neumann theorem does not constitute a threat to Weylean invariantism because one can, in fact, establish QFT's categoricity despite unitary inequivalence. I begin by articulating the type of invariantism that I am concerned with in more detail, and by describing what I take to be the general philosophical significance that Weyl attributed to categoricity. Briefly put, he took this to be a form of theoretical completeness, which led him to think that relations, rather than objects, constitute the physical content of a theory. I then turn to quantum mechanics and explain why this is complete in Weyl's sense. The explanation assumes a reading of the Stone-von Neumann theorem as a categoricity result, which also explains why QFT is considered incomplete, for the invalidation of the Stone-von Neumann theorem entails, on this reading, a failure of categoricity. Then, I discuss the claim that, despite unitary inequivalence, there may be ways of restoring QFT's categoricity. Some facts about other relations that may obtain between Hilbert space representations, like quasi-equivalence and weak equivalence, are analyzed to see if they are enough for establishing categoricity. I argue that, unless one is justified on physical grounds to discard physically significant representations, the claim is false: there is no categoricity without unitary equivalence. This points to a general problem for structuralist views of objectivity, like Weylean invariantism, a problem that had in fact been anticipated by Weyl. In contrast to such views, I believe that non-categoricity should be considered a resource, rather than a problem, but this line of thought will be pursued elsewhere.
2. Weylean invariantism Let us start by examining a requirement that Weyl thought ought to be imposed on scientific theorizing in general. This requirement is what he dubbed “concordance” (Einstimmigkeit) and defined in the following terms: “The definite value, which is assigned, in a certain individual case, to a quantity occurring in the theory, is determined on the basis of the theoretically posited connections and the contact with the perceptually given. Every such determination must lead to the same result.” (Weyl, 1949, 121.) This expresses the requirement that all measurements of a physical quantity should reach identical numerical results, if they are relative to the same theory and are based on conceptual and quantitative relations stipulated by this theory. Let's say that a scientific theory is concordant, or satisfies concordance, when such measurement results are approximately identical, i.e., within some measurement error margins. If concordant, a theory is “wellfounded” (op. cit., 185), or as recent commentators put it, “empirically grounded” (van Fraassen, 2009) or “confirmed” (Psillos, 2014). A similar requirement on scientific theorizing had been proposed by Reichenbach, and illustrated with respect to general relativity: “If, based on Einstein's theory, one calculates a light deflection of 1.7” by the sun, but finds instead 10”, then this is a contradiction, and such contradictions decide always upon the validity of a physical theory. The number 1,7” is obtained on the basis of equations and experiences on other material. The number 10” is, in principle, not obtained in a different manner, for it is by no means directly read off, but constructed from reading data with the help of quite complicated theories of the measuring instruments. One can thus say that one chain of calculations and experiences assigns to the real event the number 1,7, the other the number 10, and this is the contradiction.” (Reichenbach, 1920, 41) Scientific theorizing demands that such contradictions be discarded, that is, that all measurements of a physical quantity should
reach approximately identical results. Furthermore, Reichenbach thought that the satisfaction of this requirement was necessary for obtaining what he called “univocality” (Eindeutigkeit), i.e., a univocal correlation between the fundamental concepts of a scientific theory and their physical meaning or interpretation: “Univocality means…that a physical quantity, determined from different observational data, is expressed by [approximately] the same measure number.” (op. cit., 43). On Reichenbach's view, as I see it, this condition on measurement, i.e., precisely what Weyl called concordance, is necessary for a univocal relation between theory and physical reality: no Einstimmigkeit, no Eindeutigkeit. However, Weyl considered univocality an ideal that cannot be realized. In the winter semester of 1931, after having accepted a position as Hilbert's successor in Göttingen, Weyl taught a course on axiomatics. Speaking about completeness, he made the following comment: “We only require that [any] two contentual interpretations of a complete system of axioms be isomorphic to each other…. One designates this conception of completeness also as categoricity of the system.” (ETH manuscript Hs91a) To be sure, the idea of completeness as categoricity was already present in his 1927 monograph, Philosophie der Mathematik und Naturwissenschaften, although Weyl would use the term “categoricity” in print only in its 1949 English translation: “One might have thought of calling a system of axioms complete if the meaning of the basic concepts present in them were univocally fixed through the requirement that the axioms be valid. But this ideal cannot be realized, for the isomorphic mapping of a contentual interpretation is surely just another contentual interpretation. The final formulation is therefore this: a system of axioms is complete, or categorical, if any two contentual interpretations of it are necessarily isomorphic.” (Weyl, 1949 25) The ideal of completeness conceived of in Reichenbachian terms of univocality cannot be realized, according to Weyl, due to the fact that an axiomatized theory can be isomorphically interpreted over different domains of objects. This fact raises what has become known more recently as the problem of indeterminacy of reference (Putnam, 1980). Weyl's solution to this problem was to reconceptualize completeness in terms of categoricity. What is the significance of this solution? The existence of isomorphic interpretations of a scientific theory led Weyl to the view that relations, rather than objects, constitute the proper physical content of the theory: “A science can determine its domain only up to an isomorphic mapping. In particular it remains entirely indifferent as to the ‘essence’ of its objects. That which distinguishes the real points in space from number triples or other interpretation of geometry one can only know by immediate, living intuition. But intuition is not in itself some blessed tranquility, which it would never be able to leave behind. Rather, intuition presses on toward the chasm and adventure of cognition. However, it would be a chimera to expect cognition to reveal to intuition an essence deeper than that openly available to intuition. The idea of isomorphism designates the natural insurmountable boundary of scientific cognition.” (Weyl, 1949, 26) This view engenders a certain epistemological limitation: no scientific theory can be extended by adding a statement that expresses the difference between any isomorphic interpretations of the theory. Thus, no theory can offer scientific knowledge (i.e., what Weyl called cognition) about what makes this difference (i.e., what Weyl called the essence of objects) since this is particular to each domain of interpretation and is not invariant under isomorphic transformations. Therefore, a theory can offer scientific knowledge only if this is about relations, rather than objects, since only relations can be invariant under such transformations. On this view, all interpretations of a categorical theory are physically equivalent. Furthermore, as Weyl noted, this view has “enlightening value” for the problem of our knowledge of objective reality: such knowledge
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can be only about relations that remain invariant under isomorphic transformations. This takes us to Weyl's famous dictum: “objectivity means invariance with respect to the group of automorphisms”. As I see it, this expresses the view that objectivity can be attained only if the relations described by a physical theory are invariant under the group of automorphisms, i.e., only if the set M of all interpretations of the theory is such that for any m, m′ in M there is a structure preserving mapping s: m → m′, which is the identity mapping. However, making this type of invariance a requirement for objectivity presupposes either that there is only one domain of objects as a possible interpretation for the theory or that a unique domain among the possible ones can be identified as the domain with respect to which invariance is to be measured. That there is only one domain of objects as a possible interpretation may seem adequate for some theories, but it is not generally adequate since a physical theory can typically be interpreted over different domains of objects. If a unique domain among the possible ones can be identified as the domain with respect to which invariance is to be measured, then the unique domain either is not the domain of objects in the world, and so invariance would be insufficient for objectivity, or it is, and so invariance is redundant. Thus, I think that Weyl's invariance requirement should be rendered more general, by letting s be a bijective mapping. What this generalization says is that objectivity requires that all interpretations of the theory belong to the same isomorphism class, i.e., that objectivity requires categoricity. This claim is what I take to characterize Weylean invariantism. Attributing this view to Weyl seems justified not only in light of his reconceptualization of theoretical completeness in terms of categoricity. There is also indirect evidence that justifies this attribution, based on his skeptical remarks to the effect that objectivity cannot be attained because categoricity does not obtain. Weyl believed that a theory can be concordant, and thus wellfounded, without being objective: “The positing of the real external world does not guarantee that this constitutes itself from phenomena through the cognitive work of reason establishing concordance. For this much more is needed — that the world be ruled by simple elementary laws. […] the question about its reality mingles inseparably with the question about the reason for the lawful-mathematical harmony of the world.” (Weyl, 1949, 125) Objectivity would require, beside concordance, that the world be ruled by simple elementary laws, which as Weyl seems to indicate, entails that the world be lawful-mathematically harmonious. But he thought that this further requirement failed to be satisfied. First, he noted: “We still share [Kepler's] belief in a mathematical harmony of the universe. It has withstood the test of ever widening experience. But we no longer seek this harmony in static forms like the regular solids, but in dynamic laws.” (Weyl, 1952, 76) Mathematical harmony is determined by the dynamical laws that rule the world — its lawfulness, as it were. However, Weyl then remarked, such lawfulness does not characterize our world: “If nature were all lawfulness then every phenomenon would share the full symmetry of the universal laws of nature as formulated by the theory of relativity.” (ibid.) But, of course, not all physical quantities share the symmetry of the Poincaré group in special relativity, or the symmetry of the group of diffeomorphisms in general relativity. So nature is not all lawfulness. Thus, our belief in the mathematical harmony of the universe is unjustified. Weyl drew this conclusion from more general considerations as well: “The truth as we see it today is this: The laws of nature do not determine uniquely the one world that actually exists, not even if one concedes that two worlds arising from each other by an automorphic transformation, i.e., by a transformation which preserves the universal laws of nature, are to be considered the same world.” (Weyl, 1952, 27) The truth as he saw it is that the full symmetry of the universal laws does not physically obtain. This
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underlies Weyl's general point that concordance, although necessary, is not sufficient for objectivity. But it also emphasizes, I believe, that what was further required – invariance under automorphisms or, as I argued above, more generally, invariance under isomorphisms – does not obtain. This fact raises a problem to which Weyl did not seem to have any solution.
3. Quantum mechanics and categoricity On Weyl's conception of completeness as categoricity, quantum mechanics turns out to be a complete theory. The argument that quantum mechanics satisfies the Weylean invariantist requirement for objectivity is based on a reading of the Stone-von Neumann theorem as a categoricity result. In order to see why this reading is justified, consider a classical system with a finite number of degrees of freedom. The observables of the system, like momentum and position coordinates, are associated (in Heisenberg's formalism) with matrix operators pm and qn . Quantizing the system has these operators obey canonical commutation relations (CCRs):
pm pn − pn pm = 0 = qmqn − qnqm, pm qn − qnpm = − iℏδmn. The dynamics of the system relative to a canonical operator A is then given by
i(HAt − At H ) ∂A = ∂t
(1)
where H is the Hamiltonian operator, and ℏ is Planck's constant. In Schrödinger's formalism, the dynamics is represented by Schrödinger's equation:
i
∂ψt ∂t
(x1, x2 , …, x n) = Hψt (x1, x2 , …, x n).
(2)
where ψ is a wavefunction in the space L2(n) associated to the system's state space. With von Neumann's introduction of a new formalism, of course, the state space is associated with a Hilbert space = L2(n), which is a metrically complete, normed, vector space over the complex numbers, with a Hermitian inner product. The observables of the system are associated with self-adjoint, linear operators on , which obey corresponding CCRs:
^ [p^m , p^n ] = 0 = [q^m, q^n], [p^m , q^n] = − iℏδmnI . The expectation value of an observable O, described by the self^ ^ adjoint operator O , in a given state |ϕ〉 of the system, is 〈ϕ|O|ϕ〉. And the equivalence between representations (1) and (2) of the dynamics of the system is given by (Bratteli & Robinson, 1987, 5):
^ ^ 〈ϕt |At |ϕt 〉 = 〈ϕt |At0 |ϕt 〉. 0
0
Since is an infinite dimensional space, one normally associates to canonical operators p^m and q^n bounded unitary op^ ^ ^ ^ erators on the Hilbert space , that is Um(t ) = eipmt , and Vn(t ) = eiqnt . These are the so-called Weyl operators, which obey the Weyl form of the CCRs:
^ ^ ^ ^ ^ ^ ^ ^ Um(t )Un(s ) − Un(s )Um(t ) = 0 = Vm(t )Vn(s ) − Vn(s )Vm(t ) ^ ^ ^ ^ Um(t )Vn(s ) = eiℏstδmn Vn(s )Um(t ). These relations generate a certain type of C *-algebra, i.e., what is typically called a Weyl algebra. Now, if π : A → L( ) is a *-isomorphism from a Weyl algebra A to the set L( ) of bounded linear operators on , a faithful representation of A is a pair ( , π ). This is a norm preserving representation, where every state is a
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positive normalized linear functional on the Weyl algebra, and it is an irreducible representation if no (nontrivial) subspace of is invariant under the operators in π(A ). For any two such representations (1, π1) and (2, π2) of a Weyl algebra A , one can define the unitary equivalence relation in the following way: (1, π1) and (2, π2) are unitarily equivalent if and only if there is a ^ ^ ^ unitary operator U: 1 → 2 such that U π1(A) = π2(A)U for all A ∈ A. The Stone-von Neumann theorem, conjectured by Marshall Stone in 1930 and proved by von Neumann a year later, states that any irreducible representation ( , π ) of a Weyl algebra A is uniquely determined up to a unitary transformation (von Neumann, 1931, 577.) As Stone correctly noted, this result and its significance were already known to Weyl, who had famously claimed, in his 1928 book Gruppentheorie und Quantenmechanik, that the various formulations of the quantization of a physical system are essentially equivalent: “it can justly be maintained that the essence of the new Heisenberg-Schrödinger-Dirac quantum mechanics is to be found in the fact that there is associated with each physical system a set of quantities, constituting a non-commutative algebra in the technical mathematical sense.” (Weyl, 1928, 201) What is of interest to us here is that the theorem can be naturally read as a categoricity result, and also that Weyl could read it as such. This reading is justified by the fact that a unitary transformation is a mapping that preserves the relevant algebraic structure, i.e., the structure of the C *-algebra π(A) generated on a Hilbert space by the *-isomorphism π. More precisely, for any two irreducible Hilbert space representations (1, π1) and (2, π2), a unitary transformation is a *-isomorphism s: π1(A) → π2(A) implemented ^ by the unitary operator U: 1 → 2 such that s(π1(A)) = π2(A) fol^ lows from U π1(A) = π2(A)U^ for all A ∈ A . It is worth emphasizing that in quantum mechanics, s is always implemented by a unitary operator. A class of unitary equivalence of all irreducible Hilbert space representations of the Weyl algebra associated to the theory can thus be taken as a class of physically equivalent representations of the same algebra. This means that the states expressed as density matrices in an irreducible Hilbert space representation of the Weyl algebra are invariant. This reading of the Stone-von Neumann theorem justifies the claim that quantum mechanics realizes Weyl's ideal of completeness as categoricity and thus satisfies his requirement for objectivity. Quantum mechanics verifies Weylean invariantism.1 There are several assumptions behind this claim that should be mentioned. First, we should note that instrumental for the validity of the theorem was, of course, Weyl's replacement of unbounded operators with bounded unitary operators on Hilbert space. The canonical operators in the Heisenberg form of the CCRs can be realized only as unbounded operators, since there are no bounded linear operators on a Hilbert space, the commutator of which is a nonzero multiple of the identity operator. A mathematically consistent use of the Heisenberg from would thus require the existence of a common dense domain. The Weyl form of the CCRs sidesteps this problem. Furthermore, the algebraic structure generated by the Heisenberg form of the CCRs on a Hilbert space is not 1 Weyl's talk of “the one world that actually exists”, or an equivalence class of “worlds” determinable by universal laws, which I quoted in the previous section, may suggest that in quantum mechanics the proper valuation of M – the set of all interpretations of a theory – is the set of states expressed as density matrices in each of the Hilbert space representations of the Weyl algebra associated with the theory, rather than the set of representations, and that s is accordingly a mapping between states, rather than one between representations. Reading the Stone-von Neumann theorem as a categoricity result is in this case essential for verifying Weylean invariantism in quantum mechanics insofar as the unitary equivalence of representations is a necessary condition for the invariance of states across representations. (Thanks to one anonymous reviewer for pressing me on this point.)
preserved across Hilbert space representations, for there are such representations in which they have no solutions. Also, there exist Hilbert space representations in which the Heisenberg CCRs have solutions, but for which the relation of unitary equivalence fails (Schmüdgen, 1983). By contrast, there are no Hilbert space representations in which the Weyl CCRs lack solutions, and there are no unitarily inequivalent representations in which the Weyl CCRs have solutions. But this is true, of course, only for quantum mechanical systems, i.e., physical systems with a finite number of degrees of freedom.2 Secondly, the reading of the Stone-von Neumann theorem as a categoricity result, as a result that renders quantum mechanics a complete theory, in Weyl's sense, assumes that an irreducible Hilbert space representation of the Weyl algebra can be considered as a contentual interpretation of the theory. In other words, this reading of the theorem assumes that all (or at least part of the) physical content of the theory is located in the algebra π(A) on the Hilbert space, rather than in the Weyl algebra itself. While this raises a problem if one takes the latter as the exclusive repository of physical content, the assumption seems rather unproblematic if one restricts consideration of unitarily inequivalent representations to the faithful ones (as one surely does in the case of the Stone-von Neumann theorem).
4. No categoricity without unitary equivalence As already mentioned, the Stone-von Neumann theorem holds for a physical system with finitely many degrees of freedom, and guarantees unique quantization up to unitary equivalence. However, in QFT, just like in any quantum theory of systems with an infinite number of degrees of freedom, the theorem fails: there exists an infinity of unitarily inequivalent Hilbert space representations of the Weyl algebra associated to the theory. This means that, in general, there is no unitary transformation that preserves the relevant algebraic structure, i.e., the structure of the algebra π(A) generated on a Hilbert space by a *-isomorphism π. That is, there are irreducible Hilbert space representations, (1, π1) and (2, π2), such that there is no *-isomorphism s: π1(A) → π2(A) ^ implemented by a unitary operator U: 1 → 2. This entails that there is no class of unitary equivalence of all irreducible Hilbert space representations of the Weyl algebra associated to QFT that can be taken as a class of physical equivalence. In light of what has been discussed above, this fact suggests that QFT is non-categorical. Therefore, according to Weylean invariantism, QFT falls short of objectivity. As we have seen above, this led some, like Weyl, to believe that physical reality is “elusive”. To resist this conclusion, one needs a strategy for salvaging QFT's objectivity in the face of unitary inequivalence. One such strategy considers the quasilocal algebra arising as the C *-inductive limit of a net of local C *-algebras with common unit indexed by open bounded spacetime regions and satisfying isotony (Haag & Kastler, 1964). Objectivity is thought to be established since all representations of this algebra are taken to form a class of physical equivalence on the ground of their being weakly equivalent (see below). However, in addition to whether this is a valid ground for physical equivalence, the strategy discards as physically insignificant the so-called global observables (i.e., observables pertaining to infinitely extended spacetime regions such as total energy or total charge), since the quasilocal algebra cannot contain even bounded functions of such observables (Ruetsche, 2 As stated at the outset, only systems obeying CCRs (canonical commutation relations) are considered, but a similar discussion would consider systems that obey CARs (canonical anticommutation relations). For the latter, it is the JordanWigner theorem that guarantees unitary equivalence (Ruetsche, 2011, 62).
Please cite this article as: Toader, I. D. “Above the Slough of Despond”: Weylean invariantism and quantum physics. Studies in History and Philosophy of Modern Physics (2017), http://dx.doi.org/10.1016/j.shpsb.2017.04.004i
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2011, 134sq). Another strategy invokes certain restrictions on the range of validity of QFT and considers as physical only the Hilbert space representations within that range: unitarily inequivalent representations that pertain to long-distance, low energy, behavior (IR) are physical, but not “pathological” and may therefore be ignored; unitarily inequivalent representations that pertain to shortdistance, high energy, behavior (UV) are not physical, but mere artefacts of the mathematical formalism (Wallace, 2006). However, as has been pointed out in the literature, these latter unitarily inequivalent representations do have physical significance, insofar as they are necessary to account for a range of natural phenomena. For example, in order to explain phase transitions in quantum statistical mechanics, one typically needs a partition function displaying singularities, which require the thermodynamic limit and, thus, unitarily inequivalent representations (Ruetsche, 2003). Also, in quantum field theory, spontaneous symmetry breaking is needed to account for the mass of massive elementary particles, but this is usually thought to require unitarily inequivalent representations (Earman, 2004). Therefore, it seems fair to say that such a strategy should be avoided, if possible. Two other types of equivalence, which are often invoked in the literature, are quasi-equivalence and weak equivalence. These are defined as follows.3 Recall first that a von Neumann algebra is algebraically defined as a *-subalgebra N of the set L( ) of bounded linear operators on , such that N = N″, where N′ = {B ∈ L( ) : [B , A] = 0, for all A ∈ N} is the commutant of N , and N″ its double commutant. Two Hilbert space representations (1, π1) and (2, π2) are quasi-equivalent if and only if their associated von Neumann algebras π1(A)″ and π2(A)″ are *-isomorphic, i.e., if there is a *-isomorphism α : π1(A)″ → π2(A)″ such that α(π1(A)) = π2(A) for all A ∈ A . Note that α cannot be implemented by a unitary operator, unless one discards all reducible representations. Alternatively, (1, π1) and (2, π2) are weakly equivalent if and only if where Ker (π1) = Ker (π2), Ker (π ) = {A ∈ A π (A) ¼0}. It is a fact that all unitarily equivalent representations are quasi-equivalent, and all quasi-equivalent ones are weakly equivalent, but that not all weakly equivalent representations are quasi-equivalent, and not all quasi-equivalent ones are unitarily equivalent. The question is: can one establish QFT's categoricity by means of quasi-equivalence or weak equivalence? In other words, can any of these two relations help identify a mapping that preserves the relevant algebraic structure, and thus justifies one's taking an appropriate class of equivalence as a class of physical equivalence? The answer is that, even if one is flexible with respect to the relevant algebraic structure to be preserved, neither of these relations is sufficient for establishing categoricity. Quasi-equivalence seems to be sufficient for establishing categoricity, provided that one is flexible enough to take the relevant algebraic structure of any two Hilbert space representations (1, π1) and (2, π2) to be the structure of their von Neumann algebras π1(A)″ and π2(A)″, rather than the structure of the C *-algebras π1(A) and π2(A). For then we can take the *-isomorphism α : π1(A)″ → π2(A)″ to be implemented by a unitary operator ^ ^ ^ U: 1 → 2 such that α(π1(A)) = π2(A) follows from U π1(A) = π2(A)U for all A ∈ A . But it is a fact that we can do this only for quasiequivalent irreducible representations. A class of such representations is a class of unitary equivalence after all, since irreducibility ensures that quasi-equivalence collapses to unitary equivalence. Moreover, taking quasi-equivalent irreducible representations as a class of physical equivalence discards representations the quasi-equivalence of which cannot be unitarily 3
All subsequent definitions and results are from Bratteli and Robinson (1987).
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implemented. Thus, quasi-equivalence fails to squeeze all unitarily inequivalent representations of the Weyl algebra into one class of physical equivalence. Can weak equivalence be taken instead as a criterion of physical equivalence? Fell's theorem is usually thought to entail that all Hilbert space representations of the Weyl algebra of QFT are weakly equivalent. It follows from this that, for any two such representations, given any state ϕ expressed as a density matrix in the first representation, any finite set A of observables and any measurement error margin ϵ, there is a state ψ expressed as a density matrix in the second representation such that the expectation values of all observables in A are approximately identical (within ϵ) in the states ϕ and ψ. In other words, it is not possible to distinguish such representations by any given finite set of measurements. But although finite indistinguishability may be a necessary condition, it cannot be a sufficient condition for taking all weakly equivalent representations as one class of physical equivalence. The reason is known already from Weyl: finite indistinguishability guarantees Einstimmigkeit, i.e., that any given measurement yields approximately identical (within ϵ) expectation values to corresponding observables in weakly equivalent representations. But this, by itself, is not a guarantee of physical equivalence. Conversely, a careful analysis shows that many unitarily inequivalent representations are not weakly equivalent (Lupher, 2016), which enforces the point that weak equivalence fails to squeeze all representations into one class of physical equivalence. One recent strategy for salvaging QFT's objectivity argues that unitary equivalence is not required for physical equivalence (Baker & Halvorson, 2013). The argument is based on the fact that a ^ unitary operator, W , can be constructed that preserves transition probabilities between pure states, but fails to implement unitary equivalence. In other words, for any irreducible representations (1, π1) and (2, π2) of a Weyl algebra A , one can construct a ^ ^ ^ unitary operator W : 1 → 2 such that W π1(A) = π2(A)W , for most A ∈ A . For most, that is, because there are observables for which this relation does not hold, which is precisely the reason why unitary equivalence fails to obtain. A unitary operator that allows for unitary inequivalence meets the condition for implementing spontaneous symmetry breaking. But is it sufficient for physical equivalence? Physical equivalence, it is argued, requires that transition probabilities between pure states in any two irreducible representations be approximately identical (within ϵ). The unitary ^ operator W helps satisfy this requirement. But, as I see it, this merely restates Weyl's claim that Einstimmigkeit is a necessary condition for objectivity. His general point, however, was that this is not also a sufficient condition. In the case of QFT, what is further required is that a *-isomorphism s: π1(A) → π2(A) can be im^ plemented by a unitary operator U: 1 → 2 such that ^ ^ ^ U π1(A) = π2(A)U for all A ∈ A . But, clearly, W fails to implement s. Preservation of transition probabilities, in the absence of unitary equivalence, guarantees that QFT is a well-founded theory, but not that it is an objective one. To sum up, Weylean invariantism is adequate for quantum mechanics, as we have seen, for the Stone-von Neumann theorem can be read, in the context of this theory, as a categoricity result. But it seems inadequate for QFT, since here the theorem does not hold. In order to avoid the conclusion that this theory falls short of objectivity, while endorsing Weylean invariantism, one could appeal to quasi-equivalence or weak equivalence, rather than unitary equivalence, to restore categoricity. Such appeals are unsuccessful, however, as the above analysis briefly tried to show. Should one then share Weyl's dispiriting remarks, should one believe that physical reality is “elusive”? Or should one rather drop Weylean invariantism and articulate alternative conceptions of objectivity that do not require categoricity?
Please cite this article as: Toader, I. D. “Above the Slough of Despond”: Weylean invariantism and quantum physics. Studies in History and Philosophy of Modern Physics (2017), http://dx.doi.org/10.1016/j.shpsb.2017.04.004i
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5. Non-categoricity and modality Dropping Weylean invariantism may look like a radical move, but non-categoricity has not always been regarded as a theoretical defect. In the context of mathematical theories, for example, Zermelo argued that the non-categoricity of his system of axioms for set theory should be rather considered an advantage: the settheoretic antinomies could be dissolved, he claimed, by postulating an unlimited series of non-isomorphic interpretations (Zermelo, 1930). Restoring categoricity, as Fraenkel and von Neumann attempted to do, decreases the system's generality, reduces its range of applicability, and hinders the further development and enrichment of the theory. As a classic example supporting a view like Zermelo's, one should think of non-standard analysis, which could not have been developed had the non-categoricity of firstorder real analysis been considered a defect (Robinson, 1974). In the context of mathematical physics, one suggestion regarding the non-categoricity of QFT is to take unitarily inequivalent representations as complementary, in the sense that choosing one of them as physically significant does not necessarily entail that the others are physically insignificant (Clifton & Halvorson, 2001). More recently, Laura Ruetsche defended the view according to which the non-categoricity of a physical theory is an advantage, rather than a defect: “the manifold of interpretations functions as a theoretical resource. […] A theory that underdetermines its own interpretation is like a healthy breeding population: it has a shot at enough diversity to (under some interpretation or another) meet the variety of demands its scientific environment places on it.” (Ruetsche, 2011, 355) On her view, physical significance is partially determined by the applications of a theory and, in the context of QFT in particular, applications requiring the consideration of one unitarily inequivalent representation do not entail that others are unphysical, for it is possible that the latter are physically significant for other applications. On this approach, all unitarily inequivalent representations are thus placed “on a modal par,” i.e., just like the blades in a Swiss knife, each of which has its relevance partially determined by the demand its practical environment places on it (Ruetsche, 2003). There is a sense in which accommodating this view of unitary inequivalence can be used in explaining the nature of objective modality. For example, Steven French argued that “The fundamental structure [of the Weyl algebra, which admits of unitarily inequivalent representations] can be understood as inherently modal in the sense of encoding the full range of allowable physical possibilities. […] Thinking of Ruetsche's metaphor, actuality then emerges, depending on the representational blade that is pulled out of this structural knife as it were. Thinking in modal terms, the abstract [Weyl] algebra would have to be viewed as a structure that effectively encodes these modal features.” (French, 2014, 319) The idea of inherent modality is naturally directed against a Humean modal ontology, which ‘outsources’ modality to possible worlds and, as is well known, faces a whole battery of standard problems: quidditism & humility, quantum nonlocality, nomic indeterminacy, etc. (Berenstain & Ladyman, 2012) The idea of inherent modality is also conceived of as a plausible alternative to dispositionalism, as the latter is thought to be unable to properly accommodate the modal character of physical laws and symmetries (French, 2014). Nevertheless, the idea of inherent modality seems to be only one option, for French also makes the following suggestion: “If one wished one could adopt the possible worlds analysis of modality, insist that the actual world ‘contains’ no inherent modality and understand the structure [of the Weyl algebra] as spanning physically possible worlds, rather than being confined to the actual one.” (French, 2014, 319) Albeit pluralism seems always laudable, one should perhaps ask whether there is a
good way to decide between these two options. On the one hand, accommodating QFT's non-categoricity in a way that supports the idea of inherent modality suggests a view according to which the quantum states expressed as density matrices in a representation that is considered in a certain application context obtain in the actual world, and all states expressed as density matrices in the representations that are not considered in that context do not obtain, but are nevertheless contained, in the actual world. This seems to lead one to think of quantum states as abstract states that can be instantiated by a physical system. On the other hand, accommodating non-categoricity in a way that supports the possible worlds analysis of modality suggests a view according to which the states expressed as density matrices in a representation that is considered in a certain application context obtain in the actual world, and all states expressed as density matrices in the representations that are not considered in that context are not contained in the actual world, but in physically possible worlds. This allows one to think of quantum states as concrete states of a physical system. Assuming that this is a plausible reading of the two views of objective modality presented by French, and assuming that this reading can sufficiently be made sense of, then there is a good way to decide between them: it requires that one determine whether quantum states are concrete or abstract. In any case, it seems to me that one who defends the idea of inherent modality and, at the same time, endorses the “ontic” view that quantum states are concrete would be caught in a tight spot. But this discussion requires another paper.
6. Conclusion In the present paper, I argued that Weylean invariantism, that is the view that scientific objectivity requires categoricity, may correctly be attributed to Weyl, who took this condition to express a type of completeness of a scientific theory. This condition is satisfied by quantum mechanics, for the Stone-von Neumann theorem can be naturally interpreted as a categoricity result. However, QFT invalidates the theorem due to unitary inequivalence, which entails that either Weylean invariantism is false and should be rejected, or that categoricity can be established despite unitary inequivalence. Since the latter does not seem to be a viable option, I pointed out that one should take seriously QFT's non-categoricity, and that doing so may improve our understanding of the nature of objective modality. I think that both points deserve further attention.
Acknowledgements Thanks to audiences in Athens, Belgrade, Bristol, Budapest, Helsinki, Munich, and Paris, where my work on this paper has been presented, and especially to Aristidis Arageorgis, for very helpful comments. The Research Institute at the University of Bucharest, where the paper has been finalized, is also gratefully acknowledged for financial and institutional support.
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Please cite this article as: Toader, I. D. “Above the Slough of Despond”: Weylean invariantism and quantum physics. Studies in History and Philosophy of Modern Physics (2017), http://dx.doi.org/10.1016/j.shpsb.2017.04.004i