Macroscopic observables and the Born rule
I. Long run frequencies
Abstract
We clarify the role of the Born rule in the Copenhagen Interpretation of quantum mechanics by deriving it from Bohr’s doctrine of classical concepts, translated into the following mathematical statement: a quantum system described by a noncommutative algebra of observables is empirically accessible only through associated commutative algebras. The Born probabilities emerge as the relative frequencies of outcomes in long runs of measurements on a quantum system; it is not necessary to adopt the frequency interpretation of singlecase probabilities (which will be the subject of a sequel paper). Our derivation of the Born rule uses ideas from a program begun by Finkelstein (1965) and Hartle (1968), intending to remove the Born rule as a separate postulate of quantum mechanics. Mathematically speaking, our approach refines previous elaborations of this program  notably the one due to Farhi, Goldstone, and Gutmann (1989) as completed by Van Wesep (2006)  in replacing infinite tensor products of Hilbert spaces by continuous fields of algebras. In combination with our interpretational context, this technical improvement circumvents valid criticisms that earlier derivations of the Born rule have provoked, especially to the effect that such derivations were mathematically flawed as well as circular. Furthermore, instead of relying on the controversial eigenvectoreigenvalue link in quantum theory, our derivation just assumes that pure states in classical physics have the usual interpretation as truthmakers that assign sharp values to observables.
Dedicated to the memory of Bernd Kuckert (1968–2008)
1 Introduction
In its simplest formulation, the Born rule says that if is some quantummechanical observable with nondegenerate discrete spectrum , then the probability that a measurement of in a state yields the result is given by
(1) 
where is a normalized eigenvector of with eigenvector . In other words, if with , then . More generally, if is a selfadjoint operator on a Hilbert space with associated spectral measure , then the probability that the proposition comes out to be true if is measured in a state equals
(2) 
The Born rule provides the key link between the mathematical formalism of quantum physics and experiment, and as such is responsible for most predictions of quantum theory. In the history and philosophy of science, the Born rule (on a par with the Heisenberg uncertainty relations) is often seen as a turning point where indeterminism entered fundamental physics.^{1}^{1}1 The Born rule was first stated by Max Born in the context of scattering theory [2], following a slightly earlier paper in which he famously omitted the absolute value squared signs (though he corrected this in a footnote added in proof). The wellknown application to the position operator is due to Pauli [39]. The general formulation (2) is due to von Neumann [36, §iii.1]. See [33] for a detailed reconstruction of the historical origin of the Born rule within the context of quantum mechanics, as well as [40] for a briefer historical treatment in the more general setting of the emergence of modern probability theory and probabilistic thinking. Of course, classical physics is full of random phenomena as well. But in all known cases, their apparent random character may be retraced to ignorance about the initial state or about microscopic degrees of freedom or time scales; see, e.g., [13] and [40]. In contrast, the type of randomness to which quantum mechanics gives rise via the Born rule is generally felt to be ‘irreducible’ (in the sense of not being reducible to ignorance, not even about the Laws of Nature).^{2}^{2}2Notable exceptions are Einstein [12, pp. 129–130] and ’t Hooft [25].
Even the assumption that quantum mechanics is a correct and fundamental theory by no means implies that this feeling is correct. Indeed, although among rival interpretations the Copenhagen Interpretation is the one that arguably puts most emphasis on both the fundamental and the probabilistic character of quantum theory, a mature work by one of its founders actually contains the following passage:
‘One may call these uncertainties objective, in that they are simply a consequence of the fact that we describe the experiment in terms of classical physics; they do not depend in detail on the observer. One may call them subjective, in that they reflect our incomplete knowledge of the world.’ (Heisenberg, [22, pp. 53–54].)
This claim is in tune with one of the two main principles of the Copenhagen Interpretation,^{3}^{3}3The other one, the Principle of Complementarity, plays no role in this paper. namely Bohr’s doctrine of classical concepts. A mature and wellknown expression of this doctrine is as follows:
‘However far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms. (…) The argument is simply that by the word experiment we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account of the experimental arrangements and of the results of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics.’ (Bohr, [1, p. 209].)
Elsewhere, Bohr time and again stresses that measurement devices must be described classically ‘if these are to serve their purpose’. We take this to mean that, although such devices are ontologically quantummechanical by nature, they become a tool (in fact, the only tool) for the description of quantum phenomena as soon as they are epistemically treated as if they were classical. Thus the socalled Heisenberg cut, i.e. the borderline between the part of the world that is described classically and the part that is described quantummechanically, is epistemic or (inter)subjective in nature and hence movable; see also [43, 44]. In our opinion, this ideology provides an attractive qualitative basis for the understanding of randomness in Nature, for it preserves the fundamental difference between random phenomena in classical and in quantum physics (the given explanation of quantum probabilities as arising from the classical description of some part of the world would not make any sense if applied to classical probabilities), while discarding the notion of strictly ‘irreducible’ randomness (which is only defined through negation and quite possibly makes no philosophical sense at all).
However, little (if any) work has been done in relating this ideology to the Born rule, which in the Copenhagen Interpretation simply seems to be taken for granted as a mathematical recipe that requires no explanation. It is the purpose of the present paper to fill this gap: as we shall see, the Born rule can actually be derived from a particular instance of Bohr’s doctrine of classical concepts, provided one identifies the Born probabilities with the relative frequencies of outcomes in long runs of measurements on a quantum system.^{4}^{4}4With this limited goal it is not even necessary to mention singlecase probabilities, let alone interpret them; doing so requires a far deeper analysis, which will be the subject of a sequel paper, based on [24]. As always, the mathematical implementation of Bohr’s philosophical ideas is ambiguous; as far as his doctrine of classical concepts is concerned, we read it as saying that a quantum system described by a noncommutative algebra of observables is empirically accessible only through commutative algebras associated with .^{5}^{5}5Apart from leading to the Born rule, this reading also gives rise to a very pretty description of complementarity through the mathematical framework of topos theory; see [24]. Cf. Scheibe [43] and Howard [26] for different readings of the doctrine of classical concepts. For convenience, and in line with the modern mathematical description of quantum theory [47, 3, 19, 45, 31, 32], we assume that these algebras are in fact (unital) algebras. The simplest kind of commutative algebras associated with are its (unital) commutative subalgebras; in this paper we need a more subtle limiting procedure to ‘extract’ a commutative algebra of macroscopic observables.
Our derivation of the Born rule relies on certain ideas that were originally proposed by Finkelstein [17] and Hartle [21], whose work was continued by Ochs [38], Bugajski and Motyka [5], Farhi, Goldstone and Gutmann [15], and Van Wesep [50].^{6}^{6}6Some of these papers were not quite written in support of the Copenhagen Interpretation but rather against it, usually defending the Everettian stance. We review this development in Section 2, either incorporating or circumventing critique of the papers just listed that has been issued by a number of authors, including Cassinello and SánchezGómez [6] and Caves and Schack [7]. Their critique has been partly of a mathematical and partly of a conceptual nature, but in our opinion one of the most devastating arguments against the program in question, namely its reliance on the socalled eigenstateeigenvalue link, has not been made before.
We will show how the program of deriving the Born rule from ‘first principles’ can nonetheless be carried out if it is underwritten by Bohr’s doctrine of classical concepts (in its reading mentioned above). The mathematical formalism needed to accomplish this starts from the modern algebraic approach to the quantum theory of large systems [47, 3, 19, 23, 35, 45], which, however, we need to reformulate in order to incorporate Bohr’s doctrine in an optimal way. This reformulation is based on the unified picture provided by continuous fields of algebras [10, 29] in the description of the classical limit of quantum mechanics. This limit actually has (at least) two guises, namely the limit of Planck’s constant going to zero, and the limit of a system size going to infinity. Both can be brought under the umbrella of continuous fields of algebras; for this was done in [31], and for it was announced in [32] and will be completed in the present paper, where essential use is made of ideas of Raggio and Werner [41] and Duffield and Werner [11]. In fact, once the appropriate framework has been set up in Section 3, the derivation of the Born rule in Section 4 will turn out to be almost trivial.
This paper is part of a larger research programme, whose goal it is to interpret quantum mechanics entirely in terms of its classical limit. This is meant as a technical implementation of the Copenhagen Interpretation as originally formulated by Bohr and Heisenberg (cf. [27]), whose goal was expressed quite well by Landau and Lifshitz [30, p. 3]:
“Thus quantum mechanics occupies a very unusual place among physical theories: it contains classical mechanics as a limiting case, yet at the same time it requires this limiting case for its own formulation.”
2 The strong law of large numbers in quantum theory
Let us first review what has been achieved mathematically in [17, 21, 38, 5, 15, 50]. For simplicity, we restrict ourselves to the simple situation of repeated measurements on a twolevel (or, in current parlance, onequbit) system, i.e. with Hilbert space . Suppose we have an observable (i.e. a hermitian matrix) with eigenvalues and and corresponding orthogonal eigenstates and . A long series of measurements of in a given initial state (prepared anew for each subsequent measurement) will produce a sequence , where or . We idealize a long series of measurements as an infinite one, so that , with and the space of infinite binary sequences is denoted by . We define as the Born probability
(3) 
so that .
We first review the classical strong law of large numbers relevant to , seen as a measure space with Borel structure generated by the sets
(4) 
where and . For any , consider the probability measure on defined by and . This defines a probability measure on for which and for all . Let
(5) 
This is a Borel set. The strong law of large numbers states that
(6) 
A measure theorist will read this as is stands: has measure one with respect to . A probability theorist defines functions by , notes that the are i.i.d. random variables, and says that the sequence of functions on converges pointwise to with probability one (or almost surely) with respect to . A physicist defines an elementary proposition (or ‘yesno question’) (i.e. the characteristic function of ) on the ‘phase space’ , which is answered by yes in a pure state if , and by no otherwise. The probability measure defines a mixed state on , and (6) gives the stateproposition pairing in the case at hand as
(7) 
If, for a general yesno question and state , one initially interprets as the probability of obtaining a positive answer to in the state (or, more generally, interprets as the expectation value of an observable in a state ), then one still has to expand this interpretation by stipulating what notion of probability one is using [18, 34]. Even if a probability equals one, as in (7), one still has to declare whether or not one adopts the socalled Necessity Thesis [34] (stating that probability one implies certainty). These questions cannot be answered by the mathematical formalism.
The papers just cited attempt to extend the strong law of large numbers to the quantum case, and, not always sensitive to the last remark, draw certain conclusion about quantum mechanics from such an extension. A correct way of proceeding at least mathematically emerges from a combinination of results in [15, 50], as follows. Let be the fold tensor product of , and let be the separable component of the infinite tensor product Hilbert space (in the sense of von Neumann [37]) of that contains , where is a given (unit) vector.^{7}^{7}7Apart from the original source [37], this formalism is also explained in e.g. [15] or [14, §6.2]. The details are not relevant here, as we will replace the use of infinite tensor products of Hilbert spaces by a different formalism later on. In any case, the simplest way to define is to regard it as the Hilbert space of the GNSrepresentation of the infinite tensor product (cf. [28, §11.4]) induced by the vector state ; see Section 4 below. The unit vector , seen as a state, is the quantum analogue of the probability measure in the classical situation just reviewed. The quantum analogue of the proposition is a projection on , defined as follows. For each , define a projection on by , where the projection on acts on the ’th copy of in the infinite tensor product and all other entries are unit matrices on . Similarly, is defined by replacing by . The projections commute, and generate a complete Boolean algebra of projections on . Let be the (countably complete) Boolean algebra of Borel sets in . By Theorem 1 in [50], there is a unique homomorphism of Boolean algebras that satisfies
(8) 
for each . The projection , then, is what its notation says, i.e. the image of the Borel set under . Interpreted as a yesno question, it asks if a given measurement outcome has mean .
Let be as in (3). It is easy to show [50] that (6) implies
(9) 
Regarding the unit vector as a state (in the algebraic sense) on any von Neumann algebra of operators on containing , we can rewrite (9) in the form of the classical pairing (7), i.e.
(10) 
Indeed, since is a projection, eqs. (9) and (10) are equivalent.
Furthermore, let us define the frequency operator on by stipulating that its eigenstates are (where or ), with eigenvalues
(11) 
In words, is the relative frequency of the entry 1 in the list . Clearly, can be extended to an operator on by
(12) 
It then follows from (9) that
(13) 
with given by (3). In fact, defining
(14) 
where denotes the limit in the strong operator topology on , it can even be shown that
(15) 
where is the unit operator on .
Results of this type can be derived quite easily from the modern algebraic approach to the quantum theory of large systems [47, 3, 19, 32]; see below. For the moment, we discuss the interpretation of (15) and especially of its corollary (13).
Authors of papers like [17, 21, 15, 50] argue that, in view of (3) and the definition (12) of , eq. (13) provides a derivation of the Born rule from the socalled eigenstateeigenvalue link. This terminology, which sounds like a tautology in mathematics, is often used in the philosophy of physics; see, e.g. [4, 8]. The link in question is the postulate that if is an observable and is an eigenstate of with eigenvalue , then a measurement of in the state yields the result with certainty.^{8}^{8}8Philosophical realists adhering to the eigenstateeigenvalue link would simply say that has the value in the eigenstate , but precisely among realists it has become fashionable to deny the eigenstateeigenvalue link for the reasons mentioned in the main text [4, 8]. For example, in Bohmian mechanics position always has a sharp value, whereas in the modal interpretation of quantum mechanics the link is dropped in a more flexible way. The eigenstateeigenvalue link is, of course, a special case of the Born rule, but the whole point of the exercise is to derive the Born rule from the eigenstateeigenvalue link, rather than the other way round.
To assess the claim that (13) provides a derivation of the Born rule from the eigenstateeigenvalue link, we make three points.^{9}^{9}9We leave it to the reader to assess the more farreaching claim in [50] that (13) “nullifies any remaining objection to the manyworlds view”. Given their recent attempt to derive the Born rule in a completely different way [42, 51], this claim is apparently not even supported by adherents of the manyworlds interpretation.

Applying the eigenstateeigenvalue link to conclude from (13) that has a sharp value in the state , is inconsistent with measuring by measuring each of its components in (12) separately. For each such measurement will disturb the state; neither nor any is an eigenstate of any , not to mention all of them.^{10}^{10}10In fact, any component of the complete von Neumann tensor product containing at least one simultaneous eigenstate of all is orthogonal in its entirety to . Hence is to be measured directly. Although according to [15] this can be done in some cases, it precludes any inference of singlecase Born probabilities from (13).

The eigenstateeigenvalue link is the source of the measurement problem in quantum mechanics and hence is held to be unsound by most contemporary specialists in the foundations of quantum mechanics (see [4, 8] and references therein). Indeed, the eigenstateeigenvalue link cannot be found in the writings of Bohr and Heisenberg; it was first postulated by Dirac [9].
The first two points were also made in [6, 7], but despite our adding the third objection, we hesitate in following the authors of these papers in concluding that the program of deriving the Born probabilities from properties of the frequency operator is “flawed at every step” [7]. Indeed, by changing both the conceptual and the mathematical setting we will see that each of these objections can be met:

Changing the definition of the frequency operator from a strong operator limit on a Hilbert space (which even depends on the state ) to an element of a commutative (i.e. classical) algebra of macroscopic observables (which is independent of ) ‘stabilizes’ against perturbations. Thus, without jeopardizing our derivation and interpretation of the Born rule, the frequency operator can be measured either directly (as suggested in [15]), or in terms of repeated measurements of the underlying observable in the state . The latter procedure determines the possible values (0 or 1) of each for , upon which one takes the limit . This seems to correspond to experimental practice.

The second objection is obviated if one simply interprets the possible values of the frequency operator according to its definition, i.e. as limiting frequencies of either a single experiment on a large number of sites or a long run of individual experiments on single sites. In particular, one should refrain from making any statement about singlecase probabilities. On this view, the Born rule simply says nothing about individual experiments on single sites.^{11}^{11}11Except in an empty way, as in Popper’s socalled propensity interpretation of probability [18, 34].

Instead of relying on the controversial eigenvectoreigenvalue link in quantum theory, our derivation will just assume that pure states in classical physics have the usual interpretation as ‘truthmakers’ that assign sharp values to observables.
3 Large quantum systems and the Born rule
3.1 Continuous field of algebras
In experimental physics, theoretical predictions based on the Born rule are typically checked by performing identical experiments on a given quantum system in a given state , where is large. This situation is idealized by taking the limit . We describe this limit in a way that reorganizes the wellknown algebraic description of infinite quantum systems by quasilocal algebras [47, 3, 19] and macroscopic observables [23, 35, 41, 11, 45] into our preferred tool in the mathematical analysis of classical behaviour in quantum theory [31, 32], namely continuous fields of algebras. For the reader’s convenience we recall the latter notion, replacing the original definition of Dixmier [10] by the equivalent formulation of Kirchberg and S. Wassermann [29]. By a morphism we mean a homomorphism.
Definition 1
A continuous field of algebras over a locally compact Hausdorff space consists of a algebra , a collection of algebras , and a surjective morphism for each , such that:

The function is in for each (where is the norm in ).

The norm of is .

The algebra is a module in the sense that for any and there is an element for which for all .
A continuous section of the field is a map for which there is an such that for all .
It follows that the algebra may actually be identified with the space of continuous sections of the field: if we do so, the morphism is just the evaluation map at . The general idea is that the family of algebras is glued together by specifying a topology on the bundle (disjoint union). This topology is defined indirectly via the specification of the space of continuous sections of the bundle (cf. the Serre–Swan Theorem for vector bundles). The third condition makes a module in the sense that there exists a nondegenerate morphism from to the center of the multiplier algebra of .
This seemingly technical definition turns out to provide an attractive framework for the study of the classical limit of quantum mechanics. In the scenario , the parameter space is typically , and is the commutative algebra of functions on some classical phase space. For each , one then constructs as the algebra of quantum observables for varying (it may or may not be the case that the are isomorphic for different values of ). Continuous sections of the field then describe quantization and the classical limit of observables at one go [31]. More generally, the classical theory is ‘glued’ to the corresponding quantum theories via the continuous field structure.
3.2 Macroscopic and quasilocal observables
To describe large quantum systems and their possible classical behaviour, we use the onepoint compactification . This is homeomorphic to in the relative topology borrowed from , viz. under the map and (where is the compactification point added to ).
To derive the Born rule, we need copies of a single quantum system with unital algebra of observables (e.g., as above). From the single algebra , we are going to construct two quite different continuous fields of algebras over , called and . These fields coincide as far as their fibers above are concerned, which are given by
(16) 
Here is the spatial tensor product; see, e.g., [28, Ch. 11]. However, the two fields differ in their respective fibers above the limit point , given by
(17)  
(18) 
Here is the state space of (equipped with the weaktopology),^{12}^{12}12For example, the state space of is isomorphic as a compact convex set to the threeball : describing a state as a density matrix on , the corresponding point is given by the wellknown parametrization and the algebra in the righthand side of (18) is the inductive limit with respect to the inclusion maps given by (see below for an explicit description).^{13}^{13}13One often writes for the inductive limit , where the bar denotes norm completion. In the notation of [28, §11.4], our corresponds to with and for all .
In order to define the continuous sections of , we define, for , symmetrization maps by
(19) 
where one has copies of the unit so as to obtain an element of . The symmetrization operator is given by (linear and continuous) extension of
(20) 
where is the permutation group (i.e. symmetric group) on elements and for all . For example, is given by
(21) 
where is seen as an element of the ’th copy of in . In particular, for the frequency operator in defined by (11) is of this form, since from (12) we infer that
(22) 
More generally, for (the algebra of all bounded operators on a Hilbert space ), the operator that counts the frequency of the eigenstate of some observable upon measurements of is given by
(23) 
Definition 2
We say that a sequence with is symmetric when
(24) 
for some fixed and all . We call quasisymmetric if for any there is an and a symmetric sequence such that for all .
Physically speaking, the tail of a symmetric sequence entirely consists of ‘averaged’ or ‘intensive’ observables. which become macroscopic in the limit . Quasisymmetric sequences have the important property that they mutually commute in the limit ; more precisely, if and are quasisymmetric sequences, then
(25) 
Hence we see that in the limit the quasisymmetric sequences organize themselves in a commutative algebra, which we call the algebra of macroscopic observables of the given large system. To see that this limit algebra of macroscopic observables is isomorphic to , we complete the definition of the continuous field by defining its continuous sections.
Theorem 1
For any unital algebra , the fibers
(26) 
form a continuous field of algebras over if the algebra of continuous sections is defined as follows: a section
(27)  
of the above field is declared to be continuous if the sequence is quasisymmetric, and
(28) 
Here and is the tensor product of copies of , defined by (linear and continuous) extension of ; cf. [28, Prop. 11.4.6]. The limit (28) then exists by definition of an approximately symmetric sequence: if is symmetric with (24), one has for , so that the tail of the sequence is even independent of . In the approximately symmetric case one easily proves that is a Cauchy sequence.
Proof. To prove that is a continuous field, the main point is to show that
(29) 
if is quasisymmetric and is given by (28). This is easy to show for symmetric sequences: assume (24), so that for . By the axiom it suffices to prove (29) for , which implies and hence for all . One then has . Because of the special form of one may replace the supremum over the set of all states on by the supremum over the set of all symmetric states (see Definition 5 below), which in turn may be replaced by the supremum over the extreme boundary of . The latter consists of all states of the form [46], so that . This is actually equal to . Now the norm in is , and by definition of one has . Hence (29) follows.
Given (29), the theorem follows from [31, Prop. II.1.2.3] and the fact that the set of functions on arising in the said way are dense in (equipped with the supremumnorm). This follows from the Stone–Weierstrass theorem, from which one infers that the functions in question exhaust .
We now turn to the continuous field defined by the quasilocal observables.
Definition 3
A sequence (where , as before) is called local when for some fixed and all one has (where one has copies of the unit ), and quasilocal when for any there is an and a local sequence such that for all .
The inductive limit algebra then simply consists of all equivalence classes of quasilocal sequences under the equivalence relation when . The algebraic structure on is inherited from the quasilocal sequences in the obvious (pointwise) way, except for the norm, which is given by
(30) 
Each is contained in as a subalgebra by identifying with the equivalence class (where the zero’s are irrelevant, of course; any entry could have been chosen).
Theorem 2
For any unital algebra , the fibers
(31) 
form a continuous field of algebras over if the algebra of continuous sections is defined as follows: a section
(32)  
of the above field is declared to be continuous if the sequence is quasilocal, and
(33) 
Proof. This time, the first property in Definition 1 is immediate from (30). The other properties are either trivial or follow from [31, Prop. II.1.2.3].
4 The Born rule
To see the relevance of the above considerations to the Born rule, we first rederive (15). For any , we note that a state on defines a state on by
(34) 
which limit is easily seen to exist by first approximating a quasilocal sequence by a local one.
We take and pick a unit vector with associated pure state on . As in (34), the infinite tensor product defines a state on , which is pure by [28, Prop. 11.4.7]. Hence the associated GNSrepresentation is irreducible; we may identify its carrier space with the separable component of von Neumann’s infinite tensor product that contains as the cyclic vector of the GNSconstruction (cf. [14, Ch. 6] and also see Section 2 above).
Each operator defined by (23) lies in and hence in . Although does not exist within , one may consider a possible limit as an operator on . This limit indeed exists in the strong operator topology, and commutes with all elements of (this is easily checked to be the case for any quasisymmetric sequence). Since is irreducible, the limit operator must be a multiple of the unit, and using (34) and (23) one computes the constant as
(35) 
This generalizes (15), and also, to our mind, gives an impeccable derivation of it. The type of derivation of the Born rule reviewed in Section 2 is based on (35), but despite the fact that its mathematical status has now been clarified, it faces the conceptual problems listed in that section.
To solve these problems, we use the continuous field instead of , again with . Identifying a density matrix on with a state on in the usual way by , for a symmetric sequence with (see (21)) one easily finds
(36) 
Our key application then arises from the frequency operator (23), which amounts to the choice . In that case (36) becomes
(37) 
In particular, if is a unit vector and , defining a vector state on by , one has
(38) 
This is the Born rule, at least formally. To understand why this identification is correct also conceptually, at least in the context of the Copenhagen Interpretation, one has to realize the following. Unlike its counterpart in (35), the limit operator in (38) is by construction an element of a commutative algebra, namely the algebra of macroscopic observables attached to the fold duplication of for . According to Bohr’s doctrine of classical concepts (cf. the Introduction), any statement about the quantum system described by has to be made through commutative algebras associated to , and has to use “the terminology of classical physics”. This terminology includes the role of pure states as ‘truthmakers’, in the sense that if is a classical observable defined as a realvalued function on some phase space , then a point validates the proposition for with certainty. This is precisely what happens in (38), which uses , hence , and states that in the classical state , the observable simply has the (sharp) value . Thus one has a nonprobabilistic statement in classical physics, which expresses a probabilistic observation about quantum physics.
The specific way in which converges to as a continuous section of , as well as its relationship to (35), is clarified by the following device [31, 32].
Definition 4
A continuous field of states on a continuous field of algebras over is a family of states on , defined for each , such that is continuous on for each (i.e. for each continuous section of the field of algebras).
In the case at hand, where , this only imposes the condition
(39) 
for each continuous section of the field in question, which we take to be either or . Indeed, the relationship between these two continuous fields of algebras is most easily studied through their respective continuous fields of states.
Any state on trivially defines a continuous fields of states on by restriction, using the inclusion explained just above Theorem 2. The ensuing family of states on does not necessarily extend to a continuous field on , and  especially in the context of the Born rule  it is interesting to find examples when they do.
Definition 5
A state on is symmetric when each of its restrictions to is invariant under the natural action of the symmetric group on (under which maps an elementary tensor to ).
Such states were analyzed by Størmer [46], who proved a noncommutative version of De Finetti’s wellknown representation theorem in classical probability: any symmetric state on has a unique decomposition
(40) 
where is a probability measure on , and is defined as in (34).
Theorem 3
Proof. This is immediate from (28) and (40).^{14}^{14}14Analogous results appear in the work of Unnerstall [48, 49].
We now see that the state used at the beginning of this section is an example of Definition 5, for which the associated measure in (40) and (41) is the Dirac measure concentrated at . The states on are, of course, given by , and the function has constant value . Hence one recovers the limit (38) either from (39) or from (41), since ; the fact that these computations coincide is an illustration of Theorem 3.
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