Persistence and Nonpersistence as Complementary Models of Identical Quantum Particles

According to our understanding of the everyday physical world, observable phenomena are underpinned by persistent objects that can be reidentified (or tracked) across time by observation of their distinctive properties. This understanding is reflected in classical mechanics, which posits that matter consists of persistent, reidentifiable particles. However, the mathematical symmetrization procedures used to describe identical particles within the quantum formalism have led to the widespread belief that identical quantum particles lack either persistence or reidentifiability. However, it has proved difficult to reconcile these assertions with the fact that identical particles are routinely assumed to be reidentifiable in particular circumstances. Moreover, neither of these assertions accounts for the mathematical form of the symmetrization procedures used to describe identical particles within the quantum framework, leaving open theoretical possibilities other than bosonic and fermionic behavior, such as paraparticles, which do not appear to be realized in nature. Here we propose the novel idea that both persistence and nonpersistence models must be employed in order to fully account for the behaviour of identical particles. Thus, identical particles are neither persistent nor nonpersistent. We prove the viability of this viewpoint by showing how Feynman's and Dirac's symmetrization procedures arise through a synthesis of a quantum treatment of these models, and by showing how reidentifiability emerges in a context-dependent manner. We further show that the persistence and nonpersistence models satisfy the key characteristics of Bohr's concept of complementarity, and thereby propose that the behavior of identical particles is a manifestation of a persistence-nonpersistence complementarity, analogous to Bohr's wave-particle complementarity for individual particles.


Introduction
We ordinarily conceive of the everyday physical world as consisting of objects that bear properties and that persist through time. Developed early in life through our continual sensorimotor interaction with the physical world, this conception organizes our experience of the external world into a coherent, predictive model. In particular, persistence underwrites our ability to say that the object one is seeing now is the same as a specific object that one saw elsewhere at an earlier time. In practice, objects' gradual motion and slowly-varying characteristic properties(such as shape and color) provide the perceptual handles that enable their reidentification.
Classical physics incorporates these key notions-objects, properties, persistence, and reidentifiability-into its abstract conceptual framework at a fundamental level. Persistence is reflected in the assumption that objects can be labeled. Additionally, classical mechanics posits that objects localized to point-like regions of space-particles-are 1 In the paper in which he originally addressed the subject, Dirac gave an argument that purported to derive the fact that a system of two identical particles can occupy only symmetric or antisymmetric states [4, section 3]. The fact that this argument does not generalize to three or more particles was pointed out by Wigner [19,20]. In apparent reference to this result, Dirac subsequently speaks of the restriction to (anti-)symmetrical states as an empirical fact ('The invariance and permanence of the symmetry properties of the states means that for some particular kind of particle it is quite possible for only symmetrical or only antisymmetrical states to occur in nature. Whether this is the case cannot be decided by any general theoretical considerations, but can be settled only by reference to special experimentally determined facts about the particles in question.' [6, section 62, p 201]). In the third edition(1947) of [6], he adds:'other more complicated kinds of symmetry are possible mathematically, but do not apply to any known particles' [21, section 54, p 211]. These difficulties notwithstanding, variants of Dirac's original (1926) argument are frequently encountered in textbooks of quantum theory. As pointed out, for instance, in [14,22], these arguments are unsatisfactory in that they make implicit assumptions of an abstract nature. For example, a common argument is that a stateψ(x 1 , x 2 ) of two identical particles must be an eigenstate of the permutation operatorP which describes a 'swap' of the two particles, from which it follows that only symmetric or antisymmetric states are possible. However, it is not clear why physically realizable states should be restricted to eigenstates ofP (this does not follow from the requirement that expectation values be the same for the original and permuted state; such a requirement can be satisfied by restricting observables, A, to those that are permutation invariant, , 0 A P = [ ] , without imposing any restriction on the allowable states [14]); and, even if such a restriction is granted, the argument does not generalize to three or more particles unless one rules out the possibility of describing the physical state of a system by means of Hilbert subspaces of dimension higher than one(admitting this possibility leads to so-called paraparticles) [14,19,20]. Another common argument is to require that the probability distribution x x , 2 1 2 y | ( )| associated with two 'swapped' particles be the same as x x , 1 2 2 y | ( )|, and to argue that this impliesψ(x 2 , x 1 )=e if ψ(x 1 , x 2 ), and hencee if =±1, a condition only satisfied by symmetric or antisymmetric states. However, such an argument tacitly assumes thatf is independent ofx 1 , x 2 [22]. searches for specific hypothetical deviations('quons', for example [15]) from bosonic and fermionic behavior have been carried out, but no such deviations have yet been reported(see, for example, [16][17][18]).
The above considerations suggest that neither interpretation is satisfactory, and that a more thoroughgoing revision of our conceptual picture is necessary if we are to pinpoint the essential idea that underlies the behavior of identical particles and rigorously account for the empirical success of Dirac's symmetrization procedure. Now, as we have noted, identical particles sometimes behave as if persistent(for example, two electrons moving along distinct particle 'tracks'), and sometimes as if nonpersistent(as in Bose's photon-counting procedure). This suggests that, rather than trying to account for this behavioral diversity on the basis of persistence or nonpersistence alone, we instead attempt to combine both of these pictures in a more even-handed way.
In this paper, we develop a novel understanding of the quantum mechanical treatment of identical particles along these lines. We adopt an operational approach in which the raw data consists of identical localized events.
To be concrete, one can think of observing a fixed number of localized light-flashes of the same color at successive times. At this stage, there are no 'identical particles' as such, only identical events. We construct two distinct models of these events, namely a persistence model and a nonpersistence model. These differ in whether or not it is assumed that successive events are generated by individual persistent entities('particles'). We then show that these models can each be described within the Feynman formulation of quantum theory and be synthesized to derive the Feynman's form of the symmetrization procedure [23]. As we show elsewhere [24,25] and summarize here, this procedure can be transformed into a state-based symmetrization procedure which is empirically adequate yet differs from Dirac's procedure in form and meaning, in particular allowing for the natural emergence of reidentifiability in special cases.
We then show that the persistence and nonpersistence models, and the manner of their synthesis, satisfy the key characteristics of Bohr's concept of complementarity. Specifically, we show that these models are mutually exclusive, but that they can be synthesized to generate an empirically-adequate predictive model. On this basis, we propose that the quantal behavior of identical particles reflects a complementarity of persistence and nonpersistence, analogous to the way in which the behavior of an individual electron reflects a complementarity of particle and wave.
Finally, we exhibit a precise parallel between our proposed persistence-nonpersistence complementarity and Bohr's wave-particle complementarity. In particular, we show that the Feynman amplitude sum rule can be viewed as a synthesis of the wave and particle models of elementary constituents of matter in a manner that formally mirrors the way in which a symmetrization procedure arises through a synthesis of persistence and nonpersistence models of identical localized events. These two examples thereby illustrate how complementarity can be turned into a precise methodology for synthesizing mutually-exclusive models.
The remainder of this paper is organized as follows. In section 2, we define the concepts of persistence and nonpersistence in an operational manner. In section 3, we outline the derivation of the symmetrization procedure. In section 4, we describe our complementarity interpretation of identical particles in light of this derivation, and establish a detailed parallel with Bohr's wave-particle complementarity. We conclude in section 5 with a discussion of the broader context and some open questions such as the interpretation of the quantum field theoretic formalism in light of our complementarity interpretation of the quantum mechanical treatment of identical particles.

Operational framework
Our discussion will be based on the fundamental notions of persistence and nonpersistence. In order to place these notions on a clear footing, we begin by stepping back from the familiar theoretical frameworks of classical and quantum mechanics, and instead adopting an operational perspective. Such a perspective is helpful in identifying assumptions that are of limited validity in physical domains remote from everyday experience, but that may be too entrenched in our customary patterns of thought to be clearly and consistently perceived.
1. Position measurements of localized events and their properties. Consider a situation where a position measurement, implemented by a fine grid of detectors that tile a region of space, is performed at discrete timest 1 , t 2 , ...(see figure 1). Here and subsequently, we restrict consideration to one spatial dimension for simplicity. Suppose that only two detectors fire at each time. We can speak of each such detection as a localized event(a 'flash'). Suppose further that these detectors are capable not only of registering a localized event, but also of measuring some additional properties of this event. For concreteness, we henceforth imagine that there is just one additional property, namely color; thus, at each time, one observes two colored flashes.
Let us further suppose that observation shows that these additional properties are conserved, in the sense that the total number of flashes of each color seen at each time is the same. For example, at each time, one obtains a blue flash and a red flash. If it should be the case that both of the localized events have the same additional property values(for example, both the flashes are blue), we shall say they are identical. That is the situation that concerns us here. Finally, let us suppose that the system is isolated. Operationally, this can be established by carrying out repeated trials, and showing that the probability over the locations of the two detections att 2 are determined by the locations att 1 , a condition we refer to as closure [26].
On the basis of these observations, we say that the measurements are being performed on a 'system', namely a persistent object that is such that (i)it yields two localized events at each timet i , and (ii)the property-values(colors) of these events is conserved.

Persistence model
We now construct a model of data in which two identical localized events are registered at each time. Let us assume that there exist individual entities that persist in between these detections, and that these entities can (informally) be said to cause these detections. We shall refer to these entities as particles on the understanding that this word describes the localized, particle-like way that these entities manifest themselves upon detection, rather than implying anything about their nature between such detections. We ascribe intrinsic and extrinsic properties to these particles. The former are the same as the additional properties measured of the localized events, and their values are assumed to be constant. In this case, the flashes are the same color, so we say that there are two particles of the same color, and we assume that the color of each particle is constant. As the colors of two particles are the same, we shall say that they are identical.
On the basis of this persistence model, one can now meaningfully say that same particle is responsible for detections at two different times. Suppose that events are detected at locationsℓ 1 andℓ 2 at timet 1 , and atm 1 andm 2 att 2 . Here we adopt the labeling convention thatℓ 1 , m 1 are the left-most locations at each time.
According to the persistence model, even though the two events at each time are identical, one can say that one or the other of the following propositions is true: A='the same particle is responsible for the detectionℓ 1 att 1 and the detection atm 1 att 2 ,' or B='the same particle is responsible for the detectionℓ 1 att 1 and the detection atm 2 att 2 .' 1. Reidentifiability. If it is possible for an observer to determine which of the above propositions is true, we shall say that it is possible to reidentify each particle(see figure 2). That is, reidentification is the observational counterpart of the theoretical notion of persistence. As the particles are identical(same color), reidentification on the basis of measurement of their intrinsic properties is impossible, so that the possibility of reidentification will hinge upon additional theoretical and operational considerations.
In particular, if one makes the additional theoretical assumption that particle motion is continuous, and further assume that it is possible for an observer to make non-disturbing position measurements of arbitrary The measurements at timet 1 andt 2 yield outcomesℓ 1 , ℓ 2 andm 1 , m 2 , respectively, with the labeling convention that the left-most locations areℓ 1 andm 1 . An additional property-'color' in our example-of each event is measured, according to which the events are said to be nonidentical or identical. In the example here, the events are identical, illustrated by the two filled circles at each time. The set of these additional properties at each time is conserved, illustrated here by there being two filled circles at each time.
Right: a persistence model of the data posits that two persistent entities('particles') are responsible for these detections. Each particle is ascribed color as an intrinsic property, whose value is constant. In the example here, the particles are the same color, and are thus identical. On this model, one can say either that the same particle is responsible for detectionsℓ 1 andm 1 (proposition A) or that the same particle is responsible for 1 ℓ andm 2 (proposition B).
precision at arbitrarily high frequency, then reidentification(at arbitrarily high confidence level) of identical particles is possible on the basis of the measurement records. 2. Persistence grounds particle labels. If persistence is assumed, then either propositionA orB, as given above, is true. This provides the basis for particle labeling. Specifically, let us label'1' the particle that was atℓ 1 att 1 , and label the other particle'2'. Then, the configuration of the system att 1 is given by the ordered pair(ℓ 1 , ℓ 2 ), and the configuration at timet 2 is either(m 1 , m 2 ) or(m 2 , m 1 ). Each of the latter two ordered pairs thus reflects not only the observed particle positions att 2 , but also-as a result of the theoretical assumption of persistencesome information about the observed positions at the earlier (reference)time,t 1 . For example, the configuration(m 1 , m 2 ) specifies not only the information that there are two identical particles at locationsm 1 andm 2 (which is what is observed at t 2 ), but also that the particle that is now atm 1 was earlier atℓ 1 .
To say that reidentification is, in principle, possible, means that there exists an observer who can determine which of these configurations-(m 1 , m 2 ) or(m 2 , m 1 )-is in fact the case. But, for an observer who is not capable of reidentification, there is a gap between the theoretical level of description of the system-the configurationand the information available to that observer. For example, att 2 , the theoretical description might be the configuration(m 1 , m 2 ), but the observer would be incapable of distinguishing this from(m 2 , m 1 ).
3. Connection to classical mechanics. In classical mechanics, persistence is assumed, and an ideal observer is capable of reidentifying identical particles(provided they cannot coincide) by tracking their continuous trajectories with arbitrary precision but without causing disturbance. Therefore, the theoretical and observational descriptions coincide in the configuration r r , 1 2 ( ) of two particles.

Nonpersistence model
We can, however, construct a second model-a nonpersistence model-of the identical localized events in which one does not presume that there are individual persistent entities that underlie the individual localized detections. Rather, the two localized events at each time are regarded as a manifestation of a single abstract 'system' that persists. One is thus left with the bare data of localized events{ℓ 1 , ℓ 2 } att 1 and{m 1 , m 2 } att 2 . Repeated trials of the experiment would yield a conditional probability distribution m m Pr , , In such a model, the only persistent object is an abstract 'system' which yields two localized detection events at each measurement time. As this object is not analyzed into two separate persistent objects, the fundamental techniques that one ordinarily employs in constructing particle-based models are unavailable. For instance, in classical mechanics, one can start by positing that individual objects move uniformly if isolated, and then build up a model of a system of how two such objects interact with one another by imposing constraints in the form of conservation laws; but such a model-building strategy hinges on an analysis of the system into persistent individuals which is unavailable in the nonpersistence model.

Derivation of a symmetrization procedure
As indicated in the Introduction, the quantum treatment of identical particles brings the assumption of persistence into question. Without the assumption that identical localized events are underpinned by individual persistent entities, the possibility of creating a model of the events based on analysis into persistent entities is blocked.
On the other hand, we have observational evidence that persistence is at least approximately valid in certain situations, for example in the case of identical localized events in a bubble chamber. In order to construct a predictive model, we incorporate both of these pieces of observational evidence by formulating two models of these events-one that assumes non-persistence and the other that assumes persistence-and then posit a connection between them.

Synthesis of persistence and nonpersistence
To be specific, consider again the above experiment involving position measurements at timest 1 andt 2 . We now construct two theoretical models of this situation within the Feynman quantum formalism(see figure 3). In the persistence model, irrespective of whether or not reidentification is possible, we can say that either one or the other transition connects the observed data-a direct transition, where the particle that was atℓ 1 att 1 is found atm 1 att 2 ; or an indirect transition fromℓ 1 tom 2 . Let us denote the amplitudes of these transitionsα 12 andα 21 , respectively.
The second model, the nonpersistence model, does not presume that there are persistent entities that underlie the observed localized detections. Accordingly, the only amplitude that one can associate with the given data in this model is the transition amplitude,α, from the initial data{ℓ 1 , ℓ 2 } att 1 to the final data{m 1 , m 2 } att 2 .
The connection between these two models takes the form of the operational indistinguishability postulate (OIP), which posits that the amplitudes in the persistence model determine the amplitude in the nonpersistence model. In the case of measurements at two successive times under consideration here whereH is an unknown continuous complex-valued function to be determined. The OIP also applies to the case where one has observations at three successive times,t 1 , t 2 andt 3 . In that case, the persistence model has two possible transitions between timest 1 andt 2 , and two possible transitions between timest 2 andt 3 , so that there are four possible transitions overall from timest 1 tot 3 viat 2 . Let the amplitudes of these transitions be denotedγ 11 , γ 12 , γ 21 , andγ 22 . Then the generalization of equation (1) reads whereγ is the transition amplitude in the nonpersistence model, andG is a function to be determined. We now incorporate the fact that there are situations in which we commonly say that the particle observed now is the 'same' as one previously observed via an isolation condition. This condition stipulates that, in the limiting case that isolation obtain for one of more of the identical particles in a given system, they can be treated as a persistent subsystem for the purpose of making predictions. For example, if the electron in a hydrogen atom is effectively isolated from all other electrons in a given system, then we can infer that the same electron is responsible for the successive electron-detections in the atom. One can accordingly apply the quantum formalism to this electron as if it were a (persistent)system. Formally, for the case of two identical events, the isolation condition requires that the transition probability H , 12 21 2 a a | ( )| is the same as the probability of the persistence-model transition that has non-zero probability. For example, if the direct transition is the one with non-zero probability, then From the assumptions above, Feynman's symmetrization procedure can be derived [24]. The key idea behind the derivation is the recognition that the amplitude for a particular process in the nonpersistence model can sometimes be computed in two different ways, and, in these instances, consistency of the assumptions implies that these calculational paths must agree. Each such call for consistency leads to a functional equation.
For example, one obtains Solution of these functional equations yields , 5 12 21 a a a =  ( ) where the±sign corresponds to bosonic or fermionic behavior. This is Feynman's symmetrization rule for two particles. The above derivation generalizes naturally toN identical particles.

Probabilistic reidentifiability
If one of the transition probabilities 12 2 a | | or 21 2 a | | is much smaller than the other, then the transition probability H , 12 21 2 a a | ( )| approximates to the largest of these probabilities. In that case, one can treat the observational data as a probabilistic version of the persistence model, so that one has probabilistic reidentifiability of the particles. Such a situation obtains, for example, for two electrons in the field of view of a particle chamber, where we can roughly say that each electron travels along its own 'track', even though there is a finite probability(as computed using the persistence model) of the electrons 'swapping' between tracks.

State representation of the symmetrization procedure
The amplitude-based symmetrization procedure given above can be re-expressed in terms of states and observables, a more familiar arena for the description of quantum phenomena. Such a re-expression also facilitates a direct comparison with Dirac's symmetrization procedure. To illustrate the key ideas, it suffices to consider two particles moving in one dimension. In that case, one can [24] re-express equation (1) in terms of states as Although this relation resembles the Dirac symmetrization postulate, it has a quite different meaning. First, the states that occur on the left and right hand sides belong to different models. The function ID y on the left hand side is the state of the system as described within the nonpersistence model. This state is defined over location-space(or reduced configuration space), namely x x 1 2  , and is normalized over that space. Thus, in the expression x x , ), x i is to be interpreted as the x-value of locationi. Consequently, x x dx dx , )| is the probability of detection of two identical particle-like events, one in the interval[x 1 , x 1 +dx 1 ] and the other in[x 2 , x 2 +dx 2 ].
In contrast, the functionψ on the right hand side of equation (7) is the state of the system as represented in the persistence model. It is defined(and normalized) over the full configuration space(x 1 , x 2 ) ä R 2 . Thus, in the (ii)Right: in the nonpersistence model, it is meaningless to say that a given detection att 2 was caused by the same thing as given detection att 1 . Accordingly, the figure on the right shows the only transition amplitude,α, that one can meaningfully assign according to this model. The operational indistinguishability postulate(OIP) posits the relationα=H(α 12 , α 21 ) between the amplitudes in these models, whereH is a complex-valued function to be determined. In [24], it is shown thatα=α 12 ±α 21 , with the sign corresponding to bosonic or fermionic behavior.
expressionψ(x 1 , x 2 ), x i represents the x-value of the location of particlei. Consequently, x x dx dx , The above relation is thus to be understood as establishing a formal connection between these two models, a connection which can be used to take states from the persistence model over to the nonpersistence model.
For convenience, one can formally extend x x , 1 2 ID y ( )to the entire configuration space, to obtain state x x , 1 2 ID y  ( ), in terms of which one can rewrite the above equation as where now(x 1 , x 2 ) ranges over 2  . Such a formal extension is useful in the sense that now ID y  andψ are both defined over 2  and both formally live in the same space(a tensor product of two labeled copies of a oneparticle Hilbert space). However, the formal extension makes the reading of the labels more complicated:althoughx 1 , x 2 on the right hand side are particle labels as before, x 1 on the left is the location of the leftmost event whenever x x 1 2  , but the location of the rightmost event wheneverx 1 >x 2 . If the two particles(as viewed in the persistence model) are confined to disjoint regions on the left and right sides, thenα reduces toα 12 . Further, owing to the isolation between the particles, the surviving amplitude can be written as a product of two amplitudes, one related to each particle. Thus, in the state formulation, x wheref a , f b can be viewed as one-particle states of labeled particles which have no common support. Under these circumstances, the observablex 1 in the nonpersistence model, which by default is the x-location of the left-most event, gains the additional meaning of the x-location of particle1. Thus, in this limiting case, one recovers reidentifiability, and one has justification to model each of the particles as a distinct entity without regard for the other.

Comparison with Dirac's interpretation of the symmetrization procedure
We are now in a position to contrast the understanding of identical particles developed above with the Dirac's interpretation of his symmetrization procedure. According to Dirac, 'if a system in atomic physics contains a number of particles of the same kind, e.g. a number of electrons, the particles are absolutely indistinguishable from one another' [7, section 54]. That is, in our terminology, the system can be described within the persistence model(so that the identical particles are modeled as persistent entities), but no observations are capable of reidentifying these particles.
In order to formalize this idea, Dirac proposed that a system of identical particles be described within a labeled tensor product of single-particle subspaces by states that are symmetric or antisymmetric in the subspace labels, and that observables be restricted to those that are symmetric in these labels. According to Dirac's interpretation, in a stateψ(x 1 , x 2 ) describing two identical particles, i is a particle label, so that x i is the location of the ith particle. Accordingly, the states ID y andψ in equation (7) are interpreted as states in the same(persistence) model, and equation (7) itself is interpreted as a way of (anti-)symmetrizing the stateψ(itself obtained by solving for the eigenstates of a measurement operator symmetric in the particle labels) in order to generate a physically legitimate state, ID y .
However, as we have detailed in the introduction(see, in particular, footnote 1), Dirac's understanding of identical particles(as persistent but not reidentifiable) does not lead to his proposed formalization without the use of additional assumptions that lack physical justification. This naturally raises the question of whether Dirac's formal procedure can be interpreted in a different manner. As we have shown above, this is indeed possible. In essence, we view equation (7) not as a means of 'symmetrizing' a state within a single(persistence) model, but rather as establishing a connection between the states of the same system as described in two different models-a persistence model and a non-persistence model. The fact that we are able to derive equation (7) by formalizing this interpretation speaks in its favor.
In this connection, we briefly note that the coherency of Dirac's interpretation-specifically, the view that the labels in a symmetrized state refer to particles(a view recently dubbed factorism [12])-has previously been brought into question by several others authors [10-12, 27, 28]. As we have pointed out in the Introduction, a major difficulty with Dirac's particle-label view is that, in a system of two electrons, both of the electrons have same reduced state, even if they are associated with widely-separated hydrogen atoms. However, this implies that each electron is as likely to be found in the vicinity of one atom as the other [10,11], which is difficult to reconcile with our usual presumption that, owing to their isolation, one can treat each of these electrons as a distinct system confined to its respective atom. Given these and related difficulties, Dieks and Lubberdink conclude [10] that 'the quantum mechanical symmetrization postulates do not pertain to particles, as we know them from classical physics, but rather to indices that have a merely formal significance.' However, such a claim leaves the challenge of formulating an alternative understanding of these indices which, for instance, is capable of rendering intelligible the usual procedures for interpreting measurement operators. For example, if one applies the symmetrization procedure to the electrons in a helium atom, the measurement operator(x 1 −x 2 ) 2 is ordinarily interpreted as representing a measurement of the squared-distance between the two electrons; but it is unclear how one would justify such an interpretation if the indices1 and2 have a 'merely formal significance'.
However, according to the interpretation that we have developed above, the indices do have a physical significance, the hitherto unexpected complication being that the physical referent of an index depends upon whether the state in question is viewed as a state within the persistence model(in which case an index is a particle label) or the nonpersistence model(in which case an index is a location label). This interpretation has a number of ramifications-for instance for the understanding of the meaning of measurement operators and for the proper understanding of the entanglement of identical particles-which will be discussed elsewhere.

Complementarity of persistence and nonpersistence
As formulated by Bohr, complementarity expresses a view about the kind of theoretical model that one can formulate about microphysical phenomena. As such, it constitutes a major pillar of Bohr's interpretation of quantum theory. Bohr gave a detailed account of the notion of complementarity in his 1927 Como lecture [29], a notion which he subsequently developed in a series of papers over the course of the next thirty years. As originally described in [29](and subsequently elaborated upon in other papers, such as those reprinted in [30]), complementarity maintains that, in order to account for data observed in a given experiment on a microphysical system(such as an electron diffraction experiment), one must draw upon elements from two apparently incompatible classical pictures 2 .
We note that, after his engagement with the EPR paper in 1935, Bohr emphasized a different complementarity [31], namely that between information gained about the same microscopic object in different experimental arrangements. Although this notion of complementarity has been widely influential(it can, for instance, be formalized in terms of quantum state tomography via mutually unbiassed bases [32], and has served as a basis for some reconstructions of quantum theory [26,33]), the notion of complementarity that concerns us here is the original one described in [29]. Its key constituent ideas(which we shall subsequently illustrate) can be abstractly summarized as follows: 1. Need for two incompatible models. A theoretical understanding of experiments on microscopic systems in general requires the use of concepts drawn from two distinct models of the type that are characteristic of classical physics. These models are incompatible in the sense that some of core assumptions about the nature of physical reality made in one model are in conflict with some of the core assumptions made in the other.
2. Synthesis of models. At least some of the models' incompatibility arises from abstract assumptions which are not directly refutable via observation. Accordingly, there is sufficient latitude to combine key features of the two models into a new predictive calculus. The resulting synthetic model is not of the classical type, and has a symbolic or abstract(non-visualizable) character. Nevertheless, it provides a predictive framework for behavior that in some sense interpolates between that permitted by the original models.
3. Limiting cases. Whereas a quantitative understanding of the phenomena observed in general requires the use of the synthetic model, certain physical situations can be understood using only one of the original models. Correspondingly, in certain limiting cases, the behavior predicted by the synthetic model approximates to that predicted by one of the original models.

Wave and particle
We illustrate the above-mentioned key features of complementarity via Bohr's paradigmatic example of the wave and particle models of the electron. First, according to the particle model, the electron is a point-like entity that has a definite position at each time. In contrast, the wave model treats the electron as a delocalized wave-like object 3 . Each of these models paints a clear, visualizable conception of the electron, but are manifestly incompatible-an electron cannot 2 The following are representative quotations from [29] 3 In classical physics, one often regards a 'wave' (such as a water wave, or a wave on a string) as underpinned by local disturbances over a region of space. In such cases, 'wave' is merely a collective noun that refers to a set of synchronized local disturbances, and so has no fundamental existence in and of itself. However, in the wave model of an electron, such an underpinning is not presumed, so that 'wave' is taken to refer to an unanalyzed, delocalized object. simultaneously be a localized point-like object as well as a delocalized object. Yet, the need for both models is made plausible by the existence of experimental phenomena(such as diffraction) associated with electrons where detections are point-like whereas the distribution of detections is wave-like.
Second, the particle model's assumption that an isolated electron is continuously localized is not directly testable-the observational data consists only of point-like detections(modeled as interactions between the electron and the detector) at discrete times. Hence, although the models themselves are at odds if their core assumptions are regarded as strictly true, there may be sufficient latitude to synthesize key features of these models whilst remaining consistent with observations. Such a synthesis was first proposed by de Broglie in his composite wave-particle model of the electron, a proposal that culminated in Schroedinger's wave mechanics. In the latter synthesis, the wave aspect is reflected in an evolving wavefunction, while the particle aspect is reflected in the projection(or Born) rule, enabling the model to incorporate behavior that is characteristic of each of the original models. Unlike the wave or particle models, the synthetic model is noncommital as to the nature of the electron, and accordingly has a relatively abstract character.
Third, there are limiting situations in which the behavior of an electron is well-modeled by the particle picture. Such a situation obtains, for example, when an electron is subject to potentials that are sufficiently slowly varying in space and time, and to position measurements that are sufficiently coarse. Additionally, in such limiting situations, the synthetic model approximates to a statistical particle model.

Persistence and nonpersistence
Wave-particle complementarity suggests that the assumption of continuous localization, a notion extrapolated from everyday perception, does not enjoy absolute validity in the microscopic realm. We propose that the assumption of persistence is similarly restricted in its validity. Furthermore, we propose that the understanding of identical particle-like events requires a synthesis of complementary persistence and nonpersistence models, analogous to the way that localization('particle') and delocalization('wave') models need to be combined in order to understand individual microscopic events.
Our proposal is based on the following considerations. First, through an operational analysis of an experiment in which identical particle-like events are registered at each instant(section 2), we have seen that one can construct two distinct models. One of these assumes that successive detections are underpinned by persistent underlying entities('particles'), whilst the other assumes that no such entities exist, but rather that the events at each instant are the manifestation of a single abstract 'system'.
Second, we have demonstrated that these models can be synthesized to derive a quantum symmetrization procedure for describing the behavior of a system of identical particle-like events. This derivation not only places a formally ad hoc formal procedure on a clear operational and logical foundation, but also naturally resolves the difficulty in reconciling the assertion that identical particles are nonpersistent or not reidentifiable with their manifest reidentifiability in particular situations(section 3.2).
Third, we assert that persistence and nonpersistence are complementary descriptions of identical particle-like events on the grounds that the persistence and nonpersistence models satisfy the three key features of complementarity, as follows.
First, these two models are mutually exclusive in the sense that they make contradictory assumptions about whether or not successive individual detections are underpinned by individual persistent entities. Consequently, the statement that 'this detection was caused by the same object as a previous detection' is meaningful in the first(persistence) model but not the second.
Second, despite their contradictory nature, these models can be synthesized. That this is possible is due to the fact that the both models make claims about the nature of the entity of entities that exist in between the detections, claims that can(by their very nature) only be indirectly probed via experiment. Insofar as the synthetic model combines models that make conflicting claims as to the nature of the entity (or entities) that underpin the detections, the synthetic model cannot, in general, be interpreted as positing that the detections are or are not underpinned by persistent entities, and is, in that sense, abstract in comparison with the original models. As we have seen, the synthesis is formally realized by using the persistence model to calculate amplitudes that are then combined to generate an amplitude in the nonpersistence model. The resulting synthetic model allows for behavior intermediate between that allowed by the two original models.
Third, in the limiting case where an electron is isolated(as judged within the persistence model), it can be regarded as if persistent, and treated entirely within the persistence model. Correspondingly, in this limiting case, the synthetic model yields the same predictions as the persistence model. 4.3. Parallel between wave-particle and persistence-nonpersistence complementarities The above considerations suggest that there is a close conceptual parallel between wave-particle and the proposed persistence-nonpersistence complementarities. We now construct a more precise formal parallel between them.
One of the difficulties in clearly understanding Bohr's formulation of complementarity is that it refers to classical models, namely the 'wave' and 'particle' model, which are not precisely defined, and whose synthesis is far from transparent. For this reason, a derivation of quantum theory in which the quantum formalism can be interpreted as arising from the synthesis of two apparently contradictory pictures would be most useful. Our previous operational derivation of Feynman's formulation of quantum theory [26] supports such an interpretation.
We begin by noting that, in Feynman's formulation of quantum theory, one can interpret the Feynman amplitude sum rule as a succinct formal encapsulation of Bohr's notion of wave-particle complementarity. Specifically, in the context of an electron double-slit experiment, there are two models of the electron(see table 1, first column). In the 'particle' model, the electron is treated as a localized, particle-like entity that traverses one slit or the other in its passage from the source to a given point on the screen. In the 'wave' model, the electron is treated as a delocalized object about which all one can say is that it passes through the silts. The amplitude sum rule posits a relationship between the transition amplitudes in these two models.
In our previous reconstruction of Feynman's formulation, we have shown that the Feynman sum rule can be derived using the postulate that(roughly speaking) the amplitudes of the two possible paths in the particlemodel determine the amplitude of the 'path' in the wave-model. A benefit of this reconstruction is that the two models being synthesized are precisely defined in operational terms. For example, the 'particle' model consists solely in being able to assert that a system passed through one slit or the other in its passage from source to screen -no additional properties(such as mass, energy or momentum) or associated classical mechanical equations of motion are implicitly included in the model. Thus, we shall now interpret this derivation as showing, in precise, operational terms, how Bohr's wave-particle complementarity can be viewed as the basis of a constructive derivation of the Feynman sum rule, in the same way that we have shown that the proposed persistencenonpersistence complementarity provides the basis of a derivation of Feynman's symmetrization procedure.
As shown in the table 1, the derivation of the Feynman sum rule allows us to exhibit a precise formal parallel between the two complementarities. In each case, two models are synthesized. One of these models permits an analysis of the situation into abstract parts-'the electron passes from the source to screen via one slit or the other' or 'the identical particles that underpin the particle-like flashes make a direct or an indirect transition from one time to the next'. This analysis allows two distinct amplitudes to be defined and, in principle, calculated by making use of the Dirac-Feynman amplitude-action quantization rule [34] and an appropriate classical model of the situation. In contrast, in the other model, no such analysis is possible-all one can say is that 'the electron passes from the source to screen via the slits' or that 'two identical particle-like events occur at each of two successive times'. One can associate a transition amplitude with such an unanalyzed process, but its calculation appears impossible if one remains within the compass of this model because no corresponding classical model exists. Although the two models in each case are contradictory in their assumptions, they can be synthesized:if one posits that the two amplitudes in each analytic model determines the amplitude in the corresponding non-analytic model, it is possible to derive the form of the functional form of the relationship between the amplitudes.
An important distinction between the two complementarities, however, deserves to be noted. As shown in [26], the Feynman sum rule can be regarded as arising from a connection between two distinct experimental arrangements-one in which there are which-way detectors at each slit, the other in which there is a single large detector covering both slits [26]. That is different from the interpretation we are adopting here, namely that the sum rule is a connection between two models('particle' and 'wave') of the same arrangement in which there is a single large detector. However, in the derivation of the Feynman symmetrization procedure for identical particles described here, only one of these two options is available:the symmetrization procedure arises from the connection between two distinct models of the same experimental arrangement. Thus, the symmetrization procedure seems to express the notion of complementarity at a deeper level, namely as a synthesis of mutuallyexclusive models of the same situation.

Discussion
It is widely believed that identical particles differ from nonidentical particles in that the former lack persistence or reidentifiability. We have pointed out the deficiencies of such a view, and proposed instead that the specialness of identical particles lies in the fact that both persistence and nonpersistence models must be employed in order to cover their full range of behavior. We have proved the viability of this viewpoint by showing how the Feynman and Dirac symmetrization procedures that are employed to treat systems of identical particles can be systematically derived through a synthesis of the persistence and nonpersistence models. We have also indicated how reidentifiability emerges in a context-dependent manner.
We have further shown that the persistence and nonpersistence models, and the manner of their synthesis, satisfy the key characteristics of Bohr's concept of complementarity. On this basis, we have proposed that the quantal behavior of identical particles reflects a complementarity of persistence and nonpersistence, analogous to the way in which the behavior of an individual electron is rendered intelligible through Bohr's wave-particle complementarity. Finally, we have constructed a precise parallel between these two complementarities, which brings their conceptual similarities and dissimilarities into sharper focus.
We conclude with a few brief remarks on which we expect to elaborate elsewhere. 1. Relationship between the two complementarities. The parallel between the persistence-nonpersistence and wave-particle complementarities raises the question of whether there is a single broader perspective from which both complementarities can be seen to emerge, and indeed whether other related complementarities exist. We leave this question open, apart from noting the presence in each of an essential tension between 'whole' and 'part'. That is, in each complementarity, one model permits analysis into 'parts' (either due to the assumption of persistence or the assumption of continuous localization) whereas the other describes the situation as an unanalyzed whole.
2. Relation between the quantum mechanical and quantum field theoretic models of identical particle-like events. According to the thesis put forward here, the symmetrization procedures used in the description of identical particle-like events are a bridge between the quantum mechanical descriptions of two different models of identical particle-like events. This bridge allows one to compute evolving states(or to compute transition amplitudes in the Feynman picture) in the persistence model, and then to combine these in specific ways(as specified by the Dirac or Feynman symmetrization procedure) to yield evolving states(or transition amplitudes) in a nonpersistence model. As the nonpersistence model regards all the events recorded at each instant as manifestations of an abstract system, the number of these events is a state-determined property(rather than an intrinsic property) of this Table 1. Parallel between the complementarity interpretations of (i)amplitude sum rule(exemplified by double slit) and (ii)symmetrization procedure(for two identical 'particles'). Case(i): given detections at a source and screen, the 'particle' model posits that a particle-like object traversed either one slit or the other. The 'wave' model eschews such an assumption, treating the electron as a delocalized object('wave'). These apparently contradictory models can nevertheless be synthesized by positing a functional relationship,c=F(a, b), between the transitions amplitudes in these models. As shown in [26], the unknown function can be obtained within a broader derivation of Feynman's rules, yieldingc=a+b (Feynman's amplitude sum rule). In the limiting case where 'interference' can be neglected, a statistical classical model(p c =p a +p b ) is recovered. Case(ii): the persistence model assumes that detections are underpinned by persistent entities, an assumption eschewed by the nonpersistence model. Model synthesis is enabled by the assumption that the relationα=H(α 12 , α 21 ) holds between the amplitudes in these models. Within the Feynman framework,H can be solved [24] to yield the Feynman form of the symmetrization procedureα=α 12 ±α 21 , where the sign corresponds to fermionic or bosonic behavior. The situation is well-described by the persistence model if isolation obtains(so that the magnitude of either α 12 or α 21 is negligible); in this limit, the synthetic model predictions reduce to that of the persistence model.

Amplitude sum rule
Symmetrization procedure Two mutuallyincompatible models 'Particle' and 'Wave' models Persistence and nonpersistence models

Synthesis of models
Limiting cases When 'interference' can be neglected, a classical particle model applies; correspondingly, the synthetic model approximates to a statistical classical model(p c =p a +p b ).
If the particles are isolated from one another as viewed in the persistence model(either α 12 or α 21 can be neglected), the system is well-described by the persistence model alone; correspondingly, in the synthetic model, a | | reduces to 12 a | | or 21 a | |.
system. Accordingly, a natural generalization of the nonpersistence model considered above would allow a variable number of events to be detected at each instant, and correspondingly allow state-transformations in which the number of events can change.
One can regard the quantum field theoretic treatment of identical particle-like events as such a generalization. It is commonly assumed that, in the quantum field theoretic framework, there are no persistent entities(apart from the abstract system itself), only 'excitations'. From our standpoint, this is correct only as long as the (anti-)commutation relations amongst the creation and annihilation operators are not considered. As these relations ensure that the states in this field theoretic model agree with those of the quantum mechanical model for constant event(or particle) number, the persistence model(and the complementarity between persistence and nonpersistence) implicitly enters once the (anti-)commutation relations are imposed. This leaves open the question of how one interprets superpositions of states with differing particle numbers from the complementarity point of view.
3. Connection to everyday experience. We ordinarily assume that the appearances perceived in the present moment are underpinned by objects that: (i) persist in the time between these appearances; and (ii) assume forms that coincide with those of these appearances, not only at the moment of perception but also during the intervening intervals.
As an extrapolation from what is directly perceived, these assumptions constitute a 'theory' developed very early in life 4 , and are written into the foundation of classical physics. The above complementarities bring this extrapolation into question, at two distinct levels: (i) wave-particle complementarity brings into question the assumption that an object takes the same form between observations as it does during its appearances.
(ii) persistence-nonpersistence complementarity brings into question the more basic idea that an object exists between observations and underpins them.
Nevertheless, the synthesis of complementary models(wave and particle models; or persistence and nonpersistence models) yields a theory that fits the observations. Thus, on the one hand, one can regard complementarity as pointing to the limitations of our ordinary models of the appearances; but, on the other, as offering a constructive path to transcend these limitations.