SPACE-TIME INDIVIDUALITY AND CONFIGURATION SPACE
We have seen that individuating the particles in terms of their properties in general is problematic. The classical alternative of both individuating and distinguishing them in terms of spatio-temporal location, in conjunction with the Impenetrability Assumption (playing the role of guarantor) also faces acute difficulties. If the standard interpretation is adopted then it can be shown that the family of observables corresponding to the positions of single particles cannot provide distinguishing spatio-temporal trajectories. However, as is well known, the Bohm interpretation does allow for the introduction of such trajectories; indeed, the only observable admitted is that of position. Hence, this might seem the natural home for some form of Space-Time Individuality. Let us briefly consider this option.
On this interpretation we have a dual ontology of point particles plus 'pilot' wave, where the role of the latter is to determine the instantaneous velocities of the former through the so-called 'guidance equations'. 105 These are taken to 'complete' the standard formulation of quantum mechanics so that, in addition to the quantum state, whose development is determined by the Schrödinger equation, there is also a set of single-particle trajectories, each of which is determined by the guidance equation (plus the initial positions of the particles, of course). 106
It is part of the attraction of this approach that it retains, or appears to retain, a form of classical ontology in which the standard philosophical problems of wave-particle duality are apparently resolved by effecting a metaphysical split or division of labour, with particles traversing well-defined spatio-temporal trajectories guided by the pilot wave. This obviously meshes well with our metaphysical package of regarding the particles as individuals. 107 Indeed, a form of PII(1) can now be defended against the results of the previous section. As Belousek has emphasized, the conclusion that PII is violated is based on
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the restriction of the relevant set of properties to those that can be represented in terms of Hermitian operators. 108 This excludes the possibility of invoking dynamically relevant properties that are representable in terms of real-valued continuous functions, as in the Bohmian case. If such a possibility is allowed, then it turns out that the Indistinguishability Postulate can still be satisfied, while the particles possess distinct, dynamically relevant properties (their positions) which allow them to be distinguished and hence individuated. 109 PII is thus preserved in more or less the same way it is in the classical situation-via the spatio-temporal positions of the particles.
However, the ontological picture is not quite as straightforward as it might seem. Certain interference experiments-interpreted within this framework-seem to imply that the properties of the particles actually belong to the pilot wave rather than the particles themselves, thus undermining the classical nature of the ontology. 110 If such properties are rendered as 'non-local' as the state-dependent ones, it is difficult to see how they can be regarded as possessed by individual particles. Here a 'principle of generosity' has been proposed which assigns these properties to both the particle and the pilot wave 111 and which retains some element of classicality. Again we see how the attempt to retain a sense of individuality in this context has a cost: intrinsic properties are no longer solely 'possessed' by the particles themselves and hence the relevant sense of 'object' has been broadened in this case. 112
Further criticisms of the view that the Bohm interpretation is philosophically classical can be found in Bedard (1999). It is interesting, from the perspective of our discussion, that one of Bedard's criticisms is that the emphasis on particles and their spatio-temporal trajectories fails to account for atomic and molecular bonding, where the nature and effects of the wave field cannot be ignored. Of course, the quantum mechanical explanation of chemical bonding crucially hinges on the exchange integral which effectively embodies the Indistinguishability Postulate (it is this integral that the wave field encodes). And when it comes to understanding the latter, the advocate of the Bohm view can simply appeal to the accessibility constraints discussed previously. Hence, although the trajectories themselves might not play a role in explaining bonding, the metaphysics of individuality certainly can. 113
Nevertheless, one might wonder if some attenuated form of Space-Time Individuality might be retained within the orthodox interpretation of quantum mechanics. We recall the way in which this form of individuality is manifested within classical statistical mechanics via the phase space formulation. And, indeed, there does exist a quantum mechanical version of the phase space approach to particle statistics, although it tends not to feature in the philosophical discussions of particle individuality.
Again, it is worth recalling some history here. Schrödinger's early attempt to give a broadly classical interpretation of the new quantum mechanics foundered on the point that the appropriate space-known as configuration space-for a many-particle wave function had to be multi-dimensional. Even then, as Einstein pointed out in 1927, use of the full configuration space formed by the N-fold Cartesian product of three-dimensional Euclidean space appeared to conflict with the new quantum statistics insofar as within this full space, configurations related by a particle permutation are regarded as distinct. The standard technique, of course, is to identify points corresponding to such a permutation and thereby construct what is known as the 'reduced quotient space' formed by the action of the permutation group on the full configuration space. Appropriate quantum conditions can then be applied. 114 In the context of this kind of approach,
. the mutual indistinguishability [in the physicists' sense] of the particles is coded into the topology of configuration space itself, and the different statistical types then correspond to different choices of boundary conditions on the wave function .. 115
This obviously fits nicely with the general understanding of the Indistinguishability Postulate we have been exploring in this chapter. However, two issues immediately come up: first of all, and most importantly, to what extent do we recover a quantum version of Space-Time Individuality within this configuration approach; secondly, does it still give us the full panoply of different kinds of statistics?
That these issues are related can be seen if we think about the problem of collisions: the reduced configuration space is not in general a smooth manifold since it possesses singular points where two or more particles coincide. This leads to two technical difficulties: first, it is not clear how one might define the
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relevant Hamiltonian at such singularities 116 and, secondly, it would appear that their presence implies Bose-Einstein statistics only. The obvious, and now standard, solution is to simply remove from the configuration space the subcomplex consisting of all such coincidence points, yielding a smooth manifold. 117 Imbo, Shah Imbo and Sudarshan have shown that within this framework one can obtain not merely ordinary statistics, parastatistics and fractional or anyon statistics but even more exotic forms which they call 'ambistatistics' and 'fractional ambistatistics'. 118
On the other hand, it has been suggested that one of the advantages of the configuration space approach is that it actually excludes the possibility of non-standard statistics. 119 This claim is based on the work of Leinaas and Myrheim (1977), which apparently demonstrates that for a space of dimension 3 or greater only the standard Bose-Einstein and Fermi-Dirac statistics are possible. Brown et. al. take this to undermine the idea of symmetrization conditions being imposed as kinds of 'initial conditions':
. the (anti-)symmetrisation condition on the wave function is now seen to be related to the dimensionality of space, in contrast to the Messiah and Greenberg analysis wherein the (anti-)symmetrisation condition receives the status of a postulate. 120
If this were correct, then it would obviously undermine the kind of approach we have adopted here. However, the claim is problematic. It involves two crucial steps: the first takes into account the topology of the reduced configuration space and the basic point is that in the three-dimensional case, the reduced space (minus the singularity) is doubly connected, whereas in the case of two-dimensions it is infinitely connected. The next all-important step is to apply an appropriate quantization procedure and here Leinaas and Myrheim follow what they take to be the simplest way, namely the 'Schrödinger quantization scheme'. 121 What this means-no surprises here-is that for each point in the configuration space a one-dimensional complex Hilbert space is introduced and the state of the system is then described by a continuum of vectors
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spanning this Hilbert space; that is, ".Ψ is assumed to be a single-valued function over the configuration space, whose function value Ψ(x) at the point x is a vector in [the Hilbert space]". 122 If Ψ undergoes parallel transport around the singularity it transforms into PΨ, where P is a (familiar) linear, unitary operator acting in the Hilbert space. And-here's the crucial point-since the latter is one-dimensional, P is just the well-known phase factor expt[iy], where y is real and independent of the point x. For bosons and fermions y = o and π, respectively, but in the cases of one-dimensional and two-dimensional configuration spaces a continuum of intermediate statistics is permitted. In the three-dimensional case, however, the fact that the space is doubly connected means that the further condition must be imposed on P that P 2 = 1 so that P = ±1 and, of course, only Bose-Einstein and Fermi-Dirac statistics are allowed. In other words, non-standard statistics are eliminated not just because the relevant configuration space is doubly connected but because a standard quantization procedure incorporating one-dimensional Hilbert spaces is assumed. From the group-theoretical perspective this amounts to allowing one-dimensional representations only and so it should come as no surprise that paraand ambi-statistics cannot arise. Effectively what Leinaas and Myrheim have done is to ignore the 'kinematical ambiguity' inherent in the quantization procedure which derives from the (mathematical) fact that the set of irreducible representations of the permutation group contains not just the trivial representation manifested above, but also others corresponding to exotic statistics. 123
What about the more philosophical issue concerning particle individuality? Let us consider, first of all, the justification for the removal of the coincidence points. An obvious rationale for this would be to appeal to the Impenetrability Assumption. This in turn can be understood as due to certain repulsive forces holding between the particles. Such a conjecture has been made in the case of anyons 124 and more generally this has been taken to confer a further advantage on the configuration space approach, in the sense that
. it allows particle statistics to be understood as a kind of 'force' in essence similar to other interactions with a topological character, like the interaction between an electric and magnetic charge in three spatial dimensions, or the type of interaction in two dimensions which is responsible for the Bohm-Aharonov effect and fractional statistics. 125
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Again, we can't help but recall the relevant history, explored in Chapter 3. The suggestion that the non-classical aspects of quantum statistics reflects a lack of statistical independence and hence a kind of correlation between the particles goes back to Ehrenfest's early reflections on Planck's work and crops up again and again in the literature. And as we noted above, Reichenbach argued that such correlations-taken realistically in this sense-represent causal anomalies in the behaviour of the particles: for bosons these anomalies consist in a mutual dependence in the motions of the particles which could be characterized as a form of action-at-a-distance; for fermions, the anomaly is expressed in the Exclusion Principle if this is interpreted in terms of an interparticle force. 126 As far as Reichenbach himself was concerned, such acausal interactions should be rejected, which meant adopting the Received View.
Moving on from this issue of acausal anomalies, Brown et al. reject this understanding of impenetrability as suspiciously 'ad hoc' in the general case of indistinguishable particles. 127 Shifting to the framework of the Bohm interpretation removes this ad hocness. Since the guidance equations are first-order, the trajectories of two particles which are non-coincident to begin with will never coincide. In effect the impenetrability of the particles is built into these equations and the singularity points remain inaccessible. Thus the topological approach and de Broglie-Bohm interpretation fit nicely together:
. within the topological approach to identical particles the removal of the set . of coincidence points from the reduced configuration space . thus follows naturally from de Broglie-Bohm dynamics as it is defined in the full space .. 128
Our point is not to advocate the Bohmian approach but merely to emphasize that the configuration space approach can also provide a suitable formal framework for this package of individual particles subject to accessibility constraints (we recall Bourdeau and Sorkin's remark above about different choices of boundary condition on Ψ). However, it is also important to recognize that this topological approach offers nothing new, philosophically speaking, since one could also maintain the alternative view that the particles are non-individuals. Indeed, one of the motivations given by Leinaas and Myrheim is that within this approach one can dispense with the whole business of
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introducing particle labels and then effectively emasculating their ontological force by imposing appropriate symmetry constraints. 129 Of course, there is still the issue of the singular points, whose removal on the grounds of impenetrability considerations meshes so nicely with the metaphysics of individuality. One possibility is to tackle the problem of collisions directly. Bourdeau and Sorkin, for example, 130 focus on the Hamiltonian, which they require to be self-adjoint and in the two-dimensional case they show that for fermions, the self-adjoint extension of the Hamiltonian to cover the singularities is unique, so that collisions are strictly forbidden, whereas in the case of both Bose-Einstein and fractional statistics there are a range of alternative extensions, some of which allow collisions but some of which do not. By requiring that the wave function remains finite at the coincident point, they argue that a unique choice of Hamiltonian can then be made and it turns out that collisions are allowed only in the case of Bose-Einstein statistics. Thus, whereas for fermions it doesn't really matter whether the singular points are retained or not, for bosons and anyons, on this account, it does, since these points are either the locations of collisions in the boson case or the locations of vanishing Ψ for anyons. There is clearly more to say about these and related issues-such as the generalization of this line to higher dimensions and parastatistics, for example 131 -but our aim is just to indicate how the configuration space approach can be made consistent with both metaphysical packages, albeit at a certain cost.
There is a further cost associated with the account of quantum particles as individuals and this is revealed when we consider the implications of quantum entanglement. We shall begin by outlining the argument that such entanglement implies a form of non-individuality, before indicating how this is not, in fact, decisive.
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