It is odd that neither Cassirer nor Eddington appear to refer to the introduction of group theory into quantum mechanics by Weyl and Wigner. As Weyl noted, this hinged on the   identification of two different kinds of symmetries: spatio-temporal symmetries, such as the rotational symmetry associated with the representation of the atomic nucleus as a fixed centre of force; and permutation symmetry, or invariance under a particle permutation. Given Cassirer's understanding of spatio-temporal symmetries, and their incorporation into his epistemology, we might speculate that he saw little that was particularly new in these developments, from the structuralist perspective. Eddington, on the other hand, did make an explicit attempt to accommodate permutation invariance within his account, but he gave it a particular geometric twist as I indicated above. Weyl, however, clearly recognised the import of this new symmetry and not only explored its physical consequences in The Theory of Groups and Quantum Mechanics but later, in The Philosophy of Mathematics and Natural Sciences, considered in more detail its philosophical implications. As I have noted elsewhere, that Weyl was aware that here was a new aspect of symmetry in the world goes some way to responding to Donini's perplexity over why, in The Theory of Groups..., he appeared to have forgotten all his work on ‘relativity and invariance matters’ . It’s not a question of forgetting the latter but of focusing on a new fundamental symmetry. Mackey characterises the formal moves in terms of two, intertwined sets of developments: what can be called the 'Weyl programme', which saw group theory as a way of bringing order to the collection of principles and ad hoc rules that constituted quantum mechanics in the late 1920s, and setting the foundations of the theory on a secure basis; and the 'Wigner programme' which saw group theory as a way of bypassing the computational intractability associated with tackling the dynamics head on. Of course, both Weyl and Wigner made important contributions to each. Wigner, in particular, further extended the reach of group theory within physics by applying it both to the nucleus  and elementary particles.

This latter extension was presented by Wigner at the 1935 'Pittsburgh Symposium on Group Theory and Quantum Mechanics'  where he notes the 'unique correspondence' between possible Lorentz invariant equations of quantum mechanics and the representations of the inhomogeneous Lorentz group. Such a representation, ' ... though not sufficient to replace the quantum mechanical equations entirely, can replace them to a large extent.' It can give the change through time of a physical quantity corresponding to a particular operator, but not the relationships holding between operators at a given time. The issue then is to determine the irreducible representations of this group and these are established in Wigner's famous 1939 paper43. It is this work of Wigner's - specifically, the association of ‘elementary physical systems’ with irreducible representations of the inhomogeneous Lorentz group - which is drawn upon by Castellani in her analysis of the group theoretic constitution of objects.  According to Weyl, ‘objectivity means invariance with respect to the group of automorphisms [of space-time]’ (Weyl, 1952). On the basis of this statement, Castellani presents an ‘objectivity condition’ (for the physical description of world), namely invariance with respect to the space-time symmetry group. The issue then is to move from objectivity to objects: ‘What is of interest, from the point of view of the object question, is how this objectivity condition for the laws of physics can be used with regard to the determination of ‘objects’ within a given physical domain.’

The basis for such a move is precisely Wigner's association of an ‘elementary system’ with an irreducible representation of the space-time symmetry group, such that the set of states of the system constitutes a representation space for the irreducible representation. For quantum systems, the appropriate representation space will be the Hilbert space, of course. The labels of the irreducible representations are thus associated with values of the invariant properties characterising the systems. Two issues then arise. First of all, what this group-theoretic construction yields are classes or kinds of particles, not distinct objects.   Secondly, in addition to spatio-temporal symmetries, there is the permutation symmetry which needs to be accommodated within this approach. Let us consider each of these issues in turn. With regard to the first issue, we need something else to give ‘individual objects’; Castellani identifies this something else as 'imprimitivity'. This was originally introduced by Mackey in the 1950s , although it is implicit in Wigner's 1939 work, and it has been notably applied to the definition of physical particles by Piron. The basic idea is to use the notion of a 'system of imprimitivity' associated with a symmetry group in order to determine 'individuating' observable quantities such as position and momentum and thus move from kinds to individual objects by supplementing the above group-theoretica account. Putting things somewhat crudely, we obtain an imprimitivity system in the following way: we associate with a system, in addition to the group G, a configuration space S (strictly a Borel space) on which G acts. A projection valued measure is then defined on S (where a projection valued measure is a mapping from a Borel subset of S to the relevant projection operator) and if the projection valued measure satisfies a certain identity (Ux -1PEUx = PEx-1; where PE is a projection operator and U is a unitary representation) then the projection valued measure constitutes a ‘system of imprimitivity’ for U based on S. The importance of the system of imprimitivity associated with U is that it determines the structure of U as an induced representation. In particular, if S is transitive and L is a unitary representation of a closed subgroup of G, then the equivalence class of L is uniquely determined by the pair U,P, where P is a system of imprimitivity for U and the commuting algebra for L is isomorphic to the subalgebra consisting of all bounded linear operators that commute with all PE .

In particular, if S denotes physical space (3 dimensional, Euclidean, affine), and G is now the Euclidean group of all rigid motions of space, then the position of a particle, regarded as an ‘S valued observable’, can be described by a projection valued measure defined on S. The relevant projection operator is then the self-adjoint operator corresponding to the real-valued observable which has the value 1 when the particle is ‘in’ Borel sub-set/at a given position and 0 when it is not. If we impose the requirement that the description of the system be covariant with respect to G, then the projection operator must satisfy the identity which renders the projection valued measure a system of imprimitivity. Introducing momentum observables and applying certain group-theoretic results, one can then obtain the usual commutation relations, not by analogy with the Poisson brackets of classical mechanics but as a consequence of Euclidean invariance. Furthermore, one can show that every irreducible representation of the commutation rules is equivalent to the Schrödinger representation. The apparently special choices in the latter for representing position and momentum observables are in fact the most general ones possible subject to the commutation rules, if we assume irreducibility. On this basis, it is claimed, we can prove the isomorphism of Schrödinger wave mechanics (based on the Schrödinger epresentation) and matrix mechanics. And the results just keep on coming: if the relevant configuration space is affine, we get the Born interpretation of ||2 and an ‘illustration’ of complementarity in the sense that one can show that no single state exists in which both position and momentum can be localised sharply. As far as the current discussion is concerned, the important point arising from all this is that, ‘All we need to discuss physical events are position observables and a dynamic group.’  According to Castellani, this restores the notion of an object and thus we get the group theoretic characterisation (or for her, constitution) of not only kinds but individual objects: ‘The aim is to arrive at a definition of a particle by determining “individuating” observable quantities (such as the position and momentum) with the help of the imprimitivity systems.’ (Castellani 1998, p. 190)

Now this is not what the structuralist wants45! So let us consider the philosophical implications of the above formal moves in a little more detail. First of all, these moves have not reintroduced ‘substance’ of course . However, what we appear to have arrived at, via this long detour through group theory, is nothing less than the good ol’ ‘bundle theory’ of individual objects, according to which such objects are regarded as nothing more than a 'bundle' of properties, with spatio-temporal location typically privileged as that property which confers individuality, distinguishability and (classically at least) reidentifiability. Now the bundle theory, as usually understood, requires some form of the Principle of Identity of Indiscernibles in order to effectively guarantee individuation and as we all know, the status of this Principle is problematic in quantum mechanics (for a recent discussion see Massimi forthcoming). I'm not going to get into that discussion again, except to note the following: on the one hand, from the perspective of the configuration space approach a form of the Identity of Indiscernibles is manifested by the removal of coincidence points in the relevant configuration space. This, in turn, is effectively written into the guidance equations of Bohm theory  and thus the latter can be understood as embodying a metaphysics of individual particles46. On the other, if one were to accept the well known arguments that the Identity of Indiscernibles is at best inapplicable, at worst violated in quantum physics, one would have an ontology of bundles which aren’t tied together. This amounts to an ontology of non-individual objects described by something like qua/quasi-set theory. Our long discussion of group theory and systems of imprimitivity seems to have led us right back to where we started, namely the underdetermination between individuality and non-individuality. Is there a way the structural realist can accommodate the central insight of Mackey’s comment above without being committed to objects that are either individuals or non-individuals? A possible response is to understand imprimitivity as giving a group-theoretic grasp on the position of a 'particle' (perhaps understood as Bell’s 'beable') but to insist that this does not yield objecthood (beables don't give objects). In other words, we can buy into the whole group-theoretic analysis/reduction of 'objects' but simply resist the exportation of position, say, beyond the temporally limited domain of the immediately observable and into the realm of quantum objects as a whole47. The question now is, what kind of ontological picture does this give? First of all, position can be regarded as yielding, not individuality per se, but only a kind of ‘pseudo-individuality’, as noted above, or what Toraldo di Francia refers to as 'mock individuality’ in the sense that one can pretend the particles are individual objects at the point of measurement, as it were, but only temporarily. It is significant that this notion is articulated in the context of what can be taken as a form of structuralism48, according to which particles are regarded as 'nomological' objects in the sense that '... physical objects are today knots of properties, prescribed by physical laws' . It is in this context that Dalla Chiara and Toraldo di Francia develop their view of quantum particles as 'anonymous' in the sense that proper names cannot be attached to them, although here too there is a tension between this and the underlying structuralism. However, the important point is that pseudo-individuality allows us to refer to 'objects', without compromising our structuralism: 'This is why an engineer, when discussing a drawing, can temporarily make an exception to the anonymity principle and say: "Electron a, issued from point S, will hit the screen at P, while electron b, issued from T, will land at Q."'

Simons' approach avoids this form of criticism by replacing compresence with a form of Husserlian 'foundation relation': 'An electron must have a certain mass, charge and spin, and in addition is variably endowed with a position relative to other things and with a velocity and acceleration in particular directions at any time. When individual tropes require other individual tropes we say they are rigidly dependent or founded on these. When founding is mutual then a group of tropes must either all exist or none do. The mass, charge and spin of an electron must coexist, they require each other and form a bundle. A bundle consisting of all the tropes mutually founding one another directly or indirectly we may call a nucleus.'

Tropes may also require other tropes as members of a kind and in such cases, instead of 'founding' we have 'generic dependence', with the tropes generically required forming a 'halo'. Further reasons for rejecting compresence are derived from examples in physics. Thus, Simons argues, compresence is neither necessary nor sufficient for tropes to form a bundle: '... it is not necessary because a trope bundle may be widely distributed, as in particle pair formation where paired tropes constituting electromagnetic polarisation or spin may be vastly separated yet mutually dependent. It is not sufficient because more than one trope bundle can be compresent as when two or more electrons occupy the same shell of an atom.'

In other words, compresence cannot do the job because of quantum non-locality and indistinguishability! Replacing compresence by the above notion of rigid dependence or founding is supposed to accommodate examples like these. Of course, compresence does not disappear from the picture entirely, as it may still be concomitant. If the trope bundle theory is sufficiently 'flexible', then perhaps it can cover both pseudo-individuals and structures: a pseudo-individual is a bundle of compresent tropes, whereas a structure, or 'kind-structure', in the above sense, is a bundle of tropes which are not compresent51. This is not to say that the physics somehow requires trope theory; it may be that some form of bundle theory of individuality can do the same job, but at least we don't have substance and we don't have the Identity of Indiscernibles. What about the second issue? How are we to understand the action of the permutation group? In the final section I'd like to explore this question in the light of two options, attached to each of the 'horns' of the metaphysical underdetermination with which we started. Option 1: the particles are regarded as individual objects and permutation symmetry is either regarded as a property of the particles themselves, or, more plausibly but still problematically, as an emergent property of the assembly; Option 2: permutation symmetry reflects the metaphysically weird nature of quantum objects as non-individuals. And I shall conclude that Huggett's attempt to avoid either option leads us back to structural realism.

The Ontological Status of Permutation Symmetry

Let me begin with the first option. If permutation symmetry is regarded as an intrinsic property and intrinsic properties are understood as delineating natural kinds, then electrons, say, obeying different statistics - which is a possibility raised by violations of the spinstatistics theorem - would constitute different such kinds by virtue of possessing a different 'kind' of permutation symmetry. The particles of each such kind would still be indistinguishable, of course, and in a way that generates problems for the identity of indiscernibles and hence the bundle theory of individuality. And if we were to reject the latter and adopt some form of Transcendental Individuality, then this option of understanding permutation symmetry as an intrinsic property is still problematic, as Hilborn and Yuca have recently spelled out (preprint). First of all, the only thing that would distinguish these electrons of different kinds would be their permutation symmetry; in all other respects they are indistinguishable. If two or more such electrons were to come together in an atomic system, the question arises as to which statistics would dominate . Perhaps we would need some form of 'meta-quantum statistics' to answer it! More fundamentally, as Hilborn and Yuca emphasise, we don't measure the permutation symmetry of individual particles; more than one particle is needed. Thus, as a property in this intrinsic sense, the permutation symmetry of the individual particle can be regarded as 'empirically superfluous'. For these sorts of reasons, Hilborn and Yuca reject this option. Instead they prefer a 'holistic' perspective which can accommodate the 'emergent' quality. Permutation symmetry 'emerges' at the collective level as a property, not of the particles themselves, but of the quantum state. They argue that the possibilities of non-standard statistics and violations of the spinstatistics theorem in general can be accommodated quite naturally within this framework: 'This holistic point of view is both more faithful to the possibilities of physics (including possible violations of the spin-statistics connection) and a stronger philosophical stance. It also has the merits of simplicity and efficiency. On this account, permutation symmetry is a property of the collective state of the identical particles, not an intrinsic property to be associated with each particle.' 

However, some care needs to be taken concerning what this holistic point of view amounts to. Hilborn and Yuca understand Redhead and Teller to be arguing in its favour, particularly with regard to '[t]he fact that holism places no epistemological limitations on the observer'. Redhead and Teller themselves understand the metaphysics of this holism in terms of non-individual quanta; on such an account, permutation symmetry is a manifestation of the metaphysically peculiar nature of the quanta. However, this account of permutation symmetry as a property of the state can also be accommodated within the alternative metaphysics of particles as individuals. In this case, as I have indicated elsewhere, the state space breaks up into sub-spaces of different symmetry, with transitions between such sub-spaces suitably prohibited. And permutation symmetry is then a kind of initial condition representing a further structural characteristic of the state space. Thus we seem to face our metaphysical underdetermination once again. Huggett's recent analysis of permutation invariance (1999) can be seen as an attempt to establish an alternative to both the particles-as-individuals and particles-as-non-individuals packages, where the former takes permutation invariance to be a kind of mysterious 'brute fact' of the universe and the latter takes it to be associated with the non-individuality of quanta. The central motif of Huggett's approach is that permutation invariance should be regarded straightforwardly as a symmetry on a par with rotational symmetry, for example, and hence it is symmetry considerations, rather than either 'brute fact' or metaphysics, which explains quantum statistics. Of course, what is meant by 'a symmetry' needs to be spelled out and Huggett takes us through three such explications: First of all, permutations are covariant, in the sense - as Huggett takes it - that the permutation group has a unitary representation in the state space. However, the explanation of (non-relativistic) spin requires not just that the rotation group has a unitary representation but also, of course, that the state vectors lie within multiplets of distinct intrinsic angular momenta, 0, 1/2, 1, and so on. In other words, the representations must also be irreducible. This gives a stronger notion of symmetry, which Huggett calls 'elementary state covariance': a symmetry group is said to be elementary state covariant if and only if the particle state vectors transform according to the unitary representation of the group. The (philosophical) point then is that we now have an account of the relationship between quantum statistics and permutations which, Huggett claims, is identical to that which is given for spin and rotations in non-relativistic quantum physics: if permutations are included in the full group of symmetries and it is postulated that this group is elementary state covariant, then only those many-particle states are allowed which are appropriately symmetrized or parasymmetric. The advantages of this are three-fold: i) it provides an explanation of the state space restrictions in terms of symmetry '... without the unnecessary extra logical strength of further (possibly questionable) assumptions'; ii) it provides an understanding of symmetrization within the many-particle tensor product formalism without having to invoke the Fock space that the non-individual quanta interpretation hinges upon; and, crucially, iii) it provides a unified treatment of quantum statistics and spin in terms of a fundamental symmetry principle. Now, of course, as Huggett acknowledges, permutation symmetry is very different from rotational symmetry: a quantum system is not just covariant but invariant, in the sense that permutations are not just indistinguishable to similarly permuted observers but to all observers. Nevertheless, he argues, permutation invariance is implied by a further symmetry principle which space-time symmetries also obey, together with the formal structure of the permutation group. This further principle is what he calls 'global Hamiltonian symmetry' which implies that the relevant symmetry operator commutes with the relevant Hamiltonian. What we take the relevant Hamiltonian to cover is crucial here because, again as Huggett acknowledges, the principle would appear to be violated in the case where, for example, we have a noncentral potential term in the Hamiltonian of an atomic system, but, he insists, the symmetry is restored if we consider the 'full' Hamiltonian of system plus field, which does commute with the operators of the rotation group. As he points out, if observers are taken to be systems too, this symmetry principle is equivalent to covariance for space-time symmetries. With regard to the permutation group, of course, permutations of a sub-system are permutations of the whole system and the above 'global Hamiltonian symmetry' very straightforwardly implies permutation invariance, without any additional assumptions concerning the structure of state space.  Thus quantum statistics comes to be explained in terms of a fundamental symmetry principle. This is an attractive proposal which meshes nicely with the history of this subject. Nevertheless, one might feel that the proposed explanation contains a crucial lacuna. Consider: as Huggett acknowledges, permutation symmetry is different from space-time symmetries and permutation invariance only follows from his general symmetry principle given the particular structure of the permutation group. This generates the obvious question: why should the group structure be this way and not like that of the rotation group? Or, better perhaps, since the question could be answered by simply insisting 'that's the way the maths is', why should this particular piece of maths be applicable? One obvious answer is to say that it reflects the nature of the objects themselves, as non-individual quanta. In other words, the explanatory gap gets filled by metaphysics and we fall back to the particles-as-non-individuals view. Alternatively, we might insist that it has nothing to do with the objects themselves, which can still be regarded as individuals but is a reflection of some kind of initial condition for our universe. Of course, Huggett would be unhappy with either option and might insist that it is enough to note the general symmetry principle plus the structure of the permutation group - the explanatory buck stops there. And indeed it has to stop somewhere, so the issue comes down to that of what counts as an appropriate terminus. At this point we might recall some more history: Newton famously refused to elaborate on the metaphysics of gravity and insisted that he could explain the phenomena using his law of universal gravitation and that was enough. Leibniz, caught up in his own baroque form of metaphysics, equally famously objected that this left gravity as an 'occult' force. We'll leave it to the student of counterfactual history to wonder whether Leibniz would have been satisfied with Einstein's attempt to bring gravity back into the light! What was it that Leibniz found unsatisfactory with Newton's account and does it have anything to do with the uneasiness that one might feel at Huggett's attempt at an explanatory terminus? Perhaps it has to do with the feeling that for an explanation to be satisfactory, it has to incorporate some aspect of how the world is, from a realist perspective, or how the world could be, from that of today's anti-realist. Let's consider the analogy with spin: here one might object that the rotation group doesn't actually explain spin in the sense of accounting for its existence. Spin is a property that is attributed to objects, originally for experimental reasons, and its nature, possible values etc. came to be described group theoretically in the well known way: for massive particles the possible representations of the rotation group, as the relevant little group of the Poincaré group, yield the allowed values of spin, and this is the case in both relativistic and non-relativistic QM (the latter being the case with which Huggett is concerned), whereas for massless particles one needs to introduce parity and the explanation of the two spin states for such particles is relativistic. But the point is that what one is explaining here is not the existence of spin, as an intrinsic property, but its structure, the values it can take in particular cases and so on. In the case of quantum statistics, on the other hand, it is the existence of these statistics themselves, as expressed via permutation invariance, that we are trying to explain and we might feel justified to ask, with regard to this explanation, what is it about the world that gives rise to this phenomenon? If the spin-statistics theorem could be proven, we could follow the reductionist route and the explanation would terminate in spin, understood as an intrinsic property of things. (Of course, if the theorem could be proven, one might feel tempted to push the reduction the other way and have spin emerging as a result of the statistics, understood in one of the ways to be canvassed below.) Or we could suggest that the statistics is a holistic or emergent property of particle collectives, as indicated by Hilborn and Yuca above, but then we have to come up with an appropriate metaphysics of emergence. Or we could insist that it has to do with the peculiar metaphysical nature of the particles themselves, as non-individual objects. Or we could say that the objects are not peculiar at all, metaphysically speaking, and that permutation invariance is a reflection of certain initial or boundary conditions that pertain in our universe. All these options provide a metaphysical component and simply saying that permutation invariance is nothing more than a result of a general symmetry principle together with the structure of the permutation group seems metaphysically and hence explanatorily deficient. If we're going to take up Huggett's option and shy away from considerations of the metaphysical nature of the particles as objects, then we're going to need an appropriate metaphysics of symmetry and it is precisely this that structuralism aims to provide. This amounts to a broadening of the group-theoretical approach to elementary particles by incorporating permutation invariance, understood not in terms of an intrinsic property of the particles but, just as these properties themselves, as an aspect of the 'world-structure', if you like. In other words, the world is ultimately and metaphysically structural in nature and permutation invariance is simply one manifestation of this structure.

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