INDISTINGUISHABILITY AND INDIVIDUALITY
We shall begin by recalling the argument that leads to the conclusion that quantum particles are non-individuals, in some yet to be worked out sense. The connection between the above philosophical views and quantum mechanics was expressed very clearly by that great philosophical commentator on modern physics, Reichenbach (responding, we suggest, to physicists' reflections on this issue as sketched in the previous chapter). He argued that the impact of modern physics on the philosophy of individuality could be evaluated by examining the behaviour of the fundamental particles of physics in aggregate. 1 The examination proceeds in terms of the permutation argument presented in Chapter 2: we consider the distribution of indistinguishable particles over states-two particles over two one-particle states, say-and assume that each resulting arrangement is accorded equal probability. 2 This
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generates four possibilities:
|
(1) |
particles 1 and 2 in state |a 1> |
|
(2) |
particles 1 and 2 in state |a 2> |
|
(3) |
particle 1 in state |a 1>, particle 2 in state |a 2> |
|
(4) |
particle 1 in state |a 2>, particle 2 in state |a 1>. 3 |
We have labelled the states with superscripts 1, 2,.. This reflects the fact that the states can be distinguished and assigned labels in this way because different states are characterized by different state functions. A second set of labels can now be introduced which help to specify which particles are in which states. 4 These are the particle labels, represented by subscripts and hence (1)-(4) above can be rewritten as:
|
(1′) |
|
|
(2′) |
|
|
(3′) |
|
|
(4′) |
|
The introduction of particle labels here might be regarded as philosophically more contentious than in the case of the states, but we shall leave consideration of this until Chapter 5.
In classical statistical mechanics, as we saw in Chapter 2, (3) and (4) (or (3′) and (4′)) are counted as distinct and given equal weight in the assignment of probabilities; that is, the situation where we have one particle in each state is given a weight of two, corresponding to the two arrangements or complexions that may be formed by a permutation of the particles. 5 That a permutation of the (indistinguishable) particles is included in the count of possible arrangements is taken to imply that the particles are individuals, spelled out in terms of some underlying 'haecceity' or 'primitive thisness' or, mo 16516r178q re typically, the spatio-temporal location of the particles.
Now we can see how the examination is going to proceed in the quantum case. We shall initially restrict our attention to the two standard forms of 'statistics'-Bose-Einstein and Fermi-Dirac. In both cases the arrangement of
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one particle in each state is given a weight of one; that is, in order to get the correct results in quantum statistics (3) and (4) (or, again, (3′) and (4′)) must be counted as one and the same. 6 This is standardly taken to reflect the fact that arrangements obtained by a permutation of the particles do not feature in the relevant counting in quantum statistics.
Let us express this point a little more formally. For
simplicity we consider only one kind of spinless particle and we keep the
number of particles fixed. We further assume that the one-particle states are
in one-to-one correspondence with the rays of the Hilbert space 
made
up of all wave functions of one coordinate. For many particle systems the
Hilbert space is the joint space constructed by forming the tensor product of
the component particles' Hilbert spaces. The observables Ô of a quantum system
are then represented by Hermitian operators acting upon that system's Hilbert
space.
Now consider a system consisting of two indistinguishable
particles as above. The Hilbert space for this system is: total
= 1
2
, where the subscripts '1' and '2' label the component particles, and 1
= 2
= .
Given the state of the system represented by (3′), for example, another
state can be obtained by permuting the particles, where for our two-particle
system this state is represented by (4′). This process of permuting the
particles can be represented by a unitary operator which acts on the particle
labels and these 'permutation operators' form a group known as-surprise,
surprise-the Permutation Group. 7
That particle permutations are not counted is then understood in terms of there being no measurement that we could perform which would result in a discernible difference between permuted (final) and unpermuted (initial) states. This is expressed by the so-called 'Indistinguishability Postulate' (IP):
if a particle permutation is applied to any ket for an assembly of particles, then there is no way of distinguishing the resultant permuted ket from the original unpermuted one by any means of observation at any time. 8
It can be represented more formally by insisting that every
physical observable Ô must commute with every permutation operator 
,
.
In other words, we have that for any arbitrary state ψ, Hermitian
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operator Ô, and permutation operator 
:
The most significant observable of a quantum system is, of
course, the Hamiltonian, Ĥ, and the indistinguishability of the particles
thus requires that this be a symmetric function of the dynamical variables; 9
that is, Ĥ is invariant under the action of the permutation group of
permutations of the composite particles' labels: 
,
.
As we indicated in Chapter 3, (IP) allows for the possibility of forms of quantum statistics which are different from the 'standard' Bose-Einstein and Fermi-Dirac kind. If one wants to restrict the formalism to the latter kinds only, then a further condition, known as the 'Symmetrization Postulate' (SP) must be applied, 10 as we noted. Put simply, this dictates that states of indistinguishable particle systems must be either symmetrical or anti-symmetrical under the action of the permutation operators (corresponding to the Bose-Einstein and Fermi-Dirac cases respectively). The difference between SP and IP can be expressed as follows: SP expresses a restriction on the states for all observables, Ô in the first place; whereas IP expresses a restriction on the observables, Ô, for all states. 11
Now, IP seems to run counter to the
whole point of regarding the particles as individuals and labelling them-from
the point of view of the statistics, the particle labels are otiose. The
implication, then, is that the particles can no longer be considered to be
individuals, that they are, in some sense, 'non-individuals'. This conclusion
expresses what we have called the 'Received View': classical particles are
individuals but quantum particles are not. We have indicated how this view
emerged, historically, in Chapter 3 and it achieved further prominence in the
1960s through the work of
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Post, on the other hand, drew on the distinction between form and substance, arguing that what quantum statistics indicates is the ontological primacy of the former over the latter. As we have indicated, both the metaphor and the distinction can be traced back to Schrödinger's reflections on the implications of the new quantum mechanics, And as we shall see in the rest of the book, one can in fact go beyond mere metaphor and underpin the Received View with an appropriate logico-mathematical framework. Before we do that, however, we need to consider two broad challenges it must face. Both challenges resist the argument for the existence of a fundamental metaphysical distinction between classical and quantum particles but approach the issue from different directions.
4.1.1 Challenge No. 1: Classical Particles as Non-Individuals
The first challenge accepts that quantum particles are non-individuals but rejects the claim that this marks a crucial difference from their classical counterparts. Thus, it has been suggested that classical particles of the same kind, like their quantum counterparts, not only possess the same set of intrinsic properties, but are also 'indistinguishable' in the stronger sense indicated above; that is, they are what we have called 'non-individuals'. Various arguments have been given for what at first sight appears such an unlikely claim. One such insists that since there is no provision for classical particles to carry any 'identification marks' they cannot be considered to be individuals either. 14 Now, as we have seen, if we consider non-spatio-temporal properties only, which might include such 'marks', and if the Identity of Indiscernibles is taken to ground our understanding of individuality, then it is indeed true that classical particles violate some forms of this Principle. On this basis they might then be regarded as non-individuals. However, we don't even have to fall back to some form of haecceity or some substantivalist metaphysics to save individuality in this case: we can simply invoke the relevant set of spatio-temporal properties, together with the Impenetrability Assumption, and come up with an appropriate form of the Identity of Indiscernibles which is not violated. As we shall see, however, this option is not available for quantum particles (at least not under the standard interpretation of quantum mechanics).
An alternative argument explicitly adopts a broadly positivistic approach to particle individuality by insisting that the meaning of quantum 'indistinguishability', as expressed by IP, is determined experimentally. 15 Such non-individuality is then understood as not restricted to quantum particles, since it is taken to follow from the requirement that entropy be 'extensive'. We recall from Chapter 2 that this is the requirement that if the volume of some system (a gas, say) is divided into two halves, then the entropy of the whole system must equal the sum of the entropies of the respective halves. As we noted there, this requirement leads to the Gibbs Paradox.
The 'paradox' is resolved by simply dividing the classical expression for the number of possible complexions by N!. This is justified on the grounds that this procedure eliminates 'redundant' complexions obtained through a permutation of the particles. Now, how are we to interpret this procedure? One suggestion, following on from the above work which attempts to ground non-individuality on experimental considerations, is that it implies that we should regard classical statistical mechanics as subject to IP as well. 16 In that case, what we have up until now understood as 'classical' particles would also be viewed as 'non-individuals' and the contrast with quantum physics would have to be sought elsewhere. However, this suggestion radically misconstrues the historical situation and, by blurring the ontological character of classical particles, it also blurs the distinction between classical and quantum statistical mechanics in general. There is a more plausible alternative.
Historically, the failure of extensivity and the Gibbs Paradox in general were seen as revealing a fundamental flaw in the Maxwell-Boltzmann definition of entropy and one which is corrected by shifting to quantum statistical mechanics at the theoretical level and an understanding of the particles as quantum in nature at the ontological level. It is in this sense that quantum mechanics, in general, has been said to 'resolve' the problem. 17 In other words, the force of the argument can be turned around: what it shows is that the world is actually quantum in nature, as one would expect. What the exclusion of the permutations (by dividing by N!) is a manifestation of is precisely that the particles are not just indistinguishable in the classical sense. If this aspect is incorporated into the analysis from the word go, the so-called 'paradox' simply
end p.145
does not arise. As the author of an undergraduate textbook on statistical physics puts it:
Historically, failure to appreciate the significance of the identity of the particles in a system led to certain inconsistencies, known as the Gibbs paradox. We shall not discuss this paradox since a careful reader of this book should not be troubled by it. 18
Of course, this counter-suggestion is not decisive but one might wonder how, if classical statistics is to be subject to IP and classical particles regarded as non-individuals, classical statistics and particles respectively can still be called 'classical'? Once the division by N! is introduced (yielding the Sackur-Tetrode equation for the entropy, as we noted in Chapter 2), the effect can be thought of as rippling through the structure of classical statistical mechanics, effectively transforming it into the quantum form. In the process we appear to lose the very distinctions on which our discussion was originally based. But of course, if the above argument were to be accepted, this would provide further motivation for our programme of explicating the sense of 'non-individuality', in both the classical and quantum cases.
4.1.2 Challenge No. 2: Quantum Particles as Individuals
The second challenge accepts that classical particles are individuals, but resists the Received View by denying that it follows from quantum statistics-and, more particularly, the Indistinguishability Postulate-that quantum particles are non-individuals, in whatever sense. The conclusion expressed by the Received View depends on understanding IP simply as imposing restrictions on the set of possible observables, such that particle permutation operators cannot be included. It is this formal 'fact' which motivates commentators to adopt the metaphysics of non-individuality. However, it can also be shown that the action of IP is to divide up the relevant Hilbert space into a number of irreducible subspaces, corresponding to different symmetry types-Bose-Einstein, Fermi-Dirac, parabosonic, parafermionic and so on. These subspaces correspond to different irreducible representations of the permutation group. Thus, for example, in the case of three particles, the relevant six-dimensional subspace decomposes into irreducible subspaces invariant under the particle permutation operators, giving a one-dimensional symmetric subspace, another one-dimensional anti-symmetric subspace and
end p.146
two two-dimensional paraparticle or 'triangular' subspaces. The states for 'ordinary' particles correspond to the one-dimensional subspaces, whereas paraparticle states correspond to the multi-dimensional subspaces. 19
Furthermore, given IP, it can be shown that all matrix elements of an observable connecting different representations are zero. 20 Hence, if there exist states corresponding to subspaces of different representations then transitions between such states must be forbidden. 21 In other words, bosons 'live' in the one-dimensional symmetric subspace, fermions occupy the anti-symmetric subspace, which is also one-dimensional and paraparticles remain in the multi-dimensional subspaces.
This gives us an important perspective on IP from which it can be seen as imposing a restriction on the states of the assembly of particles such that once a particle is in a given subspace, the other subspaces-corresponding to other symmetry types-are inaccessible to it. If we consider the time evolution of the system as effected by some Hamiltonian, then IP implies that if the system starts in a subspace corresponding to a particular symmetry type, 22 then it will always remain in that subspace: bosons will always be bosons, fermions will always be fermions etc. Once a system starts to evolve in a particular subspace, the dynamics is such that it can never get out of it. From this perspective, IP can be thought of as an extra postulate, representing an accessibility constraint on the set of states of the assembly.
If we go back to that simple example of two particles distributed over two states, this alternative understanding of IP yields a completely different explanation for the reduction in statistical weight: if, in this case, the restriction is imposed that the state of the system be either symmetric or anti-symmetric then only one of the two possible states formed by a permutation of the particles is ever available to the system and so the statistical weight corresponding to the distribution of one particle in each (one-particle) state is half the classical value. In the general case, where we have more than two particles, we get more general symmetry types, as we indicated, corresponding to parastatistics (and to accommodate these types we impose IP rather than simply SP). Each such type will correspond to a respective vector of the
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Hilbert space, which will then be inaccessible to particles of a different symmetry type. From this perspective, states formed by a particle permutation are not counted not because they do not exist, but because they are simply not available to the particles of the relevant symmetry type.
With the reduction in statistical weight now explained by the inaccessibility of certain states, rather than by the non-classical metaphysical nature of the particles as non-individuals, one can continue to regard them as individuals for which certain states are now inaccessible-just because the particle labels are statistically otiose does not mean they are metaphysically so. 23 Quantum statistics is effectively recovered by regarding such states as possible but never actually realized. 24 Through this alternative explanation, this way of understanding IP allows us to retain the metaphysics of individuality for quantum particles. What is now doing all the work, as it were, is the understanding of IP as an extra requirement concerning state accessibility.
The two components of this view-the accessibility constraint and the individuality of the particles-can then be understood philosophically in ways that emphasize the commonality with the classical situation. Let us consider these components in a little more detail.
4.1.3 The Indistinguishability Postulate as an Initial Condition
The notion of state accessibility restrictions can, of course, also be found in classical statistics, albeit in a somewhat restrictive form. The most important and most obvious has to do with the available energy, but other uniform integrals of the equations of motion may exist for a particular assembly which have the effect of restricting its representative point to a certain region of phase space. With regard to thermodynamic consequences, only the energy integral is deemed significant and as imposing constraints on accessible regions of the phase space. In quantum statistics, we have these further constraints and the symmetry type of any suitably specified set of states is an absolute constant of motion equivalent to an exact uniform integral in classical terms. 25
Hence this notion of accessibility can be formulated for both classical and quantum systems but the considerations of the previous section go some way toward explicating the greater role played by symmetry in the quantum case as
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compared to the classical. 26 When it comes to ordinary statistics, the question as to what states are accessible is straightforward. In the paraparticle case it is slightly more complicated: for particles obeying Gentile's paragas statistics, the number of accessible states will simply be intermediate between the number accessible to fermions and bosons. This number will increase as the statistics becomes less fermionic and more bosonic. In the case of parastatistics, it turns out that it is a linear combination of states that now becomes accessible to the particles. 27
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