Chapter 7 Objections 61
7.3 Objection: PI is compatible with classical physics 65
I have claimed that the incompatibility between PI and MA is general—any theory which incorporates the former is incompatible with the latter. It might be objected that this assertion is refuted by the use of reduced configuration spac- es54 in classical mechanics (CM). The use of certain reduced spaces—as de- scribed below—is equivalent to the imposition of PI, and since CM is clearly compatible with a particle interpretation it must be false that PI is always in- compatible with MA.55
This objection inappropriately attributes properties of CM to the conjunction of CM and PI, which I’ll denote CM+PI. The implicit assumption is that every ‘classical theory’ is compatible with MA. More explicitly, it is simply assumed that, because CM is compatible with MA, CM+PI also must be compatible. However, when CM+PI is examined in detail, it becomes clear that this mon- strous form of CM is patently at odds with a particle interpretation. To argue for this claim, we first need to clarify what it means to incorporate PI in CM. With- out loss of generality, we can restrict our attention to a pair of particles con- strained to move in one dimension. If we ignore time-dependence, models of CM in this case are just points in R2nD. (The notation “nD” just means that
we remove the points in the ‘diagonal’ set D=f< x; y >2R2 :x=yg. We
do this because the particles are assumed to be impenetrable.) Each model in
R2nD is interpreted in a straightforward manner: one particle is said to have
the property of being located at one of the two coordinates of the point, and the other particle is said to be located at a place corresponding to the remaining coordinate.
One way we might try to impose permutation invariance on the theory of CM is by insisting on symmetric probability distributions over points in R2. To remain
classical, these distributions should be interpreted either epistemically (we just don’t know which model applies) or in terms of ensembles of two-particle sys- tems prepared in identical but coarse-grained states as would be the case in sta- tistical mechanics. The resulting theory is permutation invariant in that the probability of the system occupying a particular state is insensitive to which par- ticle has which properties. This is the notion of permutation invariance Bach
54 Or reduced phase spaces, as the case may be.
55 This objection was suggested to me both by Chris Smeenk (in discussion) and by an ano-
66 (1997) defends in his monograph on indistinguishable classical particles. But this sort of indistinguishability is insufficient for satisfying PI as it was formulated in Chapter 5. While the distributions over models are permutation invariant, mod- els (which are still points in R2nD) are not. If we permute the parts of a model
that are taken to refer to a single particle in the interpretation of the model, namely one coordinate of a pair of coordinates, then we attain a distinct model with a distinct interpretation.
Another strategy is to identify the points in the configuration that differ only by a permutation of their coordinates. That is, we can simply declare that points which differ by a permutation are equivalent in the sense that they represent the same state. We can represent this equivalence by taking the quotient of the con- figuration space R2 nD under the action of the permutation operator ¦
21. The
resulting set is denoted (R2nD)=¦
21 and is called a ‘reduced configuration
space’.56 A version of Hamilton’s laws of motion obtain on this new space, and so we can be said to have a version of CM that respects PI.
Using a reduced configuration space to represent a classical system in this way entails a radical sort of indistinguishability anathema to classical sensibilities and in contrast to the sort of harmless indistinguishability considered by Bach. As Saunders puts it, the use of a reduced phase space “…takes particle indistingui- shability all the way down to the microscopic details of individual particle mo- tions, whereas, according to Bach, it ought to concern only statistical descrip- tions (probability measures)” (Saunders 2006, 197). In fact, Bach asserts that attempting to take indistinguishability to include every property of a particle in- cluding its trajectory is incoherent. More accurately, he argues that if classical particles are indistinguishable in this sense, they cannot be said to have trajecto- ries. I won’t go through the details of Bach’s argument [see (Bach 1997, 7-8)], but I will point out that it relies on a strong notion of what it is to bear a proper- ty: a particle has a trajectory (or position in the time-independent case) only if there is a probability of 1 of measuring the particle to possess this property. While one can take issue with Bach’s particular approach, there is no escaping the fact that if one is to impose PI on CM and retain particles, one will have to be much more liberal in attributing properties to particles—it is indisputably the case that some part of the classical picture must yield if PI is to be added. As with QM, we can always accommodate PI in CM by loosening our notion of property. But even after making such an adjustment, we still encounter the same problems with property independence as we saw in the quantum case—while we can loosen our notion of ‘property’ enough to allow particles in permutation
67 invariant states to have them, there is just no way to respect PI and avoid attri- buting the same properties to all particles of a type. The points of (R2nD)=¦
21 are sets of points in R2 of the form fhx; yi;hy; xig. The two
particle roles in such a pair—the parts which individually correspond to a single particle specification—are identical. Thus, if our interpretation function is in- deed a function, the specifications attributed to each of the two particles must also be identical.
How do I know what the roles are? It is implicit in PI—the reason we quo- tiented out by the action of permutation in the first place was because PI tells us that this action must yield models with identical interpretations. Which points in the original configuration space should be identified as equivalent was deter- mined by an implicit choice of particle role, namely projection onto one dimen- sion or the other of the two-dimensional space. So for instance, the role of par- ticle 1 in the model hu; vi is just u. In the quotient space, we can extend this no- tion of projection, but the result is a pair of identical numbers for either particle. The role of particle 1 in fhx; yi;hy; xig is fx; yg. So is the role of particle 2. One might be inclined to complain at this point that I’ve made a mistake in un- derstanding the quotient space. One might think the right thing to do is to begin our physics with the reduced space and simply decline to interpret the original space from which it was constructed. This would leave us free to specify some other way of extracting particle roles from models (since PI would no longer fix the choice for us) and thus free to interpret our models as systems of particles with whatever properties we wish. But such a move is not open to us. There are only two possibilities: either the new way of extracting and interpreting particle roles also respects PI such that permutes of the new models will yield identical interpretations, or it won’t. In the former case, I can simply run my argument again and show that both particles must get the same specification. In the latter case, the theory is no longer permutation invariant and my thesis does not apply. The upshot is that it is possible to consistently impose PI on classical theories, provided we are willing to adopt a decidedly non-classical view of what it means to bear a property. However, the resulting theory of PI+CM—unlike CM alone—is strictly incompatible with MA.