The argument for QM – version 1 55 

In the preceding section, I stated the incompatibility argument in its strongest, most general form. Of course, one can also show that PI and MA are incompat- ible in the context of a particular theory and with the assumption of a particular metric for comparing particle specifications. A natural theory for which to do this is QM, the first theory to explicitly formulate and endorse PI. In this and the following section, I run the argument using two versions of the quantum formalism and two different metrics. While this is redundant in some respects,

43 _{More accurately, in order to assess whether the theory T can be interpreted as representing}

two particles in accord with MA, we need to consider a theory T’ which is effectively a one- particle version of T. I am ignoring this complication and assuming that for any scientific theory, e.g. QM, it is unambiguous which set of equations and boundary conditions are sin- gle particle theories and which are multiparticle. Cf. fn 33 in Chapter 4.

56 my intention is to illustrate both the irrelevance of one’s choice of metric and the inevitability of the conclusion no matter how liberally one reads the quan- tum formalism. The reader who is satisfied with the general argument is invited to skip one or both of these illustrations.

To see how the argument works within QM, we first need to be clear on what constitutes the theory and how models of the theory are to be interpreted. In my first approach, I will take the theory of QM to be the dynamical postulates which specify trajectories in a Hilbert space according to either the Schrödinger or Heisenberg equations of motion. For simplicity, I’ll ignore time dependence and focus just on time-independent solutions of the equations of motion for suitably specified boundary conditions. In the time-independent case, models are just rays in a Hilbert space. 44

How should these models be interpreted? What objects are specified by the models and how should we assign properties to them? In the case of a single- particle system, each ray in the Hilbert space is taken to specify one particle in the world along with its properties. If we adopt the conservative eigenvector- eigenvalue link, then our interpretive function would attribute one property to the particle for each of the self-adjoint operators for which the ray representing that particle contains an eigenvector. For instance, if we are considering a sta- tionary state of a particle in a square potential well, then (since it is stationary) its state representation must be an eigenvector of the Hamiltonian. In our interpre- tation, we would attribute to the particle an energy represented by the corres- ponding eigenvalue. However, since the state representation cannot also be an eigenvector of the position operator, this interpretive scheme would preclude us from attributing any position property to the particle in our interpretation. Assuming that particles in an interpretation are only ascribed properties for which the associated ray contains an eigenvector, then there is a one-one cor- respondence between rays in the Hilbert space and descriptions in the interpre- tations of one-particle models.45 _{Because models of QM (as I’ve construed it} here) reside in projective Hilbert spaces (the set of rays of a Hilbert space), there is a natural choice of metric for measuring the distance between models. Specif- ically, the distance between any two rays ri and rj in the projective Hilbert space is given by the Fubini-Study metric:46

44 _{This is only true if we neglect the possibility of paraparticles.}

45 _{This assumption rules out the standard approach of using reduced density operators to}

represent (mixed) single-particle states. In the next section, I will lift this restriction and em- ploy a different metric.

57 dFS(ri; rj) = arccos Ãs hÃjÁihÁjÃi hÃjÃihÁjÁi ! ,

where jÃi is any one of the vectors lying along ri and jÁi is any one of the vec- tors lying along rj. Since there is a one-one correspondence between one- particle rays and descriptions for the interpretive scheme we’ve assumed, we can transfer this metric onto the space of particle specifications. That is, the distance between any two specifications may be defined as the distance between the cor- responding rays.

If we consider the case of two-dimensional particle states, this metric makes the inconsistency argument easy to visualize. For instance, we might consider just the spin degree of freedom of a spin-½ particle. In that case, the Fubini-Study metric describes the geometry of the surface of a sphere embedded in Euclidean three-dimensional space (see Figure 6.1).47 _{Each point on the sphere corres-} ponds to a ray in the 2-D Hilbert space (which in turn corresponds to a particle specification), and the Fubini-Study metric gives the distance between two states (or two specifications) along a great circle of the sphere. To visualize the argu- ment on this sphere, pick a threshold distance ² small compared to sup(dFS(ri; rj)). Now choose a pair of rays r1 and r2 from the space of one-

particle models such that dFS(r1; r2)>2². According to MA(iii) we must be

able to find a two-particle model that—when interpreted—yields two copies of a single-particle description corresponding to a ray r¤ _{that is within ² of both r}

1

and r2. This means that r1 and r2 must lie within a disc of radius ² centered on r¤_{. Obviously, if this is the case then it cannot also be true that r}

1 and r2 are

more than 2² apart since the diameter of the disc is 2². Thus we arrive at a con- tradiction. Given PI and the Fubini-Study metric on the space of quantum models, MA(iii) is not satisfiable.

47 _{Up to a multiplicative constant, the differential form of the Fubini-Study metric is the}

metric of the ‘Bloch sphere’. For a discussion of the Bloch sphere, see, e.g. (Dickson 2007, Sec. 1.3.5; Nielsen and Chuang 2000). For a detailed discussion of the geometric interpreta- tion of the density matrix in arbitrary dimensions (though they do not use the term ‘Bloch sphere’), see (Filippov and Man'ko 2008).

58

Figure 6.1 The argument visualized for pure states on the Bloch Sphere. The disk indicates a set of specifications within a distance ε of the specification r* . If QM is to satisfy MA, it must be possible to find two additional specifications within this disk but greater than 2ε apart—a clear impossibility.