Some quantum semiholisms

I’ve been naively thinking about what a reductive physicalist quantum ontology that matches the Hilbert-space formalism in the Schroedinger picture might look like.

My first thought is something like this. “Space” is (the surface of) a sphere in a separable Hilbert space, with an inner product structure (perhaps derived from a more primitive linearity and metric structure using the polarization identity) and “the universe” is a point particle walking on that sphere.

But that description is missing crucial structure, because when described as above, all the points on the sphere are on par. Although the universe-particle was at a different location on the sphere 13 billion years ago than where it is now, there is nothing to distinguish these two points in the story, and hence nothing to ground the vast changes in the universe between then and now. What we need to do is to paint the sphere with additional structure.

There are multiple ways of having the additional structure. Here are two.

Option I. Introduce a number of additional causally impotent “point particles” living on the sphere but not moving around as “markers”, and define the rich intuitive structure of our universe from the inner-product relationships between the universe-particles and the marker-particles. Here are two variants on this option.

• (Ia): There are countably many point particles corresponding to basis vectors in some privileged countable Hilbert space basis, and these “marker-particles” are then located at a set of points on our sphere that form an orthonormal basis. For instance, if we “think of” the Hilbert space for a system of N particles as L²(R^3N), we might have a different static marker-particle for each 3N-dimensional Hermite polynomial.

• (Ib): There are uncountably many marker-particles, and they are located at a set of points of the sphere such that the closure of their span is the whole Hilbert space, but they are not orthogonal. For instance, in our N-particle case, we might think of each marker-particle as corresponding to a normalized indicator function of a subset of R^3N with non-zero Lebesgue measure, and require them to be located on our Hilbert space sphere in places which give them the “right” inner product relationships for normalized indicator functions.

Note that since what is physically significant are the inner products beween the positions of the marker-particles and the universe-particle, we need not think of the particles as having “absolute positions” on the sphere—we can have a “relationalist” version where all we need is the inner-product relationships between the particles (marker and universe). Or, if we want something more like the Heisenberg picture, we could suppose absolute positions, keep the universe particle static, and make the marker particles move. There are many variants.

Option II. We enrich the structure of our “space” (i.e., the surface of the Hilbert space sphere) by adding fundamental binary relations between points on that sphere that correspond to some privileged collection of operators (e.g., normalized projections onto subsets of R^3N with non-zero measure).

Anyway, here is an interesting feature of these two stories. On none of them do we have Schaffer-style holism. On Option I, we have an infinite number of fundamental “particles” in “space” (i.e., on our infinite-dimensional sphere), though only one of them is moving, and we may or may not have the “space” itself. On Option II, we have the two fundamental entities: the universe-particle and the sphere itself, with the universe-particle having merely positional structure, while the sphere has a complex operator structure.

We might call these stories semiholistic. Of course, there are fully holistic stories one can tell as well. But one doesn’t have to.