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.

Metaphysical semiholism

For a while I’ve speculated that making ontological sense of quantum mechanics requires introducing a global entity into our ontology to ground the value of the wavefunction throughout the universe.

One alternative is to divide up the grounding task among the local entities (particles and/or Aristotelian substances). For instance, on a Bohmian story, one could divide up 3N-dimensional configuration space into N cells, one cell for each of the N particles, with each particle grounding the values of the wavefunction in its own cell. But it seems impossible to find a non-arbitrary way to divide up configuration space into such cells without massive overdetermination. (Perhaps the easiest way to think about the problem is to ask which particle gets to determine the value of the wavefunction in a small neighborhood of the current position in configuration space. They all intuitively have “equal rights” to it.)

It just seems neater to suppose a global entity to do the job.

A similar issue comes up in theories that require a global field, like an electromagnetic field or a gravitational field (even if these is to be identified with spacetime).

Here is another, rather different task for a global entity in an Aristotelian context. At many times in evolutionary history, new types of organisms have arisen, with new forms. For instance, from a dinosaur whose form did not require feathers, we got a dinosaur whose form did require feathers. Where did the new form come from? Or suppose that one day in the lab we synthesize something molecularily indistinguishable from a duck embryo. It is plausible to suppose that once it grows up, it will not only walk and quack like a duck, but it will be a duck. But where did it get its duck form from?

We could suppose that particles have a much more complex nature than the one that physics assigns to them, including the power to generate the forms of all possible organisms (or at least all possible non-personal organisms—there is at least theological reason to make that distinction). But it does not seem plausible to suppose that encoded in all the particles we have the forms of ducks, elephants, oak trees, and presumably a vast array of non-actual organisms. Also, it is somewhat difficult to see how the vast number of particles involved in the production of a duck embryo would “divide up” the task of producing a duck form. This is reminiscent of the problem of dividing up the wavefunction grounding among Bohmian particles.

I am now finding somewhat attractive the idea that a global entity carries the powers of producing a vast array of forms, so that if we synthesize something just like a duck embryo in the lab, the global entity makes it into a duck.

Of course, we could suppose the global entity to be God. But that may be too occasionalistic, and too much of a God-of-the-gaps solution. Moreover, we may want to be able to say that there is some kind of natural necessity in these productions of organisms.

We could suppose several global entities: a wavefunction, a spacetime, and a form-generator.

But we could also suppose them to be one entity that plays several roles. There are two main ways of doing this:

1. The global entity is the Universe, and all the local entities, like ducks and people and particles (if there are any), are parts of it or otherwise grounded in it. (This is Jonathan Schaffer’s holism.)

2. Local entities are ontologically independent of the global entity.

I rather like option (2). We might call this semi-holism.

But I don’t know if there is anything to be gained by supposing there to be one global entity rather than several.