Bohm’s interpretation of quantum mechanics has two ontological components: It has the guiding wave—the wavefunction—which dynamically evolves according to the Schrödinger equation, and it has the corpuscles whose movements are guided by that wavefunction. Brown and Wallace criticize Bohm for this duality, on the grounds that there is no reason to take our macroscopic reality to be connected with the corpuscles rather than the wavefunction.
I want to explore a variant of Bohm on which there is no evolving wavefunction, and then generalize the point to a number of other no-collapse interpretations.
So, on Bohm’s quantum mechanics, reality at a time t is represented by two things: (a) a wavefunction vector |ψ(t)⟩ in the Hilbert space, and (b) an assignment of values to hidden variables (e.g., corpuscle positions). The first item evolves according to the Schrödinger equation. Given an initial vector |ψ(0)⟩, the vector at time t can be mathematically given as |ψ(t)⟩ = Ut|ψ(0)⟩ where Ut is a mathematical time-evolution operator (dependent on the Hamiltonian). And then by a law of nature, the hidden variables evolve according to a differential equation—the guiding equation—that involves |ψ(t)⟩.
But now suppose we change the ontology. We keep the assignment of values to hidden variables at times. But instead of supposing that reality has something corresponding to the wavefunction vector at every time, we merely suppose that reality has something corresponding to an initial wavefunction vector |ψ0⟩. There is nothing in physical reality corresponding to the wavefunction at t if t > 0. But nonetheless it makes mathematical sense to talk of the vector Ut|ψ0⟩, and then the guiding equation governing the evolution of the hidden variables can be formulated in terms of Ut|ψ0⟩ instead of |ψ(t)⟩.
If we want an ontology to go with this, we could say that the reality corresponding to the initial vector |ψ0⟩ affects the evolution of the hidden variables for all subsequent times. We now have only one aspect of reality—the hidden variables of the corpuscles—evolving dynamically instead of two. We don’t have Schrödinger’s equation in the laws of nature except as a useful mathematical property of the Ut operator described by the initial vector). We can talk of the wavefunction Ut|ψ0⟩ at a time t, but that’s just a mathematical artifact, just as m1m2 is a part of the equation expressing Newton’s law of gravitation rather than a direct representation of physical reality. Of course, just as m1m2 is determined by physical things—the two masses—so too the wavefunction Ut|ψ0⟩ is determined by physical reality (the initial vector, the time, and the Hamiltonian). This seems to me to weaken the force of the Brown and Wallace point, since there no longer is a reality corresponding to the wavefunction at non-initial times, except highly derivatively.
Interestingly, the exact same move can be made for a number of other no-collapse interpretations, such as Bell’s indeterministic variant of Bohm, other modal interpretations, the many-minds interpretation, the traveling minds interpretation and the Aristotelian traveling forms interpretation. There need be no time-evolving wavefunction in reality, but just an initial vector which transtemporally affects the evolution of the other aspects of reality (such as where the minds go).
Or one could suppose a static background vector.
It’s interesting to ask what happens if one plugs this into the Everett interpretation. There I think we get something rather implausible: for then all time-evolution will disappear, since all reality will be reduced to the physical correlate of the initial vector. So my move above is only plausible for those no-collapse interpretations on which there is something more beyond the wavefunction.
There is also a connection between this approach and the Heisenberg picture. How close the connection is is not yet clear to me.