Monday, August 7, 2023

A deterministic collapsing local quantum mechanics without hidden variables beyond the wavefunction

I will give a really, really wacky version of quantum mechanics as a proof of concept that if one wants, one can have all of the following:

1. Compatibility with experiment

2. Determinism

3. Collapse

4. No “hidden variables” beyond the wavefunction: the wavefunction encompasses all the information about the world

5. Locality

6. Schroedinger evolution between collapes.

Here’s the idea. We suppose that the Hilbert space for quantum mechanics is separable (i.e., has a countable basis). A separable Hilbert space has continuum-many vectors, so each quantum state vector can be encoded as a single real number. We suppose, further, that collapse occurs countably many times over the history of the universe. We can now encode all the times and outcomes of the collapses over the history of the universe as a single real number: the outcome of a collapse is a quantum state vector, encodable as a real number, the time of collapse is of course a real number, and a countable sequence of pairs of real numbers can be encoded as a single real number.

We now consider the wavefunction ψ of the universe. For simplicity, consider this as a function on R3n × R where n is the number of particles (if the number of particles changes over time, we will need to tweak this). Say that x ∈ R3n is rational provided that every coordinate of it is a rational number. We now add a new law of nature: ψ(x,t) has the same value for every rational x and every time t, which value encodes the history of all the collapses that ever happen in the history of the universe.

Since standard quantum mechanics does not care about what happens to the wavefunction on sets of measure zero, and the set of rational points of R3n has measure zero, this does not affect Schroedinger evolution between collapses, and so we have 6. We also clearly have 2, 3 and 4. If we suppose a prior probability distribution on the collapses that fits with the Born rule, we get 1. We also have 5, since any open region of space that contains an experiment will also contain the real number encoding the collapse history.

Of course, this is rather nutty. It just shows that because the wavefunction has more room for information than just the quantum state vector—the quantum state vector can be thought of as an equivalence class of wavefunctions differing on sets of measure zero—we can stuff the hidden variables into the wavefunction. Those of us who think that the state vector is the real thing, not the wavefunction, will be quite unimpressed.