I haven’t closely read the details on the hypothetical experiments yet, but I want to comment on the technical details of the quantum mechanics at the beginning.
In quantum mechanics, probabilities of mutually exclusive events still add: . However, things like “particle goes through slit 1 then hits spot x on screen” and “particle goes through slit 2 then hits spot x on screen” aren’t such mutually exclusive events.
This may seem like I’m nit-picking, but I’d like to make the point by example. Let’s say we have a state where . If we simply add the complex amplitudes to try to calculate , we get 0; in actuality, we should get as we expect from classical logic.
Here’s where I bad-mouth the common way of writing the Born rule in intro quantum material as and the way I’d been using it. By writing the state as and the event as we’ve made it look like they’re both naturally represented as vectors in a Hilbert space. But the natural form of a state is as a density matrix, and the natural form of an event is as an orthogonal projection; I want to focus on events and projections. For mutually exclusive events and with projections and , the event has the corresponding projection .
So where’s the adding of amplitudes? Let’s pretend I didn’t just say states are naturally density matrices and let’s take the same state from above and an arbitrary projection corresponding to some event. The Born rule takes the following form:
This is notably not just an contribution plus a contribution; the other terms are the interference terms. Skipping over what a density matrix is, let’s say we have a density matrix . The Born rule for density matrices is
Now this one is just a sum of two contributions, with no interference.
This ended up longer and more rambling than I’d originally intended. But I think there’s a lot to the finer details of how probabilities and amplitudes behave that are worth emphasizing.
Everett argued in his thesis that the unitary dynamics motivated this:
He made the analogy with Liouville’s theorem in classical dynamics, where symplectic dynamics motivated the Lebesgue measure on phase space.