Born rule from counting states

Ovidiu Cristinel Stoica

Published 2022-09-18Version 1

I give a very simple derivation of the Born rule by counting states from a continuous basis. More precisely, I show that in a continuous basis, the contributing basis vectors are present in a state vector with real and equal coefficients, but they are distributed with variable density among the eigenspaces of the observable. Counting the contributing basis vectors while taking their density into account gives the Born rule without making other assumptions. State counting yields the Born rule only if the basis is continuous, but all known physically realistic observables admit such bases. The continuous basis is not unique, and for subsystems it depends on the observable. But for the entire universe, there are continuous bases that give the Born rule for all measurements, because all measurements reduce to distinguishing macroscopic pointer states, and macroscopic observations commute. This allows for the possibility of an ontic basis for the entire universe. In the wavefunctional formulation, the basis can be chosen to consist of classical field configurations, and the coefficients $\Psi[\phi]$ can be made real by absorbing them into a global U(1) gauge. For the many-worlds interpretation, this result gives the Born rule from micro-branch counting.