Algorithmic Randomness, Quantum Physics, and Incompleteness

孝太郎 只木

Kohtaro Tadaki

Euan J. Squires

Klaas Landsman

Certain prima facie deterministic theories, notably Bohmian mechanics and kinetic theory, require ``random'' initial conditions in order to achieve their goals. I analyze recent proposals that this idea should be implemented using algorithmic randomness, seen as a mathematical sharpening of the typicality arguments originally used to understand these theories. On the one hand my analysis includes a critique of these arguments and hence a defense of this sharpening. On the other hand, I argue that Chaitin's incompleteness theorems for random sequences make it impossible for a dynamical theory to either process given random initial conditions beyond some finite (but not arbitrary) number of their digits, or else to state such conditions with arbitrary precision without compromising their randomness (these are two sides of the same coin). This challenges the ability of such theories to make any other predictions than those that hold typically in the usual sense of the word (like macroscopic statements), and moves their alleged determinism high up the metaphysical ladder.

Jeffrey A. Barrett, Eddy Keming Chen, Josiah Lopez-Wild

Appeals to typicality are common in physics, but it is often unclear what it means for a physical state to be typical relative to a probability measure, and correspondingly unclear what a law that appeals to typicality asserts. Here we consider how one might characterize physical typicality using ideas from the theory of algorithmic randomness. As a concrete example, we show how taking a physical state to be typical relative to a computable measure when it is Martin-Löf random allows one to formulate the distribution postulate in Bohmian mechanics as a statistical constraining law of the theory. Using a toy model, we show how this constraint guarantees the standard Born statistics for computable experimental protocols. Algorithmic Bohmian mechanics (aBM) thus illustrates how algorithmic randomness may be used to provide precise content to a statistical law.

Deepak Chahal, Preethi Ravichandran Kumar