On the Complexity of Measurement in Classical Physics

Abstract

If we measure the position of a point particle, then we will come about with an interval [a _n , b _n ] into which the point falls. We make use of a Gedankenexperiment to find better and better values of a _n and b _n , by reducing their relative distance, in a succession of intervals [a ₁, b ₁] ⊃ [a ₂, b ₂] ⊃ ... ⊃ [a _n , b _n ] that contain the point. We then use such a point as an oracle to perform relative computation in polynomial time, by considering the succession of approximations to the point as suitable answers to the queries in an oracle Turing machine. We prove that, no matter the precision achieved in such a Gedankenexperiment, within the limits studied, the Turing Machine, equipped with such an oracle, will be able to compute above the classical Turing limit for the polynomial time resource, either generating the class P/poly either generating the class BPP//log*, if we allow for an arbitrary precision in measurement or just a limited precision, respectively. We think that this result is astonishingly interesting for Classical Physics and its connection to the Theory of Computation, namely for the implications on the nature of space and the perception of space in Classical Physics. (Some proofs are provided, to give the flavor of the subject. Missing proofs can be found in a detailed long report at the address http://fgc.math.ist.utl.pt/papers/sm.pdf .)