Hilbert's Incompleteness, Chaitin's Ω Number and Quantum Physics

To explore the limitation of a class of quantum algorithms originally proposed for the Hilbert’s tenth problem, we consider two further classes of mathematically nondecidable problems, those of a modified version of the Hilbert’s tenth problem and of the computation of the Chaitin’s Ω number, which is a representation of the Gödel’s Incompletness theorem. Some interesting connection to Quantum Field Theory is pointed out, but a direct generalisation of the quantum algorithms cannot satisfy, among others, the requirement of finite energy. Introduction Quantum physical processes have been considered for the purpose of computation for some time, and more recently some are proposed in [1] for certain class of mathematical noncomputables associated with the Hilbert’s tenth problem and, equivalently, the Turing halting problem [2]. There is, however, a hierarchy of noncomputable/undecidable problems [3] of which the Turing halting problem is at the lowest level. Another well-known nondecidable result is the Gödel’s Incompleteness Theorem about the incompatibility of consistency and completeness of Arithmetics, see [4] for a readable account. As the mathematical property of being Diophantine, which is at the heart of the Turing halting problem, is not a sufficient condition for that of being arithmetic [5], which concerns the Gödel’s, the quantum algorithm cited above has no direct consequence on the Gödel’s theorem. Notwithstanding this, in order to explore the limitation of our algorithm we consider two further classes of mathematically non-decidable problems. We find some connection between Quantum Field Theory and the Chaitin’s Ω number which intimately links to the Gödel’s theorem. Along the way we also consider a modified version of the Hilbert’s tenth ∗email: [email protected]