Indeterminism and Undecidability

Abstract

The aim of this paper is to argue that the (alleged) indeterminism of quantum mechanics, claimed by adherents of the Copenhagen interpretation since Born (1926), can be proved from Chaitin's follow-up to Gödel's (first) incompleteness theorem. In comparison, Bell's (1964) theorem as well as the so-called free will theorem-originally due to Heywood and Redhead (1983)-left two loopholes for deterministic hidden variable theories, namely giving up either locality (more precisely: local contextuality, as in Bohmian mechanics) or free choice (i.e. uncorrelated measurement settings, as in 't Hooft's cellular automaton interpretation of quantum mechanics). The main point is that Bell and others did not exploit the full empirical content of quantum mechanics, which consists of long series of outcomes of repeated measurements (idealized as infinite binary sequences): their arguments only used the long-run relative frequencies derived from such series, and hence merely asked hidden variable theories to reproduce single-case Born probabilities defined by certain entangled bipartite states. If we idealize binary outcome strings of a fair quantum coin flip as infinite sequences, quantum mechanics predicts that these typically (i.e. almost surely) have a property called 1-randomness in logic, which is much stronger than uncomputability. This is the key to my claim, which is admittedly based on a stronger (yet compelling) notion of determinism than what is common in the literature on hidden variable theories. Contents 1 Introduction: Gödel and Bell 2 Randomness and its unprovability 3 Rethinking Bell's theorem 4 Are deterministic hidden variable theories deterministic? 5 Conclusion and discussion A The Born rule B 1-Randomness C Bell's theorem and free will theorem References

Bohmian Mechanics is Not Deterministic

“…12 This is Corollary 3.4.2 in Landsman [3], which is Corollary 4.2 of the version arXiv: abs/ 2003. 03554.…”

“…First, as explained in Landsman [2,3], in the narrow context of deterministic hidden variable theories, refined mathematical methods are available (namely from the theory of algorithmic randomness) that make an appeal to a bipartite setting unnecessary: since my argument does not rely on entanglement, we may simply work with a quantum coin toss (realized for example as a spin-z measurement on a spin- 1 2 particle in the state (1, 1)∕ √ 2 , or optically using suitable polarizers). The settings, possible contexts, and quantum state of the experiments are then fixed to be the same for each experiment in a long run, so that only the hidden state (i.e.…”

“…14 This notion of randomness (in this case of binary sequences) arose in the 1960s when ideas from probability theory were combined with the theory of algorithms and computation (à la Turing). I already motivated its use in quantum theory (and elsewhere) in Landsman [3], and will return to this motivation in Sect. 4 below in connection with the notion of typicality used by the Bohmians.…”

“…Partly to set the stage and partly to briefly discuss other aspects of determinism I will not go into here any further, let me summarize the formalism of Bohmian mechanics. 3 For simplicity, consider a system of N non-relativistic spinless particles with masses m 1 , … , m N , moving in ℝ 3 , so that the classical configuration space is ℝ 3N . Suppose the classical Hamiltonian, defined on the associated phase space T * (ℝ 3N ) ≅ ℝ 6N , is given by 2 Here 'causal' is synonymous to 'deterministic'.…”

“…Similarly, Holland's well-known book The Quantum Theory of Motion is subtitled: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics. 3 For details see e.g. Holland [13], Dürr and Teufel [12], Bricmont [14], and Goldstein [15].…”