We present a commentary on the famous 1964 paper of John Bell that rules out the entire class of underlying hidden variable theories for quantum mechanics that are local.
John S Bell 1987 Cambridge: Cambridge University Press vii + 212 pp price £25 ISBN 0 521 33495 0 This book is a collection of John Bell's papers dealing explicitly with the foundations of quantum mechanics (or, as he puts it, ' quantum philosophy '), plus one paper on the pedagogy of special relativity. The first two are the classic papers of 1 966 and 1 964, which respectively demolished the 'impossibility proofs' concerning hidden variables and proved the famous theorem on the incompatibility of the predictions of local hidden-variable theories with those of quantum mechanics.
Heisenberg deduced the famous uncertainty principle which shows that there exist several conjugated quantities that can never be measured precisely at the same time. Further, Copenhagen interpretation gives a new perspective about quantum mechanics, which claims that particles do not have properties like position or momentum until people measure it. Hence, the famous EPR paradox was proposed to question the realism and locality of Quantum mechanics, which leads to the Hidden Variable explanation. However, this theorem was proved wrong by John Bell in 1964 with Bell Inequality. In addition, Hidden Variable Interpretation was developed by De Broglie and Bohm, they came up with the Bohmian Mechanics, which can be considered as Non-local Hidden Variable theorem. This interpretation gives physical meaning to waves. Unlike Copenhagen Interpretation, this theorem claims that particles do have a determined position. Some people may argue that it can be proved contradicted by using the Turing method (self-reference). Because if the algorithm represented by physics law described Hidden Mechanics to determine particles’ state, then the future is determined and predictable. Therefore, another algorithm can be established based on that, which will lead to “Liar Paradox”. This article will briefly introduce the uncertainty principle and some interpretations about quantum mechanics. Furthermore, this article will combine some ideas in Alan Turing’s Halting Problem to the universe of non-local hidden variables as a thought experiment, which involve self-reference, to give an interesting result of locality and determinism.
John Bell showed that a big class of local hidden-variable models stands in conflict with quantum mechanics and experiment. Recently, there were suggestions that empirically adequate hidden-variable models might exist which presuppose a weaker notion of local causality. We will show that a Bell-type inequality can be derived also from these weaker assumptions. 1. Introduction2. The EPR-Bohm experiment3. Local causality4. Bell's inequality from separate common causes4.1A weak screening-off principle4.2Perfect correlation and ‘determinism’4.3A minimal theory for spins4.4No conspiracy5. Discussion