A proposal to get some common-sense intuition for the paradox of the double-slit experiment

Abstract

We argue that the double-slit experiment can be understood much better by considering it as an experiment whereby one uses electrons to study the set-up rather than an experiment whereby we use a set-up to study the behaviour of electrons. We also show that the concept of undecidability (like e.g. occurs in G\odel's theorem) can be used in an intuitive way to make sense of the double-slit experiment and the quantum rules for calculating coherent and incoherent probabilities. We meet here a situation where the electrons always behave in a fully deterministic way, while the detailed design of the set-up may render the question about the way they move through the set-up experimentally undecidable. It is very important to make a distinction in quantum mechanics between the determinism of nature (Einstein) and the decidability of a question within an experimental set-up (Bohr). The former is about the absolute truth of an answer to a yes or no question, and follows binary logic (true or false), the latter about what an experimental set-up can decide and tell about the truth of that answer, and follows ternary logic (true, false or undecidable). Binary and ternary logic are incompatible. The viewpoints of Bohr and Einstein are thus operating on different levels and it is only by confusing these two levels that these two viewpoints seem to be irreconcilable. We show that the expression $\psi_{1} + \psi_{2}$ for the wave function of the double-slit experiment is numerically correct, but logically flawed. It has to be replaced in the interference region by the logically correct expression $\psi'_{1} + \psi'_{2}$, which has the same numerical value as $\psi_{1} + \psi_{2}$, such that $\psi'_{1} + \psi'_{2} = \psi_{1} + \psi_{2}$, but with $\psi'_{1} = {\psi_{1} +\psi_{2}\over{\sqrt{2}}} \,e^{\imath {\pi\over{4}} } \neq \psi_{1}$ and $\psi'_{2} = {\psi_{1} +\psi_{2}\over{\sqrt{2}}}\,e^{-\imath {\pi\over{4}}}\neq \psi_{2}$. Here $\psi'_{1}$ and $\psi'_{2}$ are the correct contributions from the slits to the total wave function $\psi'_{1} + \psi'_{2}$. We have then $p = |\psi'_{1} + \psi'_{2}|^{2} = |\psi'_{1}|^{2} + |\psi'_{2}|^{2} = p_{1}+p_{2} $ such that the paradox that quantum mechanics (QM) would not follow the traditional rules of probability calculus disappears. The paradox is rooted in the wrong intuition that $\psi_{1}$ and $\psi_{2}$ would be the true physical contributions to $\psi'_{1} + \psi'_{2} =\psi_{1} + \psi_{2}$ like in the case of waves in a water tank. The solution proposed here is not {\em ad hoc} but based on an extensive analysis of the geometrical meaning of spinors within group representation theory and its application to QM. Working further on the argument one can even show that an interference pattern is the only way to satisfy simultaneously two conditions: The condition obeying binary logic (in the spirit of Einstein) that the electron has only two mutually exclusive options to get to the detector (viz. going through slit S$_{1}$ or going through slit S$_{2}$) and the condition obeying ternary logic (in the spirit of Bohr) that the question which one of these two options the electron has taken is experimentally undecidable. A very important element in the analysis is the problem of the non-existence of {\em common} probability distributions for different experimental set-ups. And this is a recurrent theme in other situations that are felt as paradoxical. The CHSH inequality used in the experiments of Aspect {\em et al.} clashes with quantum mechanics because it is based on the assumption that there would exist a {\em common} probability distribution for the hidden variables in the different experiments that have to be made to determine the various quantities that intervene in the inequality. That common probability distribution for measured quantities does not exist is well known from quantum mechanics itself (because the operators that come into play do not commute and therefore do not have common eigenstates, i.e. common probability amplitudes). But this non-existence of common probability distributions applies also to certain variables in the set-ups of the different experiments and the latter fact can also be explained by purely classical reasoning.