# A proposal for the solution of the paradox of the double-slit experiment

Abstract : The solution for this paradox proposed here is not {\em ad hoc} but based on the author's reconstruction of quantum mechanics, based on the geometrical meaning of spinors within the representation theory of the rotation group SO(3) and the homogeneous Lorentz group SO(3,1). We argue that the double-slit experiment can be understood much better by considering it as an experiment whereby the particles yield information about the set-up rather than an experiment whereby the set-up yields information about the behaviour of the particles. The probabilities of QM are conditional, whereby the conditions are defined by the macroscopic measuring device. They are not uniquely defined by the interaction probabilities in the point of the interaction. Furthermore, a particle can interact coherently or incoherently with the matter of this experimental device. Coherent interactions do not leave any information behind about the passage of the particle through the set-up, while incoherent interactions do. When the particles interact incoherently with the set-up the answer to the question through which slit they have moved is experimentally decidable. When they interact coherently the answer to that question is experimentally undecidable. But the particles always behave in a fully deterministic way. We explain the different viewpoints of Bohr and Einstein in terms of two different choices of axiomatic systems ${\sf{TL}}$ (Bohr) and ${\sf{TL+D}}$ (Einstein) to deal with the experimental undecidability of the question through which slit the particle has travelled. Bohr takes the undecidability as is. Einstein adds an axiom that we know nevertheless that the particle has only two mutually exclusive options to travel through the device, even if the interactions do not create any information about this issue. Both axiomatic systems are internally consistent and contradiction-free, such that it is logically flawed to attack one axiomatic system from the stronghold of the other one after promoting it to a touchstone of absolute truth. We show that the expression $\psi_{3}= \psi_{1} + \psi_{2}$ for the wave function of the double-slit experiment is numerically correct, but logically flawed. Within the axiomatic system ${\sf{TL}}$ it is just meaningless to decompose $\psi_{3}$. Within the axiomatic system ${\sf{TL+D}}$ 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 $\psi'_{1} = e^{\imath \pi/4 } (\psi_{1} +\psi_{2})/\sqrt{2} \neq \psi_{1}$ and $\psi'_{2} = e^{-\imath \pi/4 } (\psi_{1} +\psi_{2})/\sqrt{2} \,\neq \psi_{2}$. This reflects the change of the definition of the conditional probabilities due to the change of the boundary conditions and the axiomatic system. Here $\psi'_{1}$ and $\psi'_{2}$ are the correct (but experimentally unknowable) contributions from the slits to the total wave function $\psi_{3} = \psi'_{1} + \psi'_{2}$. 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.
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https://hal-cea.archives-ouvertes.fr/cea-01383609
Contributor : Gerrit Coddens <>
Submitted on : Monday, June 7, 2021 - 12:47:44 PM
Last modification on : Thursday, July 8, 2021 - 3:33:21 AM

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Coddens-double-slit-symm-3.pdf
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• HAL Id : cea-01383609, version 6

### Citation

Gerrit Coddens. A proposal for the solution of the paradox of the double-slit experiment. 2021. ⟨cea-01383609v6⟩

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