Authors: Jean Louis Van Belle
This paper discusses Feynman’s derivation of the Hamiltonian matrix in the famous Caltech Lectures on Quantum Mechanics, which is illustrative of the mainstream interpretation of what probability amplitudes may or may not represent. We refer to this argument as Feynman’s Time Machine because the “apparatus” that is considered in the derivation is, effectively, the mere passage of time. We show Feynman’s argument is ingenious but, at the same time, deceptive. Indeed, the substitution – for what Feynman refers to as “historical and other reasons” – of real-valued coefficients (K) by pure imaginary numbers (-iH/ħ) effectively introduces the periodic functions (complex-valued exponentials) that are needed to obtain sensible probability functions. The division by Planck’s quantum of action also amounts to an insertion of the Planck-Einstein relation through the backdoor. The argument is, therefore, typical of similar arguments: one only gets out what was already implicit or explicit in the assumptions. The implication is that two-state systems can be described perfectly well using classical mechanics, i.e. without using the concepts of state vectors and probability amplitudes. This paper, therefore, complements earlier deconstructions of some of Feynman’s arguments, most notably his argument on 720-degree symmetries – which we referred to as “the double life of -1” – as well as the reasoning behind the establishment of the boson-fermion dichotomy. This paper may, therefore, conclude our classical or realist interpretation of quantum mechanics.
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