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arXiv:physics/0603140

Majorana and the path-integral approach to Quantum Mechanics

Salvatore Esposito

Published Mar 17, 2006• 1 citation
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Abstract

We give, for the first time, the English translation of a manuscript by Ettore Majorana, which probably corresponds to the text for a seminar delivered at the University of Naples in 1938, where he lectured on Theoretical Physics. Some passages reveal a physical interpretation of the Quantum Mechanics which anticipates of several years the Feynman approach in terms of path integrals, independently of the underlying mathematical formulation.

DOI
10.48550/arxiv.physics/0603140

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We give, for the first time, the English translation of a manuscript by Ettore Majorana, which probably corresponds to the text for a seminar delivered at the University of Naples in 1938, where he lectured on Theoretical Physics. Some passages reveal a physical interpretation of the Quantum Mechanics which anticipates of several years the Feynman approach in terms of path integrals, independently of the underlying mathematical formulation.
From January to March 1938 – as we have seen in Chapter 1 – Majorana delivered his only lectures on quantum mechanics at the University of Naples, where he obtained a position as a full professor of theoretical physics. He prepared a set of lecture notes [26] for his students (see Section 13.5), and, among the original manuscripts of these lectures, we find some additional spare papers that cannot be considered as notes for academic lectures, even for an advanced course such as that taught by Majorana. Instead, as suggested in Refs. [328, 329], they probably refer to a seminar or conference held at the University of Naples, whose main topic was likely the theoretical interpretation of the molecular bonding in the framework of quantum mechanics; the notes were prepared by Majorana for his own personal use. Scientific interest in this dissertation, however, focuses not on its main topic, nor on the calculations (which are practically absent from the manuscript), but rather on the interpretation given by Majorana to key concepts of the quantum theory, i.e. the concept of quantum state and the direct application of the quantum theory to the particular case of molecular bonding. The manuscript indeed discloses a peculiar cleverness on the part of Majorana in referring to this pivotal argument of quantum mechanics, and also reveals the concept was at least ten years ahead of its time. This link had been already noted earlier by Nicola Cabibbo [330], who saw in the Majorana manuscript a vague and approximate anticipation of the idea underlying the Feynman interpretation of quantum mechanics in terms of path integrals. A more analytic study has revealed, however, some intriguing surprises, upon which we will focus here. In addition to the historical interest of Majorana's manuscript, we stress also the particularly powerful didactic method used by him on the given subject: his original presentation of the quantum-mechanical problems is exceedingly useful in teaching now those or related topics. For this reason, in the Appendix we will report the entire text (translated from the Italian) as prepared by Majorana.
The path integral formulation in quantum mechanics corresponds to the first quantization since it is just to rewrite the quantum mechanical amplitude into many dimensional integrations over discretized coordinates xnx_nxn​. However, the path integral expression cannot be connected to the dynamics of classical mechanics, even though, superficially, there is some similarity between them. Further, the field theory path integral in terms of many dimensional integrations over fields does not correspond to the field quantization. We clarify the essential difference between Feynman's original formulation of path integral in QED and the modern version of the path integral method prevailing in lattice field theory calculations, and show that the former can make a correct second quantization while the latter cannot quantize fields at all and its physical meaning is unknown.
This year is the centenary of the birth of Ettore Majorana, one of the major Italian physicists of all times. In this note we briefly sketch a few biographical details about Ettore Majorana and introduce and discuss the main points of Majorana's 10th article. In his article Majorana explicitly considers quantum mechanics as an irreducible statistical theory because the theory is not able to describe the time evolution of a single particle or atom in a precise environment at a deterministic level. This lack of determinism at the level of an elementary physical system motivated him to suggest a formal analogy between statistical laws observed in physics and in the social sciences. We hope the occasion of the centenary of the birth of Ettore Majorana will be useful to remember and to reconsider not only his exceptional achievements in theoretical physics but also his fresh and original views on the role of statistical laws in physics and in other disciplines such as the social sciences.
Efforts to give an improved mathematical meaning to Feynman's path integral formulation of quantum mechanics started soon after its introduction and continue to this day. In the present paper, one common thread of development is followed over many years, with contributions made by various authors. The present version of this line of development involves a continuous-time regularization for a general phase space path integral and provides, in the author's opinion at least, perhaps the optimal formulation of the path integral.