We review the non-trivial issue of the relativistic description of a quantum mechanical system that, contrary to a common belief, kept theoreticians busy from the end of 1920s to (at least) mid 1940s. Starting by the well-known works by Klein-Gordon and Dirac, we then give an account of the main results achieved by a variety of different authors, ranging from de Broglie to Proca, Majorana, Fierz-Pauli, Kemmer, Rarita-Schwinger and many others. A particular interest comes out for the general problem of the description of particles with \textit{arbitrary} spin, introduced (and solved) by Majorana as early as 1932, and later reconsidered, within a different approach, by Dirac in 1936 and by Fierz-Pauli in 1939. The final settlement of the problem in 1945 by Bhabha, who came back to the general ideas introduced by Majorana in 1932, is discussed as well, and, by making recourse also to unpublished documents by Majorana, we are able to reconstruct the line of reasoning behind the Majorana and the Bhabha equations, as well as its evolution. Intriguingly enough, such an evolution was \textit{identical} in the two authors, the difference being just the period of time required for that: probably few weeks in one case (Majorana), while more than ten years in the other one (Bhabha), with the contribution of several intermediate authors. Majorana's paper of 1932, in fact, contrary to the more complicated Dirac-Fierz-Pauli formalism, resulted to be very difficult to fully understand (probably for its pregnant meaning and latent physical and mathematical content): as is clear from his letters, even Pauli (who suggested its reading to Bhabha) took about one year in 1940-1 to understand it. This just testifies for the difficulty of the problem, and for the depth of Majorana's reasoning and results.
The birth of quantum mechanics was driven by the basic principles of the theory of relativity. Indeed, in 1923 L. de Broglie was the first [1] to exploit Lorentz invariance in order to formulate the well-known relations between the energy/momentum of a particle and the frequency/wave-vector of the associated wave. According to P.A.M. Dirac [2], even the subsequent formal development of de Broglie's ideas did not led E. Schrödinger [3] to write down first his most famous (non-relativistic) equation but, rather, the relativistic wave equation now known after O. Klein and W. Gordon [4] (which, for some time, was referred to as the relativistic Schrödinger equation). The original reasoning by Schrödinger started from the relativistic energymomentum relation for an electron,
and then assumed that the electron would be described by a wave-function ψ(x, t) satisfying the equation obtained by making in (1) the replacements
from which the de Broglie’s relations came out for a plane wave e i(k•x-ωt) . The resulting, first discovered, wave equation was then the relativistic Klein-Gordon equation:
Schrödinger then abandoned his relativistic wave equation since it gave the wrong predictions for the fine structure of the hydrogen atom, but later realized that the non-relativistic approximation to his relativistic equation, the proper Schrödinger equation, led to some correct results despite the original relativistic formulation. Schrödinger and others soon recognized that the source of discrepancy between the relativistic wave equation and observations was the neglect of the spin of the electron (the Klein-Gordon equation describes spin 0 particles) but, as well-known, we have to wait until 1928 when Dirac discovered [5] how to incorporate the spin of the electron in wave mechanics, in a consistent, relativistically invariant manner. The enormous success of the Dirac theory after the discovery of the positron in 1932 [6], and especially its quantum field theory formulation and incorporation in quantum electrodynamics, almost entirely obscured other subsequent formulations and generalizations of relativistic wave equations for particles with different value of the spin. Indeed, the early history of such equations is quite rich and intriguing, and extends over almost two decades. In the present paper we give (in the following section) an historical account of the different formulations of relativistic wave equations appeared in the literature till the end of 1940s, together with the physical motivations for them. Subsequent elaborations were mainly aimed at achieving mathematical improvements, and will not be considered here. It is mandatory to recall that relativistic wave mechanics derives its physical relevance just from its incorporation into quantum field theory (sometimes referred to as “second quantization”): as later realized, indeed, a relativistic quantum theory of a fixed number of particles is an impossibility. While some mentions about the quantum field theory justification of the equations proposed will be added occasionally, given their relevance in the subsequent developments, we will not dwell upon this topic which is far beyond our aim (for several aspects, see, instead, Ref. [7]). We will refer chiefly to the original papers (primary sources), since no exhaustive historical reviews are known; see however the partial but beautiful, though dated, review in Ref. [8] (see also [9]). While such papers tell us the known, though not widespread, story on this subject, in Sect. 3 we focus on several important results achieved by the Italian physicist Ettore Majorana [10] [11] but not published by him. They are contained in some booklets with his personal research notes, which only recently have been published [12] [13], and clearly testify (once more [14]) how he anticipated key results from different authors obtained over the subsequent years. In this respect, it is as well interesting to point out how results published by Majorana were received by his contemporaries, since they were fully understood only years later (a first illuminating example being just the mentioned Ref. [9]). From the correspondence of W. Pauli, then, we will be able in Sect. 4 to directly follow the study of the seminal paper published by Majorana in 1932 [15], that occupied Pauli for about one year. Finally, in Sect. 5 we summarize the results reviewed and give our conclusions.
The original rejection by Schrödinger (and others) of Eq. ( 3) as the correct quantum equation describing an electron was based, as recalled above, on the comparison of its prediction with the accurate experimental spectroscopic data on the hydrogen atom, and such discrepancy was correctly attributed to the non consideration of the spin of the electron. However, when Dirac approached the problem of making a relativistic theory of the spinning electron, no reference to such discrepancy was made, but rather he focused on another theoretical problem, namely tha
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