Searching for an equation: Dirac, Majorana and the others

Abstract

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came back to the general ideas introduced by Majorana in 1932, is discussed as well, and, by making recourse also to unpublished documents by Majorana, we are able to reconstruct the line of reasoning behind the Majorana and the Bhabha equations, as well as its evolution.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.Comment: amsart, 34 pages, no figure

Dirac equation based on the vector representation of the Lorentz group

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Conformal relativity with hypercomplex variables

“…The tensor can be used to parameterize the sixteen parameters of the group of general linear transformations GL(4, R), which is acting on the right hand side of Eq. (16).…”

On wave equations for the Majorana particle in (3+1) and (1+1) dimensions

“…In general, the relativistic wave equation considered to mathematically describe a rst quantized Majorana particle (an electrically neutral fermion in (3+1) dimensions that is its own antiparticle) is the Dirac equation with a real Lorentz scalar potential together with the socalled Majorana condition [1,2]. The latter condition assumes that the Dirac wave function is equal to its respective charge-conjugate wave function, i.e., Ψ = Ψ C ; this is regardless of the representation of the gamma matrices that one chooses when writing the Dirac equation.…”