Comparing Pauli and Weinberg proof of the spin-statistics theorem

Full text: I shall present two different versions for the proof of the spin-statistics theorem, one by Pauli and the other by Weinberg, using different hypothesis. I'll make a parallel between them, comparing and stressing the differences they present. The spin-statistics theorem, originally presented by Pauli in 1940 [The connection between spin and statistics. Physical Review, 82, 914-927], can be enunciated as: There is a strict connection between spin and quantum statistics, such that all semi-integer spin particles (fermions) respect the Fermi-Dirac statistics, and all integer spin particles (bosons) respect the Bose-Einstein statistics. Fermi-Dirac statistics respects the Pauli Exclusion Principle: identical particles are not allowed to occupy the same energy state. Bose-Einstein statistics, on the contrary, doesn't. Finding a simple and elementary proof of the connection between statistics and spin has been a great challenge for scientists over more than six decades. Pauli wasn't able to prove the theorem using only the relativistic request of micro causality. He also had to call upon the positive energy statement [M. Massimi, M. Redhead -Weinberg proof of the spin-statistics theorem. Studies in History and Philosophy of Modern Physics, 34, 621-650. (2003)]. Micro causality is the requirement that two physical measurements made in different points x and y be mutually independent, if these two measurements were made with spatial distance. That is the same as saying that no signal with a velocity smaller than light's can get between these two points. But these conditions for causality are feasible only when one is dealing with a field whose components can be measured at any point in space-time. However, fields as Dirac fields are not measurable in this sense. And so, in 1965, Steven Weinberg managed to prove the theorem in a simpler way: using just the requirement of micro causality, in an article that proposed the calculation of Feynman rules for particles with any number of spin [Feynman rules for any spin. Physical Review, 133, B1318-B1332]. According to him, Pauli proof is direct only when it comes to integer spin particles, but indirect for semi-integer particles. Weinberg simplification resides on the fact that he considers the condition of micro-causality just as being necessary for the Lorentz invariance of the S matrix [The quantum theory of fields vol.I. Cambridge: Cambridge University Press. (1995)]. The microcausality requirement used by Weinberg shows the spin-statistics connection in a direct way both for integer and semi-integer spins.(author)