Jean-Marie Souriau, symplectic geometer

I have met Souriau several times in Marseille. He has successfully applied geometry to dynamics; in particular, he has proposed a solution to the Dirac problem. This means that one can find a map from the algebra of functions on phase space, to the set of differential operators on a space of functions, so that the Poisson bracket of two functions is mapped to the commutator of the two corresponding operators. Unfortunately, the space of wave-functions (on which the differential operators act) is a set of functions on phase space, not the set of functions of the position only. Thus, Souriau's map does not solve the quantisation problem, and is now known as pre-quantisation. Irving Segal proposed another, non-canonical map, and suggested that the correct quantisation would be got by limiting the wave-functions to a subset, on which the canonical operators (position, momentum) formed an irreducible operator algebra, and on which the spectrum of the energy was positive. In my paper I suggest that this can only be achieved for a quadratic Hamiltonian (essentially, a system of harmonic oscillators). In that case, Segal's constraint leads to the Bargmann-Segal realisation of the harmonic oscillator; later, this was shown to be the same as Kostant's holomorphic induction. For the non-linear fields, as proposed by Segal, I showed that the theory is not Lorentz invariant, contrary to the claim in Segal's paper. I got the idea that Segal's quantisation is non-covariant from the fact that it differed from Souriau's, which solves the Dirac problem and so is covariant. Segal was kind enough to show me his private calculation, in which the commutators of the (Hamilton-Jacobi) generators of the Lorentz group are calculated, and shown to be "correct". Unfortunately, Segal starts with the time-like convention for the Lorentz metric, and at the end, adopts the space-like convention which allows him to cancel a whole unwanted set of non-linear functionals, instead of doubling them up, as is correct. As a result of this accidental fact, he was convinced that he must have chosen the space-like metric at the beginning. The extra terms, which he wanted to cancel, come from the difference between Segal's quantisation rules and Souriau's. Thus I conclude that Segal's 1960 paper is not correct.