At the turn of the last century there were a number of questions concerning the atomic and molecular levels in the physical world. That was the motivation of Heisenberg, Schroedinger, Dirac, Weyl, Wigner, etc. in developing quantum mechanics, and they were duly "solved." The contributions fell into two categories: the analytic approach of Heisenberg, Schroedinger, Dirac, et al, and the group theory approach of Weyl, Wigner, et al. People in the first set even had a name for the second set - the "gruppenpest" (or the "pestilence of groups").
But the analysts had problems that compounded as time went on that led to the difficulty of obtaining relativistic formulations, to the concept of the Dirac sea of electrons, to a break between classical mechanics and quantum mechanics, to quantum field theory at a point, etc. We shall review the literature of the time showing what prominent physicists thought concerning these problems, as well as giving the good points of the theory.
On the other hand, the group theorists went their merry way with rigorous mathematical proofs of their various results. There weren't any arguments that I could find that were against what they did and were given by prominent physicists. This was either because putting physics in terms of group theory was too radical and abstract for them or because mistakes of the group theorists were not made. We shall review what the group theorists did.
We shall then take the various problems that have been around for the past 85 years and show how the phase space approach solves almost all of them. Examples of the difficuties that remain include solving the many body problem which predates the era of quantum theory, determining a number of constants, determining why there are apparently only Fermions and Bosons, etc.
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