A brief survey of the mathematics of quantum physics

The mathematics of quantum physics started from matrices and from differential operators. It inspired the theory of linear operators in Hilbert space and of unitary representation for symmetry groups and spectrum generating groups. The Dirac bra-ket formalism led first to Schwartz's theory of distributions and then to its generalization, the Rigged Hilbert Space (RHS) or Gelfand triplet. This Schwartz-RHS provided the mathematical justification for Dirac's continuous basis vector expansion and for the algebra of continuous observables of quantum theory. To obtain also a mathematical theory of scattering, resonance and decay phenomena one needed to make a mathematical distinction between prepared in-states and detected observables (“out-states”). This leads to a pair of Hardy RHS's and using the Paley-Wiener theorem, to solutions of the dynamical equations (Schrödinger or Heisenberg) given by time-asymmetric semi-groups, expressing Einstein causality.