The Marvelous Consequences of Hardy Spaces in Quantum Physics

Abstract

Dynamical differential equations, like the Schrödinger equation for the states, or the Heisenbergequa tion for the observables, need to be solved under boundary conditions. The original boundary condition of von Neumann, the Hilbert space axiom, required that the allowed wave functions are Lebesgue square integrable. This leads by a mathematical theorem of Stone-von Neumann to the unitary group evolution meaning the time t extends over $-\infty\,<\,t\,<\,+\infty$. Physicists do not use Lebesgue integrals but followed a different path usinga lmost exclusively the Dirac formalism and well-behaved (Schwartz) functions. This led the mathematicians to Schwartz- Rigged Hilbert spaces (Gelfand triplets), which are the mathematical core of Dirac’s bra-ket formalism. This is insufficient for a theory that includes resonance and decay phenomena, which requires analytic continuation in energy E in order to accommodate exponentially decayingG amow kets, Breit-Wigner (Lorentzian) resonances, and Lippmann-Schwinger kets. This leads to a pair of Rigged Hilbert Spaces of smooth Hardy functions, one representing the prepared states of scatteringe xperiments (preparation apparatus) and the other representingd etected observables (registration apparatus). A mathematical consequence of the Hardy space axiom is that the time evolution is asymmetric given by the semi-group, i.e., $t_0\,\leq\,t\,<\,+\infty$, with a $t_0$. What would the meaningo f that $t_0$ be?

Mathematics Subject Classification (2010). 81-06; 81P16; 81R99; 34L10.