The bra and ket formalism in extended Hilbert space

Abstract

The concept of bra‐vector is endowed with a precise mathematical meaning by taking a Hilbert‐space $\mathcal{H}$ (whose elements are the ket‐vectors) and embedding it into a larger bra‐space $\mathcal{L}$ . The space $\mathcal{L}$ is constructed by fitting together all the different spaces with ``negative'' norm corresponding to equipping operators D in $\mathcal{H}$ . It is shown that the formal manipulations of Dirac's formalism become theorems on the resulting extended Hilbert space structure $\mathcal{H}\subset \mathcal{L}$ . On $\mathcal{L}$ we define topologies and bra‐adjoints of operators in $\mathcal{H}$ and investigate their properties. We show how these results can be used in deriving rigorous versions of the typical relations involving distorted waves in time‐independent scattering theory. By applying this formalism to Fock space we obtain an extended Fock space framework suitable for the rigorous formulation of the concept of a field at a point in configuration or momentum space.