Galilean Symmetry, Measurement, and Scattering as an Isomorphism between Two Subalgebras of Observables

A model of quantum-mechanical scattering is given in terms of the Abelian subalgebras of observables which are associated with the initial preparation and final measurement procedures. The scattering is described by an operator, the $S$ matrix, which defines an isomorphism between these two subalgebras. By using the concept of presymmetry for Galilean-invariant systems, these algebras are shown to be generated by the generalized momenta ′P which are related to the dynamical situation by $H{P}_{\mathrm{ac}}=\frac{{\left|{\mathrm{P}}^{\prime }\right|}^2}{2\mu }$. Here $H$ is the total Hamiltonian effective at the time of measurement, $P_{\mathrm{ac}}$ is the projection onto its absolutely continuous spectrum, and $\mu$ is the mass of the particle. The $S$ matrix which is defined in this way gives the same formal expressions for the differential cross sections as the usual one, and it is automatically unitary. Furthermore, since the differential cross section is defined as a relative probability in terms of the results of the two measurement procedures, it can be finite even when the classical cross section is infinite. Scattering can therefore be defined in this model for spherically symmetric potentials which decrease as slowly as $r^{-\epsilon }$, for any $\epsilon >0\mathrm{}$, provided that the potential is repulsive at large separations between the particles.