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Journal of Mathematical Physics

Particles and simple scattering theory

J. V. Corbett

Published Feb 1, 1975• 1 citation
Publisher

Abstract

A necessary and sufficient condition for a pair of subspaces to be the in and out scattering subspaces for a simple scattering system is obtained. It involves the existence of certain representations of the Galilean presymmetry group on these subspaces. The physical interpretation is that, in scattering, particles remain as particles even in the presence of the interaction.

DOI
10.1063/1.522536

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A necessary and sufficient condition for a pair of subspaces to be the in and out scattering subspaces for a simple scattering system is obtained. It involves the existence of certain representations of the Galilean presymmetry group on these subspaces. The physical interpretation is that, in scattering, particles remain as particles even in the presence of the interaction.
In this paper we give a complete theory of the scattering operator on a rigorous mathematical foundation for a certain class of quantum mechanical systems called, 'simple scattering systems'. These systems are characterized by three properties formulated in Section 3. The relation of these properties to the physical content of scattering is explained. We then proceed to define the wave operator and the scattering operator and prove their existence with mathematically rigorous methods. No assumptions are needed regarding the occurrence of bound states and the location of their energy with respect to the continuum region of the total energy operator. In Section 4 we demonstrate the existence of a certain integral representation of the scattering operator which has often been used as a starting point of the iteration approximation. Certain general properties of scattering systems are obtained in Section 5. This leads to necessary conditions for systems to be scattering systems as well as to a new definition of the scattering operator. The paper concludes with a derivation of the scattering cross-section from the scattering operator. The improvement of this derivation over the older ones sonsists in the special care employed for stating the assumptions which must go into such a derivation and the avoidence of improper functions or other artificial devices.
The mathematical theory of the scattering operator is developed for the general scattering systems involving an arbitrary number of channels. It includes as a special case the theory for 'simple scattering systems' given in an earlier paper. The scattering system is defined as a quantum mechanical system which satisfies certain asymptotic and completeness conditions given in Section 4. The existence of the S-operator as well as its unitary property is then a rigorous mathematical consequence of this property. A crucial step in these deductions is the orthogonality theorem for the left projections of the wave operators which is proved in Section 5. In the last Section 7, we discuss the various ways of introducing the 'in' and 'out'-operators and their relation to the S-operator.
The major themes of quantum mechanics are the physical interpretations of interactions in bound systems and particle scatterings. Stability of atoms and molecules is necessary in order that materials in nature could have any given determined physical and chemical properties, according to which these materials are distinguished and identified. Standard scattering theory has originally been introduced in nuclear physics, where the interactions are of short range, so that in the asymptotic region, the aggregates can be considered as being fully free. The required behaviours of the total scattering state are determined by the very nature of the interactions between the widely separated aggregates in the asymptotic region of scattering. In the old literature a terminological difference has sometimes been made between the notions of scattering and collision.
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 SSS 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 \ensuremath{'}P which are related to the dynamical situation by $H{P}_{\mathrm{ac}}=\frac{{|{\mathrm{P}}^{\ensuremath{'}}|}^{2}}{2\ensuremath{\mu}}$. Here HHH is the total Hamiltonian effective at the time of measurement, Pac{P}_{\mathrm{ac}}Pac​ is the projection onto its absolutely continuous spectrum, and $\ensuremath{\mu}$ is the mass of the particle. The SSS 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 ${r}^{\ensuremath{-}\ensuremath{\epsilon}}$, for any $\ensuremath{\epsilon}>0$, provided that the potential is repulsive at large separations between the particles.
This paper is devoted to the study of the Galilei group and its representations. The Galilei group presents a certain number of essential differences with respect to the Poincaré group. As Bargmann showed, its physical representations, here explicitly constructed, are not true representations but only up-to-a-factor ones. Consequently, in nonrelativistic quantum mechanics, the mass has a very special role, and in fact, gives rise to a superselection rule which prevents the existence of unstable particles. The internal energy of a nonrelativistic system is known to be an arbitrary parameter; this is shown to come also from Galilean invariance, because of a nontrivial concept of equivalence between physical representations. On the contrary, the behavior of an elementary system with respect to rotations, is very similar to the relativistic case. We show here, in particular, how the number of polarization states reduces to two for the zero-mass case (though in fact there are no physical zero-mass systems in nonrelativistic mechanics). Finally, we study the two-particle system, where the orbital angular momenta quite naturally introduce themselves through the decomposition of the tensor product of two physical representations.