A covariant position operator. Application to a system Of two particles

Domenico P. L. Castrigiano

Mario Bunge, Andrés J. Kálnay

Nicola Pinamonti, M. Toller

We study the positive-operator-valued measures on the projective real line covariant with respect to the projective group, assuming that the energy is a positive operator. This problem is similar to the more complicated problem of finding the positive-operator-valued measures on the compactified Minkowski space-time covariant with respect to the conformal group. It also describes a time-of-arrival observable for a free particle covariant with respect to linear canonical transformations.

Eugene Feenberg

Patricia Ann Kalyniak

The problem of describing a quantum mechanical system is considered. For a system which is invariant under the Poincaré (inhomogeneous Lorentz) transformations this description is provided by the generators of those transformations, which satisfy the usual commutation relations. General expressions for the centre of mass position and internal angular momentum operators, in terms of these generators, are obtained. The generators of the Poincaré transformations are written in terms of the fundamental dynamical variables for several systems. The systems considered are those consisting of a single free spinless particle, a single free particle with spin, a single free Dirac particle, n noninteracting particles, and ft interacting particles. In each case, the centre of mass position and internal angular momentum are given in terms of the fundamental dynamical variables of the system. For the first two systems listed above, these two operators are found to be equal to the Cartesian coordinates and spin of the particle, respectively. In the case of the Dirac particle, these operators are seen to be related to the Cartesian coordinates and spin of the particle via the Pryce-Foldy-Wouthuysen transformation. Following Bakamjian and Thomas, interaction is introduced to the n particle system via a single operator which depends only on internal variables. The condition of "asymptotic covariance" of the scattering operator is discussed for two particle scattering. The scattering operator for a two particle system with no bound states and with Poincaré generators of the Bakamjian-Thomas form is seen to be asymptotically covariant.

Alessandro Toigo

Given a topological group $G$ and a unitary representation $U$ of $G$, we consider the problem of classifying the positive operator measures which are based on a $G$-homogeneous space $X$ and covariant with respect to the representation $U$. We completely solve the problem in two cases: 1) when $G$ is abelian; 2) when $G$ is not abelian, $X$ has compact stbiliser in $G$ and the representation $U$ is irreducible. We then give an application of the above results to the characterisation of covariant position and momentum observables in quantum mechanics, and to the determination of those covariant position and momentum observables that are jointly measurable.