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Consequences of space-time invariances in quantum mechanics and direct interaction theories

University of British Columbia

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.

Text

2010-02-26

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http://hdl.handle.net/2429/21125

Physics

University of British Columbia

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