Positive energy dynamics and scattering theory for directly interacting relativistic particles

Journal Article · Thu May 01 04:00:00 EDT 1980 · Ann. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1016/0003-4916(80)90182-7· OSTI ID:5238366

Ruijsenaars, S N.M.

Starting from the tensor product of N irreducible positive energy representations of the Poincare group describing N free relativistic particles with arbitrary spins and positive masses, we construct an interacting positive energy representation by modifying the total 4-momentum operator. We first make a transformation to a Hilbert space on which the free total 4-momentum operator equals the product of a dimensionless center-of-mass 4-vector ((vertical-barkvertical-bar/sup 2/+1)/sup 1/2/, k) and a free ''reduced Hamiltonian'' H/sub r//sup 0/, which is a positive operator acting only on internal variables, and then replace H/sub r//sup 0/ by an interacting reduced Hamiltonian H/sub r/ =H/sub r//sup 0/+V, where V commutes with the Lorentz group and is such that H/sub r/ is a positive operator. The resulting product form is shown to imply that the wave operators intertwine the free and interacting representations so that the S-operator is Lorentz invariant. From a physical point of view the scheme is related to the framework first introduced by Bakamjian and Thomas, in which the Hamiltonian and boost generators are modified, but the above procedure makes a mathematically rigorous discussion much simpler. In the spin-zero case we introduce a natural generalization of the pair potentials of nonrelativistic N-particle Schroedinger theory to the present relativistic setting, study its scattering theory, and point out some problems that do not have analogs at the nonrelativistic level. In the spin-1/2 case we propose, inspired by the Dirac equation, explicit reduced Hamiltonians to describe atomic energy levels and present arguments making plausible that their eigenvalues are in closer agreement with the experimental data that their non-relativistic counterparts. We also consider extensions to arbitrary spin and, in the spin-1/2 case, coupling of a quantized radiation field.

• Research Organization: Joseph Henry Laboratories of Physics, Princeton University, Princeton, New Jersey 08544
• OSTI ID: 5238366
• Journal Information: Ann. Phys. (N.Y.); (United States), Journal Name: Ann. Phys. (N.Y.); (United States) Vol. 126:2; ISSN APNYA
• Country of Publication: United States
• Language: English

Related Subjects

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71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
BANACH SPACE
DIFFERENTIAL EQUATIONS
DIRAC EQUATION
EIGENVALUES
ENERGY RANGE
EQUATIONS
FIELD THEORIES
FOURIER TRANSFORMATION
FUNCTIONS
HAMILTONIANS
HILBERT SPACE
INTEGRAL TRANSFORMATIONS
IRREDUCIBLE REPRESENTATIONS
LIE GROUPS
LORENTZ GROUPS
MANY-BODY PROBLEM
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MECHANICS
ONE-DIMENSIONAL CALCULATIONS
POINCARE GROUPS
QUANTUM FIELD THEORY
QUANTUM MECHANICS
QUANTUM OPERATORS
RELATIVISTIC RANGE
SCHROEDINGER EQUATION
SPACE
SYMMETRY GROUPS
THREE-DIMENSIONAL CALCULATIONS
TRANSFORMATIONS
WAVE EQUATIONS
WAVE FUNCTIONS