Scalar Product in Two-Particle Relativistic Quantum Mechanics

摘要

We construct, in the framework of two-particle relativistic quantum mechanics, a Poincare invariant scalar product and the corresponding physical Hilbert space of states. This is achieved by finding a tensor current of rank two satisfying two independent conservation laws, relative to particles 1 and 2, respectively. Then, the scalar product is obtained by integrating the current over two three-dimensional space-like hypersurfaces. The hermiticity of the Poincare group generators is ensured by the fact that the kernel of the current is translation invariant and covariant. A simple expression of the scalar product is obtained when one chooses for the two space-like hypersurfaces two constant parallel hyperplanes. The positivity of the norm is in general ensured if appropriate conditions are imposed on the shape and strength of the interaction potential. We also construct the star operators in the manifestly covariant formalism with constraints, for the expressions of operators. (ERA citation 12:027765)