Structure of the State Space in a Convergent Relativistic Model Theory of Interacting Particles
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概要
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The structure of the state space of a model system of interacting particles is investigated from the viewpoint of the Lorentz transformations. The model system is one of two examples which have been previously obtained by the author without use of the Lagrangian method as direct solutions of the set of the fundamental commutator equations for the ten generators (P_μ,M_<ρσ>) of the Poincare group. The investigation is focussed on the one-particle and the twoparticle sectors. The physical one-particle states |p>_<(1)> are shown to obey the expected transformation law U^dagger(τ,Λ)|p>_<(1)>=exp(-ip_μτ_μ)|Λp>_<(1)>. It is proved that also the two-particle bound states and the two-particle continuum states with "standing wave" boundary condition satisfy respectively the transformation law U^dagger(τ,Λ)|p>^^^<(b)>><(2)> and U^dagger(τ,Λ)|q,q′>^<(st)>_<(2)>=exp(-i(q_μ+q_μ′)τ_μ)|Λq,Λq′>~<(st)gt;_<(2)>. On the contrary the two-particle continuum states |q,q′>^<(±)>_<(2)> with "incident plane wave+outgoing (or ingoing) scattered wave" boundary condition turn out not to obey such a simple transformation law. It is shown, however, that by an appropriate superposition of the states |q,q′>^<(+)>_<(2)> or |q,q′>^<(-)>_<82)> one can construct new state vectors ‖q,q′≫^<(+)>_<(2)> and ‖q,q′≫^<(-)>_<(2)> which obey the simple transformation law U^dagger(τ,Λ)‖q,q′≫^<(±)>_<(2)>=exp(-i(q_μ+q_μ′)τ_μ)‖Λq,Λq′≫^<(±)>_<(2)>. The unitary S matrix with the correct transformation property is obtained in a closed form.
- 理論物理学刊行会の論文
- 1970-05-25
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