Relativistic Bound State under the Hooke-Type Action-at-a-Distance

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Abstract
The classical relativistic mechanics of a system consisting of two point particles bound by the Hooke-like action-at-a-distance is established. Relativistic invariance of this interaction is ensured by the prescription that it works between “conjugate” points on the world lines of the two particles, which perform relativistic rigid motion. The conservation laws of 4-momentum and angular momentum are given, and they lead to a linear relation between spin and squared mass of the bound system. Solutions to the fundamental equations of theory are explicitly obtained. Also it is shown that in the non-relativistic limit the theory approaches the usual Hooke-type theory. Quantization of this theory leads to a bilocal wave equation associated with two subsidiary conditions, and this is suitable for the representation of mesonic states lying on a Regge trajectory. Extension to the case where N point particles are mutually bound by the same mechanism is allowed only if N ≦3, and the N = 3 case corresponds to baryons.

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