Abstract
The semigroup of trajectories in Minkowski space-time and its induced representations are constructed as a generalization of the Galilei case. They describe relativistic pointlike particles and yield the free propagator as a path integral in the space of trajectories parametrized by a fifth parameter. This non physical propagator in a five-dimensional space is integrated over the fifth parameter to yield the physical propagator in Minkowski space. Thereafter, this notion is applied to a model of extended particles with internal Poincaré symmetry and moving in an external Minkowski space. The geometrical structure is of Hilbert bundles and the interaction is introduced as a connection. The propagator is a path integral with respect to either the internal and external trajectories and reduces to a product of an internal and an external propagator when the interaction is ignored, just as has been found in a previous work with representations of the group rather than those of the semigroup.
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Smida, A., Hachemane, M. & Hamici, AH. Geometro-Differential Conception of Extended Particles and the Semigroup of Trajectories in Minkowski Space-Time. Foundations of Physics 28, 1367–1381 (1998). https://doi.org/10.1023/A:1018831011256
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DOI: https://doi.org/10.1023/A:1018831011256