A group theory derivation of the relativistic path integral and the “history-string” dynamics

We derive the Feynman path integral for relativistic elementary particles using group theory considerations. We apply an approach in which choosing a symmetry group (or semigroup) allows deriving the kinematics and dynamics of a particle including the state space and the propagator from it. The quantum properties of a particle appear from intertwining two representations of the symmetry (semi)group, one of which describes local properties of the particle and the other describes the particle as a whole. The path-integral-like dynamics appears when the symmetry semigroup has a structure similar to that of the relativistic analogue of the Galilei group (in which the Lorentz-invariant “proper time” plays the role of time) with translations replaced with the semigroup of trajectories (parameterized paths). The classical action in the weight functional of the path integral is defined by this semigroup up to couplings to gauge and/or gravitational fields. The obtained formalism is suitable for describing not only pointlike particles but also nonlocal objects of the “history-string” type, which, as previously shown, allow explaining quark confinement.