Group-theoretical derivation of path integrals

The dynamics of nonrelativistic particles in the form of a Feynman path integral is derived from group-theo'retical considerations. A group-theoretical approach is used, this making it possible to construct the quantum theory of an elementary particle on the basis of its symmetry group. The quantum properties of the particle arise from the intertwining of two representations of the symmetry group, one of which describes the local properties of the particle, and the other the particle as a whole. This approach is appIied to the generalized Galileo semigroup, which is obtained from the ordinary Galileo group by replacing the translation subgroup by a semigroup of trajectories (parametrized paths). As a result, the propagator of the particle in an external electromagnetic or gauge field is derived in the form of a path integral. The integration measure, including the weight factor $\exp(iS)$, is uniquely determined by the requirement of invariance.