Feynman’s Path Integrals as Evolutionary Semigroups

We show that, for a class of systems described by a Lagrangian L(x, ˙ x, t) = 1 2 ˙ x 2 − V (x, t) the propagator K x ′′ , t ′′ ; x ′ , t ′ = e i ¯ h t ′′ t ′ dtL(x, ˙ x,t) D [x(t)] can be reduced via Noether's Theorem to a standard path integral multiplied by a phase factor. Using Henstock's integration technique, this path integral is given a firm mathematical basis. Finally, we recast the propagator as an evolutionary semigroup. Here we deal with systems described by a Lagrangian of the form (m ≡ 1) L(x, ˙ x, t) = 1 2 ˙ x 2 − V (x, t) (1) The propagator for this system is given by the path integral K x ′′ , t ′′ ; x ′ , t ′ = e i ¯ h t ′′ t ′ dtL(x, ˙ x,t) D [x(t)] (2)