On Feynmann’s ‘integral over all paths’

Abstract

A definition of Feynmann’s ‘integral over all paths’ is given in precise mathematical terms, namely, it is shown, subject to some restrictions on the nature of the Lagrangian, that the ratio of the ‘integral’ to a certain normalization factor is uniquely determined by postulating: (i) the composition law, (ii) the possibility of taking any factor independent of the paths outside the integration over paths, and (iii) that the quantity in question should not involve any arbitrary ‘universal’ constants of the dimensions of inverse time. The nature of the normalization factor, referred to above, is also examined. Finally, a method of parametrization of paths is introduced, and it is shown that Feynmann’s ‘integral over all paths’ can, alternatively, be defined in terms of this parametrization.