Path integration and the functional measure

Abstract

The functional measure for the Feynman path integral is investigated, and it is argued that nontrivial measure factors should not be automatically discarded as is often done. The fundamental hypothesis of path integration is stated in its Hamiltonian formulation and is used, together with the Faddeev-Popov ansatz, to derive the general form of the canonical functional measure for all gauged or ungauged theories of integer spin fields in any number of space-time dimensions. This general result is then used to calculate the effective functional measures for scalar, vector, and gravitational fields in more than two dimensions at energies low compared to the Planck mass. It is shown that these results indicate the self-consistency and plausibility of the canonical functional measure over other functional measures and suggest an important relationship between bosonic and fermionic degrees of freedom. The canonical functional measure factors associated with fields of half-integer spin and with auxiliary fields are also derived.