  Feynman's Lagrangian path integral was an outgrowth of Dirac's vague surmise
that Lagrangians have a role in quantum mechanics. Lagrangians implicitly
incorporate Hamilton's first equation of motion, so their use contravenes the
uncertainty principle, but they are relevant to semiclassical approximations
and relatedly to the ubiquitous case that the Hamiltonian is quadratic in the
canonical momenta, which accounts for the Lagrangian path integral's "success".
Feynman also invented the Hamiltonian phase-space path integral, which is fully
compatible with the uncertainty principle. We recast this as an ordinary
functional integral by changing direct integration over subpaths constrained to
all have the same two endpoints into an equivalent integration over those
subpaths' unconstrained second derivatives. Function expansion with generalized
Legendre polynomials of time then enables the functional integral to be
unambiguously evaluated through first order in the elapsed time, yielding the
Schroedinger equation with a unique quantization of the classical Hamiltonian.
Widespread disbelief in that uniqueness stemmed from the mistaken notion that
no subpath can have its two endpoints arbitrarily far separated when its
nonzero elapsed time is made arbitrarily short. We also obtain the quantum
amplitude for any specified configuration or momentum path, which turns out to
be an ordinary functional integral over, respectively, all momentum or all
configuration paths. The first of these results is directly compared with
Feynman's mistaken Lagrangian-action hypothesis for such a configuration path
amplitude, with special heed to the case that the Hamiltonian is quadratic in
the canonical momenta.
