## Getting Path Integrals Physically and Technically Right

**Authors:** Steven Kenneth Kauffmann

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 Schrödinger 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.

**Comments:** 18 pages, Also archived as arXiv:0910.2490 [physics.gen-ph].

**Download:** **PDF**

### Submission history

[v1] 2 Dec 2009

[v2] 8 Aug 2010

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