 Fresnel convolution of free-particle time slices

ID: fresnel-convolution-of-free-particle-time-slices

Completing the square in the integral over an intermediate position shows that two free-particle propagators compose to the propagator for the sum of their positive time intervals. For equal slices, the quadratic identity is

\frac{( x - x _{0} ) ^{2}}{k} + (x_{1} - x)^{2} = \frac{k + 1}{k} (x - \frac{x _{0} + k x _{1}}{k + 1})^{2} + \frac{( x _{1} - x _{0} ) ^{2}}{k + 1} .

The Fresnel integral contributes a factor proportional to

k / (k + 1)

; multiplying successive factors gives

1/ N

. A consistent convergence regulator fixes the phases of these square roots.

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