  In the positron-electron annihilation process, finite deviations from the
standard calculation based on the Fermi's Golden rule are suggested in recent
theoretical work. This paper describes an experimental test of the predictions
of this theoretical work by searching for events with two photons from positron
annihilation of energy larger than the electron rest mass ($511\,{\rm keV}$).
The positrons came from a ${\rm {}^{22}Na}$ source, tagging the third photon
from the spontaneous emission of ${\rm {}^{22}{Ne}^*}$ de-exitation to suppress
backgrounds. Using the collected sample of $1.06\times 10^{7}$
positron-electron annihilations, triple coincidence photon events in the signal
enhanced energy regions are examined. The observed number of events in two
signal regions, $N^{\rm SR1}_{\rm obs}=0$ and $N^{\rm SR2}_{\rm obs}=0$ are,
within a current precision, consistent with the expected number of events,
$N^{\rm SR1}_{\rm exp}=0.86\pm0.08({\rm stat.})^{+1.85}_{-0.81}({\rm syst.})$
and $N^{\rm SR2}_{\rm exp}=0.37\pm 0.05({\rm stat.})^{+0.80}_{-0.29}({\rm
syst.})$ from Fermi's golden rule respectively. Based on the $P^{(d)}$
modeling, the 90% CL lower limit on the photon wave packet size is obtained.
