  A conjecture connected with quantum physics led N. Katz to discover some
amazing mixed character sum identities over a field of $q$ elements, where $q$
is a power of a prime $p>3$. His proof required deep algebro-geometric
techniques, and he expressed interest in finding a more straightforward direct
proof. Such a proof has been given by Evans and Greene in the case $q \equiv 3
\pmod 4$, and in this paper we give a proof for the remaining case $q \equiv 1
\pmod 4$. Moreover, we show that the identities are valid for all
characteristics $p >2$.
