  We study Schwinger mechanism for gluon pair production in the presence of
arbitrary time-dependent chromo-electric background field $E^a(t)$ with
arbitrary color index $a$=1,2,...8 in SU(3) by directly evaluating the path
integral. We obtain an exact expression for the probability of non-perturbative
gluon pair production per unit time per unit volume and per unit transverse
momentum $\frac{dW}{d^4x d^2p_T}$ from arbitrary $E^a(t)$. We show that the
tadpole (or single gluon) effective action does not contribute to the
non-perturbative gluon pair production rate $\frac{dW}{d^4x d^2p_T}$. We find
that the exact result for non-perturbative gluon pair production is independent
of all the time derivatives $\frac{d^nE^a(t)}{dt^n}$ where $n=1,2,....\infty$
and has the same functional dependence on two casimir invariants
$[E^a(t)E^a(t)]$ and $[d_{abc}E^a(t)E^b(t)E^c(t)]^2$ as the constant
chromo-electric field $E^a$ result with the replacement: $E^a \to E^a(t)$. This
result may be relevant to study the production of a non-perturbative
quark-gluon plasma at RHIC and LHC.
