We study the Schwinger mechanism in QCD in the presence of an arbitrary time-dependent chromo-electric background field $E^a(t)$ with arbitrary color index $a$=1,2,...8 in SU(3). We obtain an exact result for the non-perturbative quark (antiquark) pr
oduction from an arbitrary $E^a(t)$ by directly evaluating the path integral. We find that the exact result is independent of all the time derivatives $frac{d^nE^a(t)}{dt^n}$ where $n=1,2,...infty$. This result 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: $E^a rightarrow E^a(t)$. This result relies crucially on the validity of the shift conjecture, which has not yet been established.
We perform path integral for a quark (antiquark) in the presence of an arbitrary space-dependent static color potential A^a_0(x)(=-int dx E^a(x)) with arbitrary color index a=1,2,...8 in SU(3) and obtain an exact non-perturbative expression for the g
enerating functional. We show that such a path integration is possible even if one can not solve the Dirac equation in the presence of arbitrary space-dependent potential. It may be possible to further explore this path integral technique to study non-perturbative bound state formation.
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.
We calculate Lorentz-invariant and gauge-invariant quantities characterizing the product $sum_a D_R(T^a) F^a_{mu u}$, where $D_R(T^a)$ denotes the matrix for the generator $T^a$ in the representation $R=$ fundamental and adjoint, for color SU(3). We
also present analogous results for an SU(2) gauge theory.
