  In a recent work we introduced a nonlocal version of the
Nambu--Jona-Lasinio(NJL) model that was designed to generate a quark
self-energy in Euclidean space that was similar to that obtained in lattice
simulations of QCD. In the present work we carry out related calculations in
Minkowski space, so that we can study the effects of the significant vector and
axial-vector interactions that appear in extended NJL models and which play an
important role in the study of the $\rho$, $\omega$ and $a_1$ mesons. We study
the modification of the quark self-energy in the presence of matter and find
that our model reproduces the behavior of the quark condensate predicted by the
model-independent relation $<\bar qq>_{\rho} = <\bar
qq>_0(1-\sigma_N\rho_N/f_{\pi}^2m_{\pi}^2 +...)$, where $\sigma_N$ is the
pion-nucleon sigma term and $\rho_N$ is the density of nuclear matter. (Since
we do not include a model of confinement, our study is restricted to the
analysis of quark matter. We provide some discussion of the modification of the
above formula for quark matter.) The inclusion of a quark current mass leads to
a second-order phase transition for the restoration of chiral symmetry. That
restoration is about 80% at twice nuclear matter density for the model
considered in this work. We also find that the part of the quark self-energy
that is explicitly dependent upon density has a strong negative Lorentz-scalar
term and a strong positive Lorentz-vector term, which is analogous to the
self-energy found for the nucleon in nuclear matter when one makes use of the
Dirac equation for the nucleon. In this work we calculate the nucleon self
-energy in nuclear matter using our model of the quark self-energy and obtain
satisfactory results.
