  We propose a "matter-antimatter coexistence method" for finite-density
lattice QCD, aiming at a possible solution of the sign problem. In this method,
we consider matter and anti-matter systems on two parallel ${\bf R}^4$-sheets
in five-dimensional Euclidean space-time. For the matter system $M$ with a
chemical potential $\mu \in {\bf C}$ on a ${\bf R}^4$-sheet, we also prepare
the anti-matter system $\bar M$ with $-\mu^*$ on the other ${\bf R}^4$-sheet
shifted in the fifth direction. In the lattice QCD formalism, we introduce a
correlation term between the gauge variables $U_\nu \equiv e^{iagA_\nu}$ in $M$
and $\tilde U_\nu \equiv e^{iag \tilde A_\nu}$ in $\bar M$, such as $S_\lambda
\equiv \sum_{x,\nu} 2\lambda \{N_c-{\rm Re~tr} [U_\nu(x) \tilde
U_\nu^\dagger(x)]\} \simeq \sum_x \frac{1}{2}\lambda a^2 \{A_\nu^a(x)-\tilde
A_\nu^a(x)\}^2$ with a real parameter $\lambda$. In the limit of $\lambda
\rightarrow \infty$, a strong constraint $\tilde U_\nu(x)=U_\nu(x)$ is
realized, and the total fermionic determinant is real and non-negative. In the
limit of $\lambda \rightarrow 0$, this system goes to two separated ordinary
QCD systems with the chemical potential of $\mu$ and $-\mu^*$. On a
finite-volume lattice, if one takes an enough large value of $\lambda$, $\tilde
U_\nu(x) \simeq U_\nu(x)$ is realized and there occurs a phase cancellation
approximately between two fermionic determinants in $M$ and $\bar M$, which is
expected to suppress the sign problem and to make the lattice calculation
possible. For the obtained gauge configurations of the coexistence system,
matter-side quantities are evaluated through their measurement only for the
matter part $M$. By the calculations with gradually decreasing $\lambda$ and
their extrapolation to $\lambda=0$, physical quantities in finite density QCD
are expected to be estimated.
