Toward the lattice QCD calculation at finite density, we propose "matter-antimatter coexistence method", where matter and anti-matter systems are prepared on two parallel ${\bf R}^4$-sheets in five-dimensional Euclidean space-time. We put a matter system $M$ with a chemical potential $��\in {\bf C}$ on a ${\bf R}^4$-sheet, and also put an anti-matter system $\bar M$ with $-��^*$ on the other ${\bf R}^4$-sheet shifted in the fifth direction. Between the gauge variables $U_��\equiv e^{iagA_��}$ in $M$ and $\tilde U_��\equiv e^{iag \tilde A_��}$ in $\bar M$, we introduce a correlation term with a real parameter $��$. In one limit of $��\rightarrow \infty$, a strong constraint $\tilde U_��(x)=U_��(x)$ is realized, and therefore the total fermionic determinant becomes real and non-negative, due to the cancellation of the phase factors in $M$ and $\bar M$, although this system resembles QCD with an isospin chemical potential. In another limit of $��\rightarrow 0$, this system goes to two separated ordinary QCD systems with the chemical potential of $��$ and $-��^*$. For a given finite-volume lattice, if one takes an enough large value of $��$, $\tilde U_��(x) \simeq U_��(x)$ is realized and phase cancellation approximately occurs between two fermionic determinants in $M$ and $\bar M$, which suppresses the sign problem and is expected 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 $M$. The physical quantities in finite density QCD are expected to be estimated by the calculations with gradually decreasing $��$ and the extrapolation to $��=0$. We also consider more sophisticated improvement of this method using an irrelevant-type correlation.

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