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We discuss that the deviation of the Kugo-Ojima color confinement parameter$u(0)$ from -1 in the case of quenched lattice simulation and the consistencywith -1 in the case of full QCD simulation could be attributed to the boundarycondition defined by fermions inside the region of $r<1$fm. By using the domainwall fermion propagator in lattice simulation, we show that the chiral symmetrybreaking in the infrared can become manifest when one assumes that theleft-handed fermion on the left wall and the right-handed fermion on the rightwall are correlated by a self-dual gauge field. The relation between theinfrared fixed point of the running coupling measured in lattice simulations,the prediction of the BLM renormalization theory, the conformal field theorywith use of the t'Hooft anomaly matching condition in non-SUSY supersymmetrictheory and the quaternion real condition are discussed.

Chiral Symmetry and BRST Symmetry Breaking, Quaternion Reality and the Lattice Simulation