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We formulate Dirac-Kaehler fermion action by introducing a new Cliffordproduct with noncommutative differential form on a lattice. Hermiticity of theDirac-Kaehler action requires to choose the lattice structure having bothorientabilities on a link. The Kogut-Susskind fermion and the staggered fermionactions are derived directly from the Dirac-Kaehler fermion formulated by theClifford product. The lattice QCD action with Dirac-Kaehler matter fermion isalso derived via an inner product defined by the Clifford product.Comment: 1+49 pages, 5 figures. v2: references added, and minor modification

DIRAC–KÄHLER FERMION FROM CLIFFORD PRODUCT WITH NONCOMMUTATIVE DIFFERENTIAL FORM ON A LATTICE