Dirac–kähler Fermion From Clifford Product With Noncommutative Differential Form on a Lattice

Abstract

We formulate Dirac-Kähler fermion action by introducing a new Clifford product with noncommutative differential form on a lattice. Hermiticity of the Dirac-Kähler action requires to choose the lattice structure having both orientabilities on a link. The Kogut-Susskind fermion and the staggered fermion actions are derived directly from the Dirac-Kähler fermion formulated by the Clifford product. The lattice QCD action with Dirac-Kähler matter fermion is also derived via an inner product defined by the Clifford product.

Noncommutativity approach to supersymmetry on the lattice: Supersymmetric quantum mechanics and an inconsistency

“…(16). Their supersymmetry transformations ǫ 1 Q 1 and ǫ 2 Q 2 obey the Leibniz rule, provided the noncommutativity relations (13), (14) and (15) hold with…”

N=4 twisted superspace from Dirac–Kähler twist and off-shell SUSY invariant actions in four dimensions

“…where P ± = 1 2 (1 ± γ 5 ) and the D-K fermion Ψ is defined in (2.23) as 74) where i = 1, 2, 3, 4. The action (5.71) can be transformed as follows:…”

“…It is well known that the Dirac-Kähler fermion mechanism is fundamentally related to the lattice formulation [64][65][66] [67,68]. In fact recently N = 2 twisted superspace in two dimensions has been successfully formulated on a lattice with an introduction of mild non-commutability [69][70][71][72][73][74] for lattice difference operator and twisted supercharges [63]. It is strongly suggested that N = 4 twisted superspace formalism in four dimensions is important to formulate four-dimensional SUSY on a lattice.…”

Formulation of supersymmetry on a lattice as a representation of a deformed superalgebra

“…It was shown that the naïve fermion formulation where the continuum differential operators in the Dirac action is naïvely replaced by the lattice difference operator can be spin diagonalized and leads to the staggered fermion formulation [48] which is shown to be essentially equivalent [49,50] to Kogut-Susskind fermion formulation [51]. The equivalence of the staggered fermion formulation and the Dirac-Kähler fermion has been proved exactly with an introduction of mild noncommutativity between differential forms and fields [16]. This means that all these lattice fermion formulations are equivalent where the mild noncommutativity seems to play an important rôle .…”