Dirac–Kähler equation
In theoretical physics, the Dirac–Kähler equation is the geometric analogue of the Dirac equation that can be defined on any pseudo-Riemannian manifold using the Laplace–de Rham operator. In four-dimensional flat spacetime, it is equivalent to four copies of the Dirac equation that transform into each other under Lorentz transformations, although this is no longer true in curved spacetime. The geometric structure gives the equation a natural discretization that is equivalent to the staggered fermion formalism in lattice field theory, making Dirac–Kähler fermions the formal continuum limit of staggered fermions. The equation was  discovered by Erich Kähler in 1962. Mathematical overview. In four dimensional Euclidean spacetime a generic fields of differential forms is written as a linear combination of sixteen basis forms indexed by formula_2, which runs over the sixteen ordered combinations of indices formula_3 with formula_4. Each index runs from one to four. Here formula_5 are antisymmetric tensor fields while formula_6 are the corresponding differential form basis elements Using the Hodge star operator formula_8, the exterior derivative formula_9 is related to the codifferential through formula_10. These form the Laplace-de Rham operator formula_11 which can be viewed as the square root of  the Laplacian operator since formula_12. The Dirac–Kähler equation is motivated by noting that this is also the property of the Dirac operator, yielding This equation is closely related to the usual Dirac equation, a connection which emerges from the close relation between the exterior algebra of differential forms and the Clifford algebra of which Dirac spinors are irreducible representations. For the basis elements to satisfy the Clifford algebra formula_14, it is required to introduce a new Clifford product formula_15 acting on basis elements as Using this product, the action of the Laplace-de Rham operator on differential form basis  elements is written as To acquire the Dirac equation, a change of basis must be performed, where the new basis can be packaged into a matrix formula_18 defined using the Dirac matrices The matrix formula_20 is designed to satisfy formula_21, decomposing the Clifford algebra into four irreducible copies of the Dirac algebra. This is because in this basis the Clifford product only mixes the column elements indexed by formula_22. Writing the differential form in this basis transforms the Dirac–Kähler equation into four sets of the Dirac equation indexed by formula_24 The minimally coupled Dirac–Kähler equation is found by  replacing the derivative with the covariant derivative formula_26 leading to As before, this is also equivalent to four copies of the Dirac equation. In the abelian case formula_28, while in the non-abelian case there are additional color indices. The Dirac–Kähler fermion formula_29 also picks up colour indices, with it formally corresponding to cross-sections of the Whitney product of the Atiyah–Kähler bundle of differential forms with the vector bundle of local colour spaces. Discretization. There is a natural way in which to discretize the Dirac–Kähler equation using the correspondence between exterior algebra and simplicial complexes. In  four dimensional space a lattice can be considered as a simplicial complex, whose simplexes are constructed using a basis of formula_30-dimensional hypercubes formula_31 with a base point formula_32 and an orientation determined by formula_2. Then a h-chain is a formal linear combination The h-chains admit a boundary operator formula_35 defined as the (h-1)-simplex forming the boundary of the h-chain. A coboundary operator formula_36 can be similarly defined to yield a (h+1)-chain. The dual space of chains consists of formula_30-cochains formula_38, which are linear functions acting on the h-chains mapping them to real numbers. The  boundary and coboundary operators admit similar structures in dual space called the dual boundary formula_39 and dual coboundary formula_40 defined to satisfy Under the correspondence between the exterior algebra and simplicial complexes, differential forms are equivalent to cochains, while the exterior derivative and codifferential correspond to the dual boundary and dual coboundary, respectively. Therefore, the Dirac–Kähler equation is written on simplicial complexes as The resulting discretized Dirac–Kähler fermion formula_43 is equivalent to the staggered fermion found in lattice field theory, which can be seen explicitly by an explicit change of basis. This equivalence shows that the continuum Dirac –Kähler fermion is the formal continuum limit of fermion staggered fermions. Relation to the Dirac equation. As described previously, the Dirac–Kähler equation in flat spacetime is equivalent to four copies of the Dirac equation, despite being a set of equations for antisymmetric tensor fields. The ability of integer spin tensor fields to describe half integer spinor fields is explained by the fact that Lorentz transformations do not commute with the internal Dirac–Kähler formula_44 symmetry, with the parameters of this symmetry being tensors rather than scalars. This means that the Lorentz transformations mix different tastes together and the Dirac  fermions are not strictly speaking half-integer spin representations of the Clifford algebra. They instead correspond to a coherent superposition of differential forms. In curved spacetime, the Dirac–Kähler equation no longer decomposes into four Dirac equations. Rather it is a modified Dirac equation acquired if the Dirac operator remained the square root of the Laplace operator, a property not shared by the Dirac equation in curved spacetime. This comes at the expense of Lorentz invariance, although these effects are suppressed by powers of the Planck mass. The equation also differs in that its zero modes on a compact manifold  are always guaranteed to exist whenever some of the Betti numbers vanish, unlike for the Dirac equation which never has zero modes on a manifold with positive curvature.