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 Dmitri Ivanenko and Lev Landau in 1928[1] and later rediscovered by Erich Kähler in 1962.
[2] In four dimensional Euclidean spacetime a generic fields of differential forms is written as a linear combination of sixteen basis forms indexed by
, which runs over the sixteen ordered combinations of indices
are the corresponding differential form basis elements Using the Hodge star operator
which can be viewed as the square root of the Laplacian operator since
The Dirac–Kähler equation is motivated by noting that this is also the property of the Dirac operator, yielding[3]
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
defined using the Dirac matrices The matrix
This is because in this basis the Clifford product only mixes the column elements indexed by
Writing the differential form in this basis transforms the Dirac–Kähler equation into four sets of the Dirac equation indexed by
leading to As before, this is also equivalent to four copies of the Dirac equation.
, while in the non-abelian case there are additional color indices.
also picks up color 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 color spaces.
[4] 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
, 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
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[6] The resulting discretized Dirac–Kähler fermion
is equivalent to the staggered fermion found in lattice field theory, which can be seen explicitly by an explicit change of basis.
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
[7] This means that the Lorentz transformations mix different spins 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.
dimensional surfaces, the Dirac–Kähler equation is equivalent to
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.
[9] 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, being given by the harmonic forms, unlike for the Dirac equation which never has zero modes on a manifold with positive curvature.