@article{70665cbfa1a94d9a8926b3f6b3c861ec,

title = "Geometric fermions",

abstract = "We study the K{\"a}hler-Dirac equation which linearizes the laplacian on the space of antisymmetric tensor fields. In flat space-time it is equivalent to the Dirac equation with internal symmetry and on the lattice it reproduces Susskind fermions. The KD equation in curved space-time differs from the Dirac equation by coupling the gravitational field to the internal symmetry generators. This new way of treating fermionic degrees of freedom may lead to a solution of the generation puzzle but is in conflict with the equivalence principle and with Lorentz invariance on the Planck-mass scale.",

author = "T. Banks and Y. Dothan and D. Horn",

note = "Funding Information: In the early 1960's, the mathematician E. Kfihler \[1\]i ntroduced a transcription of the Dirac equation as a set of equations for antisyrrimetric tensor fields. This K~ihler-Dirac (KD) equation has been largely ignored by physicists, but has recently been studied by several authors in connection with lattice fermions \[2,3\].I ndeed, as we will show below, KD fields are a natural framework for understanding Susskind's \[4\] lattice fermions. Moreover, the KD equation may be generalized to any riemannian or pseudo-riemannian manifold. This was shown by Graf \[5\]w ho suggested that KD fields might be more fundamental than Dirac spinors. This is an appealing idea because it conforms to Einstein's philosophy of associating all physical fields with geometrical objects. It has one immediate consequence that we are familiar with from lattice models: the appearance of replicas of spinor fields with identical quantum numbers. We take up the idea that K~ihler's geometric fermions are the fundamental fermi fields, and investigate the possibility that the replication is associated with the generation structure of the fermion spectrum. We show that the KD equation in a gravitational field deviates from the conventional generalization of the Dirac equation and argue that it may play an important role in fLxing the fermion mass matrix. The KD Research supported in part by the Israel Commission for Basic Research.",

year = "1982",

month = nov,

day = "25",

doi = "10.1016/0370-2693(82)90571-8",

language = "English (US)",

volume = "117",

pages = "413--417",

journal = "Physics Letters B",

issn = "0370-2693",

publisher = "Elsevier",

number = "6",

}