"An effective two dimensionality" cases bring a new hope to the Kaluza-Klein[like] theories

Abstract

One step towards realistic Kaluza-Klein[like] theories and a loop hole through the Witten's "no-go theorem" is presented for cases which we call an effective two dimensionality cases: In $d=2$ the equations of motion following from the action with the linear curvature leave spin connections and zweibeins undetermined. We present the case of a spinor in $d=(1+5)$ compactified on a formally infinite disc with the zweibein which makes a disc curved on an almost $S^2$ and with the spin connection field which allows on such a sphere only one massless normalizable spinor state of a particular charge, which couples the spinor chirally to the corresponding Kaluza-Klein gauge field. We assume no external gauge fields. The masslessness of a spinor is achieved by the choice of a spin connection field (which breaks parity), the zweibein and the normalizability condition for spinor states, which guarantee a discrete spectrum forming the complete basis. We discuss the meaning of the hole, which manifests the noncompactness of the space.

26 pages, 1 figure, an addition which helps to clarify the assumptions and their consequences (the discreteness of spectrum, the massless solution of one handedness,..)

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• Spinor states on a curved infinite disc with non-zero spin-connection fields
• New way of second quantized theory of fermions with either Clifford or Grassmann coordinates and spin-charge-family theory
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• Families of spinors in $d=(1+5)$ with zweibein and two kinds of spin connection fields on an almost $S^2$
• Properties of families of spinors in $d=(5+1)$ with zweibein of an almost $S^2$ and two kinds of spin connection fields, allowing massless and massive solutions in $d=(3+1)$