Spinor-type objects in metric-affine space-time

Abstract

A generalized spinor structure is constructed as a GL(2, C) principal fiber bundle over the metric-affine manifold. Relations between vectors and spinors are established explicitly through the generalized Pauli and Dirac matrices. An expression for generalized spinor connection is obtained. The Dirac equation in metric-affine space-time is written in the form representing an interaction of spinors with nonmetricity explicitly. A possible physical meaning of nonmetrical objects is discussed in brief.

An integrable geometrical foundation of gravity

“…The gravitational coupling of the spinors thus turned out to be, in contrast to the naive expectation, equivalent to GR. Yet, the semi-Hermitian connection with the imaginary piece might link the electromagnetic phase to spacetime geometry, see however [1,37,38,41,42].…”

“…We note that when the potential σ vanishes, a shear transformation will not affect the spin connection. By plugging these transformations (41) into the expressions for the curvatures (35), we can obtain the behaviour of the latter under the action generated by (38). The translations modify only the torsion,…”

“…The metric g is "neutral" to non-metricity, whilst the f In the presence of non-metricity, the map (66) is not conserved: ∇ασ βȦ B = L αβ µ σµȦ B . It seems to be possible to generalise this to ∇ασ βȦ B = (L αβ µ + N αβ µ )σµȦ B , where N α(βµ) = 0, but where N α[βµ] = 0 would then appear in the Dirac equation [38]. g This gives 1-1 correspondence with the affine connection, which has double the number of independent components of the spinor connection.…”

“…In particular, though the q enters precisely as the electromagnetic potential into the Dirac equation, there is no obvious electromagnetic interpretation for this phase coupling. However, Poberii has pointed out that one could enhance the symmetry with complex rescalings of the spinor metric, to the effect that matter fields could be assigned with the desired conformal weights [38].…”

Geometrical Origin of a Cosmological Term

“…This has also been shown to be true from a spinor analysis. [7] This means that the scalar field, as long as we stick to minimal coupling, does not couple to spin one half particles. 2 What this boils down to is this: the scalar field will contribute to the curvature of space, but does not produce a direct force on particles.…”