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Baela
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What is the expression for the covariant derivative of a Weyl spinor?
Supergravity transformations in curved spacetime.malawi_glenn said:Under what transformation? Are we working in curved, or in flat space-time?
In what model for supergravity?Baela said:Supergravity transformations
A Weyl spinor is a type of spinor in quantum field theory and representation theory of the Lorentz group, which represents a massless fermion. Weyl spinors are two-component objects that transform under the (1/2, 0) or (0, 1/2) representations of the Lorentz group, corresponding to left-handed and right-handed spinors, respectively.
The covariant derivative for Weyl spinors is a generalization of the usual derivative that takes into account the curvature of spacetime and gauge fields. It ensures that the derivative of a spinor transforms covariantly under local Lorentz transformations and gauge transformations, maintaining the spinor's properties under these symmetries.
The covariant derivative of a Weyl spinor is defined using the spin connection for spacetime curvature and the gauge connection for internal symmetries. For a left-handed Weyl spinor ψ, the covariant derivative is given by Dμψ = ∂μψ + ½ ωμabσabψ + igAμψ, where ωμab is the spin connection, σab are the generators of the Lorentz group in the spinor representation, g is the gauge coupling, and Aμ is the gauge field.
The covariant derivative is crucial for Weyl spinors because it allows the formulation of physical theories, such as the Standard Model of particle physics, in a way that is consistent with both general relativity and gauge symmetries. It ensures that equations involving spinors are invariant under local Lorentz transformations and gauge transformations, which is essential for maintaining the consistency and predictive power of the theory.
While the covariant derivative of both Weyl and Dirac spinors incorporates the spin connection and gauge fields, the main difference lies in the representation of the spinors. Weyl spinors are two-component objects transforming under either the (1/2, 0) or (0, 1/2) representation of the Lorentz group, whereas Dirac spinors are four-component objects that combine both left-handed and right-handed components. Consequently, the covariant derivative