Abstract
The quasiclassical theory of massless chiral fermions is considered. The effective action is derived using time-dependent variational principle which provides a clear interpretation of relevant canonical variables. As a result their transformation properties under the action of Lorentz group are derived from first principles.
Highlights
The present paper is devoted to the further study of the covariance problem for chiral fermions
[27] Skagerstam noticed that the existence of position operator with standard transformation properties and commutation rules would imply that the massless irreducible representation of Poincare group accommodates all 2λ + 1 chiralities, which is not the case
In the weak field regime where particle number is approximately conserved we can look for the effective one-particle dynamics which should be formulated in terms of coordinate and momenta
Summary
Let us start with free massless left-handed Weyl fermions. The relevant wave function obeys Weyl equation σμ∂μφ = 0,. On the other hand it is convenient because such a combination of spinors enters Lorentz invariant definition of scalar product ( being the space integral of zeroth component of current) and action functional I, eq (2.10) below (resulting from invariant action functional for Dirac particle in the limit m → 0 with one chirality deleted) Another choice of normalization would only influence the actual form of wave packet profile c( p ); the normalization condition enters explicitly, for example, the expression for coordinate. (2.2) and (2.15), together with our assumption concerning the semiclassical character of the wave packet, yield the following transformation rule for xc (valid on-shell, i.e. provided the Weyl equation (2.1) holds, cf [18]). Eqs. (2.23), (2.29) and (2.30), together with the assumption concerning the shape of wave packet, imply again the transformation rule (2.19)
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