arXiv:1801.09447v4 [hep-th] 13 Aug 2018 More about the covariance of chiral fermions theory K. Andrzejewski 1) , Y. Brihaye 2) , C. Gonera 1) , J. Gonera 1) , P. Kosi´ nski 1) * , P. Ma´ slanka 1) 1) Department of Computer Science, Faculty of Physics and Applied Informatics, University of Lodz, Poland 2) Physique-Math´ ematique, Universit´ e de Mons-Hainaut, Mons, Belgium Abstract The quasiclassical theory of massless chiral fermions is considered. The effective action is derived using time-dependent variational principle which provides a clear in- terpretation of relevant canonical variables. As a result their transformation properties under the action of Lorentz group are derived from first principles. 1 Introduction In recent years a renewed interest is observed concerning chiral anomalies, their physical consequences (in particular, for kinetic theory) and relations to topology and Berry curvature [1]-[15]. For many purposes (e.g. the dynamics in weak and slowly varying external fields) one can rely on semiclassical approximation. In particular, within this approximation the Weyl equation for helicity − 1 2 massless charged fermions is replaced by the semiclassical dynamics summarized in the action functional I =   ( q +  A) · dx − (ε + A 0 )dt +  α( q ) · d q  , (1.1) where  q denotes gauge-invariant (kinetic) momentum,  α( q ) is the vector potential describing the Berry monopole in momentum space while ε = | q| +  q ·  B 2| q| 2 . (1.2) Eq. (1.1) can be derived from Weyl Hamiltonian by making the semiclassical approxima- tions to path-integral representation of transition amplitude [10]. The obvious trouble with eq. (1.1) is that it lacks manifest Lorentz symmetry (in the absence of external fields) or Lorentz covariance, in spite of the fact that it is derived from covariant Weyl equation. In Ref. [10] a modified transformation rule under Lorentz boosts has been proposed for the particle dynamical variables, involving O() corrections, which leaves invariant the dynamics described by the action (1.1). It reduces to the standard Lorentz form for spinless particles. The modified rules involve helicity dependent terms and close only on-shell. * Corresponding author, e-mail: pkosinsk@uni.lodz.pl 1