Abstract
The coadjoint orbit method is applied to the construction of Hamiltonian dynamics of massless particles of arbitrary helicity. The unusual transformation properties of canonical variables are interpreted in terms of nonlinear realizations of Poincare group. The action principle is formulated in terms of new space–time variables with standard transformation properties.
Highlights
Triangle anomalies, chiral fermions and Berry curvature in momentum space, their interrelations and role played in various physical phenomena helicity1 2 by the action functional S=p + eA · x − | p | +eΦ − α · pdt (1)involving the vector potential α(p) describing the Berry monopole in momentum space
To derive the symmetry transformations we note that our symmetry is a dynamical one: the Hamiltonian belongs to the Lie algebra of symmetry group and, in general, does not Poisson-commute with other generators
We see that the straightforward quantization of the Hamiltonian system built on the coadjoint orbits characterized by ζμζμ = 0, wμwμ = 0 yields irreducible representations corresponding to massless particles of arbitrary helicity s
Summary
Triangle anomalies, chiral fermions and Berry curvature in momentum space, their interrelations and role played in various physical phenomena. Let us note that some of the apparently paradoxical features of Lorentz transformation laws for particles with nonzero spin (massive case) or helicity (massless case) appear to be unavoidable consequences of the group structure and basic conservation laws It has been noticed long time ago [26] that the generators of Poincare symmetry for massless particle of nonzero helicity cannot be constructed out of canonical variables obeying standard canonical commutation rules and having standard transformation properties; if it were possible, the helicity would acquire more than one value within irreducible representation of Poincare group. Another nice argument in favour of “exotic”transformation has been given in Ref. This provides alternative way of looking at the “unusual”transformation properties of dynamical variables representing massless particles
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