Null-plane dynamics of particles and fields

Abstract

The focus of this review of null-plane dynamics is the fundamental principle of quantum theory that states must form a linear manifold with a positive scalar product. The intent is to provide an integrated overview of diverse aspects of null-plane dynamics of particles and fields. Hamiltonian particle dynamics is based on the construction of nontrivial representations of the Poincaré group on finite tensor products of single-particle spaces, and on finite direct sums of such tensor products. The structure of the space of states and the representations of a kinematic subgroup are independent of the dynamics. The dynamics is specified by Hamiltonian operators which are the Poincaré generators outside the kinematic subgroup. Fock-spaces are infinite direct sums of tensor products of single particle spaces. Fock-space representations of Lagrangean field theories can be formulated as limits of Hamiltonian many-body dynamics. The required cutoff can preserve the symmetry of the kinematic subgroup but destroys the full Poincaré invariance. The nontrivial questions involve the existence of limits as the cutoff is removed. Covariant wave functions arise either as solutions of covariant wave equations, or as matrix elements of products of covariant field operators. The linear manifold of states is a manifold of equivalence classes of covariant functions. The dynamics appear in the nontrivial inner product and all Poincaré transformations are kinematic. Null-plane restrictions of the covariant functions may provide a unitary map of the covariant constraint dynamics into null-plane Hamiltonian dynamics.