Speaker
Description
Among the three prominent forms of relativistic Hamiltonian dynamics introduced by Dirac, the point form is the least popular one. A very attractive feature of the point form is the clear separation of interaction-dependent and interaction-independent Poincare generators, as already noticed by Dirac. The interaction-dependent generators generate the subgroup of space-time translations, whereas the interaction-independent generators are responsible for the subgroup of Lorentz-transformations. The latter fact makes boosts and spin addition simple. The main reason for the minor popularity of the point form is that the quantization hypersurface is curved and not a hyperplane as in the instant and the front form. But as long as one is only interested in relativistic quantum mechanics, i.e. a relativistic invariant quantum theory for a fixed (or at least restricted) number of particles, this does not matter at all. The formal problem is just to find a representation of the Poincare algebra on an appropriate Hilbert space. For interacting systems this is accomplished by means of the Bakamjian-Thomas construction. The resulting interacting theory is relativistic invariant and it allows for instantaneous interactions as well as for particle production and annihilation, if one sets up an appropriate multichannel framework. But it has a drawback: it violates, of course, microcausality and even the weaker requirement of macrocausality (often also termed as cluster separability). The solution of the cluster problem has been formally given by Sokolov and later on by Coester and Polyzou and consists in unitary transformations by means of, so called, packing operators. We will give applications of the point-form approach to the calculation of electroweak hadron form factors within constituent quark models and we will show, how pion-cloud effects can be treated within such a framework. Also the question of cluster-separability restauration will be addressed.
Cluster separability is, of course, no problem if one goes over to a local quantum field theory. But then one has to work with an intrinsic many-body theory. In the canonical formalism one has to cope with a curved quantization hypersurface, if one wants to do point-form quantum field theory. What we were able to show is that one ends up with the usual Fock-space representation of Poincare generators for a free spin-0 and spin-1/2 quantum fields if one quantizes on a space-time hyperboloid using the usual momentum state basis. It can even be shown that one recovers the usual time-ordered perturbation theory. What is still unclear is, whether quantization on a space-time hyperboloid could be of advantage when solving interacting QFTs. Our insights concerning point-form QFT will be shortly discussed.
