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In this article the methods of canonical analysis and quantization that were reviewed in the first part of the series are applied to the case of the Dirac field in the presence of electromagnetic interaction. It is shown that the quantization of electrodynamics, which begins with a given Lagrangian and ends by perturbative calculation of scattering probability amplitudes, can be performed in the way that does not employ Poincar\'e symmetry of space--time at any stage. Also, the causal structure is not needed., Comment: 20 pages, 3 figures

An elementary presentation of the methods for the canonical quantization of constraint systems with Fermi variables is given. The emphasis is on the subtleties of the construction of an appropriate classical bracket that could be consistently replaced by commutators or anti--commutators of operators, as required by canonical quantization procedure for bosonic and fermionic degrees of freedom respectively. I present a consequent canonical quantization of the Dirac field, in which the role of Poincar\'e invariance is made marginal. This simple example provides an introduction to the Poincar\'e--free quantization of spinor electrodynamics in the second part of the paper., Comment: 20 pages, no figures

In standard quantum field theory, the one-particle states are classified by unitary representations of the Poincar\'e group, whereas the causal fields' classification employs the finite dimensional (non-unitary) representations of the (homogeneous) Lorentz group. A natural question arises - why the fields are not allowed to transform nontrivially under translations? We investigate this issue by considering the fields that transform under the full representation of the Poincar\'e group. It follows that such fields can be consistently constructed, although the Lagrangians that describe them necessarily exhibit explicit dependence on the space-time coordinates. The two examples of the Poincar\'e-spinor and the Poincar\'e-vector fields are considered in details. The inclusion of Yang--Mills type interactions is considered on the simplest example of the U(1) gauge theory. The generalization to the non-abelian case is straightforward so long as the action of the gauge group on fields is independent of the action of the Poincar\'e group. This is the case for all the known interactions but gravity., Comment: 18 pages, no figures

In standard quantum field theory, the one-particle states are classified by the unitary representations of the Poincar\'e group, whereas the causal fields' classification employs the finite-dimensional (non-unitary) representations of the (homogeneous) Lorentz group. We investigate the possibility of constructing fields that transform under the full representation of the Poincar\'e group. We show that such fields can be consistently constructed, although the Lagrangians that describe them exhibit explicit dependence on the space-time coordinates. The inclusion of gravity within the framework of the Poincar\'e gauge theory is then discussed. A new feature that occurs is that the translational gauge fields enter the covariant derivative of matter fields. The Poincar\'e-gauge approach works still well and leads to interesting consequences. The detailed discussion of the Dirac field is presented and the relation to the earlier accounts on Poincar\'e-spinors is drawn. Another example that is considered is the Poincar\'e-vector field. The presentation has a partly didactic character and is addressed to all the readers who are interested in the rudiments of quantum field theory and the gauge description of gravity., Comment: 14 pages

We consider the geometric structures on the moduli space of static finite energy solutions to the 2+1 dimensional unitary chiral model with the Wess-Zummino-Witten (WZW) term. It is shown that the magnetic field induced by the WZW term vanishes when restricted to the moduli spaces constructed from the Grassmanian embeddings, so that the slowly moving solitons can in some cases be approximated by a geodesic motion on a space of rational maps from $\CP^1$ to the Grassmanian., Comment: 19 pages. Final version, to appear in the Journal of Geometry and Physics