Deterministic Physical Theories from Geometric Viewpoint

We analyze deterministic physical theories (theories where there exists a well-defined evolution operator describing the change of state with time).Quantum mechanics is a deterministic theory in this sense. Our goal is two-fold:we would like to reach a deeper understanding of quantum mechanics and we want to go beyond it.We start with classical theories where not all observables can be measured.If our devices are unable to distinguish some states we say that there exist redundant states. Then one can eliminate redundant states; this procedure leads to theories generalizing quantum mechanics. These considerations lead naturally to the geometric approach to physical theories where the starting point is the set of states (geometric theories). For a large class of geometric theories, we can prove that the interaction with a random environment leads to decoherence.As in quantum mechanics, decoherence permits us to calculate probabilities. If the group of automorphisms of the theory contains a commutative subgroup that can be identified with a group of space-time translations we can define elementary excitations of a translation-invariant stationary state (=particles and quasi-particles) and analyze their scattering in terms of inclusive scattering matrix closely related to inclusive cross-sections. To show that it is convenient to make calculations in the geometric approach, we describe the formalism of L-functionals. We apply it to the infrared problem in QED