Quantum Logic

Quantum theory is one of the greatest achievements in twentieth-century. And an important task in quantum theory is to find the axioms of quantum theory.To do this, the so called quantum logic theory was constructed. Quantum logic has a long history, and there were many results of quantum logic.Effect algebra was introduced before about ten years. It describes the unsharp quantum measurement, and plays an important role in quantum logic theory.In this paper, we discuss effect algebra and related topics. The main results are the following:1. Effect algebra is a commutative algebraic structure. To get a non-commutative algebraic structure.we introduced the concepts of left difference and right difference, and we also introduced the concept of the Bi-difference sets, the concept of pseudo-difference poset, and proved that pseudo-difference poset is equivalent to pseudo-effect algebra. We also get a representation theorem for Bi-difference sets.2. Ideal and filter are two important concepts in quantum logic. From the order structure and algebraic structures of effect algebra, there are many definitions of ideals and filters. We give the connection between these concepts. We proved that in pseudo-effect algebra, the filters and local filters are equivalent, ideals and local ideals and generalized ideals are equivalent.3. If effect algebra is also a lattice, we call it to be lattice effect algebra. Since lattice effect algebra has an effect algebra structure and also a lattice structure, there are definitions of ideal and filter in the sense of lattice and ideal and filter in the sense of effect algebra. So a question arises: are these definitions coinciding? We partially solved this problem, we get that, for lattice effect algebra, the lattice filter is really stronger than effect algebra filter. We also proved that, the necessary and sufficient condition for a lattice ideal to be effect algebra ideal is that the lattice effect algebra becomes an orthomodular lattice.4. To build the Stone theory in effect algebra, we need to study the ideal extend problem. We discuss the one point extend problem of ideals, and get some results.5. The topology of effect algebra is very important, but it is difficult to study. We introduced the topology induced by ideals of effect algebra, named ideal topology, and we show that the operations are continuous with respect to the ideal topology. We find that ideal topology has many good properties.6. Measurement theory, especially quantum measurement theory, is important subject of quantum theory. We give a representation theory of a-algebra, and improve the Brooks-Jewett theory on effect algebra.