A Quantum Logic Of Normal Operators

Quantum mechanics is one of the most important achievements of physics in20 th Century.And it is a main branch of modern physics.It give rises to a huge change in both conception and ideas of physics.Moreover,physic itself has changed a lot.Quantum logic is a math branch which developed with the movement of mathematical axiom of the quantum theory.It has a history of 80 years with a great deal of results.As we all know,in the microscopic world,particle motion is in accordance with the Schrodinger equation which is linear.In that case the equation forms a linear space.On the other hand,physical background requires that there are projection and inner product the linear space.So we can find realistic meaning from the theory about Hilbert space.For example,the values measured in the experiment must be real numbers,then the spectrum of operators we use to represent observable quantities should be real numbers,and the self-adjoint operators satisfy this requirement.Moreover,the basic assumption of quantum mechanics indicates that the evolution with time of closed quantum systems can be characterized by a unitary operator.For one given unitary operator,there must be a period of evolution of a closed single qubit system during some time which can be described by this unitary operator.In this context,as a tool in quantum observation and quantum computation,the unitary operator can be used to calculate some physical relations of two or more quantum systems.In this thesis,we mainly discuss the quantum logic including both self-adjoint operators and Unitary operators,that is,the set of normal operators.Chapter1 introduces the background of this subject and what scholars in China and abroad have studied in the related fields of this subject in recent years.Chapter2 introduces some basic knowledge,especially some definitions and simple properties of algebraic structure.In Chapter3,we define the vertical relationship between two operators in the set of normal operators in Hilbert space.In that case we then define the binary relationship and partial operation and get a quantum logic constructed by all normal operators.Chapter4 introduces two kinds of partial order and study its properties in the quantum logic as well as the structure of its subjects including one of those partial order.Finally,we study the relationship between them.