A Quantum Logic Including Unbounded Observables

Microscopic exists in the microscopic world, and the objects of the microscopic world is known as microscopic particles, quantum mechanics is a subject which describes the movement of particles. Quantum mechanics is a part of the quantum theory, and it was began in 1990 when the Planck announced his epoch-making quantum concept. According to the basic hypothesis of quantum mechanics, each isolated quantum system is related to a Hilbert space. The system is completely described by the state vector, which is a unit vector of the system’s state space. The microscopic particles can be described by the self-adjoint operators,and most of the self-adjoint operators are unbounded, such as the position and momentum. It made us begin to study the unbounded self-adjoint operators on the Hilbert space. In the 1930 s, von Neumann introduced the affiliated relationship to solve the difficulties when dealing with unbounded self-adjoint operators. The establishment of the unbounded self-adjoint operators and the consummation of the theory provide an effective mathematical tool for us to solve some modern quantum mechanics problems.The behavior of measurement must be accompanied by some interference for the object When the measurement arrived at the micro level, and we usually can’t ignored the interference. To describe the system of quantum physics whose main characteristics is the Heisenberg’s commutation relation, mathematicians have established a quantum logic instead of classical logic. The quantum logic have developed for more than eighty years.Firstly, in this paper, we introduce a binary relation by using the affiliated relationship on S(?), which is made up of all the self-adjoint operators. Then we take advantage of the spectral theorem to obtain that: if there are two self-adjoint operators which are affiliated with ?, then the closure of their sum is alsoself-adjoint, and it is affiliated with ?; After that, we present the definition of vertical of unbounded self-adjoint operators, and show four conclusions which are equivalent to the vertical self-adjoint operators. Then we established a quantum logic including unbounded observables S(?),),( 0? by using the vertical relationship,At last, we present an orthogonal order ? on S(?), and obtain that ? is a partial order on S(?), then we study some important properties about these two orders, and compare the new order ? with the usual order.