Hyper-operators And Non-commutative Calculus In Quantum Physics

1920' saw the founding of quantum mechanics, which changed revolutionarily the human concept of matter and motion. In classical physics the state of a system at time t is described by all its general coordinates and conjugate momenta , and the evolution of system state is governed by the canonical equationsMotion is deterministic. While in quantum mechanics the state of a system is described by a wave function, which is a non-zero square modulus integrable complex value function defined on the configuration space of the system. The square modulus of the wave function is proportional to the probability density of finding the system at each point in the configuration space. The microscopic system has a wave-particle duality. Quantal description is probabilistic. The evolution of the wavefunction obeys the Schr?dinger equation. The mathematical framework of classical mechanics is symplectic geometry; while the mathematical framework of quantum mechanics is the theory of linear operators on Hilbert spaces. In the literature, however, the algebraic aspect of the operator theory has received more attention than the analytic aspect. What's new? The first part of the thesis is devoted to transplanting analysis on Banach spaces well-established by mathematicians to operator functions in quantum physics. Because the argumentof an operator function is not a number, but a linear operator on the state-vector space(namely), the values at of operator functionand its derivatives are mathematical objects of different kinds: The essence of differentiation is local linearization of mappings. To meet the need of non-commutative analysis in quantum physics, we studied systematically the properties of linear spaces of various hyper-operators. We established the notation systems for various hyper-operators. Construction of various hyper-operators in terms of ordinary operators is given. It is pointed out that the operations of all kinds of hyper-operators can be reduced to the operations of ordinary operators. This part of the thesis provides tools for doing non-commutative analysis in quantum physics. We found , however, that the concept of hyper-operator itself is very important for quantum physics , group representation and Lie algebras. In the third part of the thesis, we illustrated this with examples.Chapter 1 presents the concepts of normed linear spase, Banach spase, inner product spase and Hilbert space. In a 'pure' linear space one cannot tell which of two linearly independent vectors is smaller. In order to talk about the limitation of a vector series, one has to introduce the norm of a vector. This leads to the concept of a normed linear space. In order that every Cauchy series has a limit, we have to confined ourselves to complete normed linear spaces (Banach spaces). In an inner product space one always defined the norm of a vector as . So an inner product space is automatically a normed linear space, and a Hilbert space is automatically a Banach space. We paid special attention to the complex linear space consisting of all linear mappings from dimensional complex inner product space into dimensional complex inner product space . The Hermitian conjugate of linear mapping is defined as follows. . Then the inner product of two vectors in is defined as. This is a definition independent of the bases of and . Now we can take as a complex Hilbert space. It is pointed out that if is an orthonormal basis of,and is an orthonormal basis of , then is an orthonormal basis of.Chapter 2 presents first the theory of analysis on Banach spaces well-established by mathematicians. Let be Banach spaces, a non-empty open subset of , and a mapping of into . Suppose . If for all such that , we have , where is a linear mapping frominto independent of , while is a higher order infinitesimal to . It can be shown that this is unique. We will call the derivative of at , and denote it by , and say is differentiable at . Then . When is differentiable at every point of...