R.P. Feynman showed F.J. Dyson a proof of the Lorentz force law and the homogeneous Maxwell equations, which he obtained starting from Newton's law of motion and the commutation relations between position and velocity for a single nonrelativistic particle. We formulate both a special relativistic and a general relativistic versions of Feynman's derivation. Especially in the general relativistic version we prove that the only possible fields that can consistently act on a quantum mechanical particle are scalar, gauge and gravitational fields. We also extend Feynman's scheme to the case of non-Abelian gauge theory in the special relativistic context. † This article has been published in

We solved the Schrödinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus T n = R n /Λ is defined as a quotient of the Euclidean space R n by an arbitrary n-dimensional lattice Λ. The lattice is not necessary either cubic or rectangular. The magnetic field is also arbitrary. However, we restrict ourselves within potential-free problems; the Schrödinger operator is assumed to be the Laplace operator defined with the covariant derivative. We defined an algebra that characterizes the symmetry of the Laplacian and named it the magnetic algebra. We proved that the space of functions on which the Laplacian acts is an irreducible representation space of the magnetic algebra. In this sense the magnetic algebra completely characterizes the quantum mechanics in the magnetic torus. We developed a new method for Fourier analysis for the magnetic torus and used it to solve the eigenvalue problem of the Laplacian. All the eigenfunctions are given in explicit forms.

In this paper, a new numerical algorithm based on the discrete element method is presented for analyzing the dynamic problems under impact loading. Based on the basic principle of continuum mechanics, a connective model for orthotropic media is derived using disk elements. It is also extended to a bilinear hardening elastic-plastic model for calculating the plastic deformation in metals. Moreover, Mohr-Coulomb type failure criterion is used to judge the failure of concrete, and a contact discrete model is added in the algorithm. So the algorithm can calculate not only the impact problems of continuum and non-continuum, but also the transient process from continuum to non-continuum. The wave propagation in orthotropic planes under impact loading is numerically simulated. Through comparing the results with those computed by other numerical methods and examining the stability of the numerical solution, the accuracy and efficiency of the algorithm are discussed. In addition, the transient respondences of a steel warhead penetrating a concrete disc harrow is simulated, and three kinds of basic damage forms of concrete disc harrow under different penetration velocities of warhead are summarized.

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The isoholonomic problem in a homogeneous bundle is formulated and solved exactly. The problem takes a form of a boundary value problem of a variational equation. The solution is applied to the optimal control problem in holonomic quantum computer. We provide a prescription to construct an optimal controller for an arbitrary unitary gate and apply it to a k-dimensional unitary gate which operates on an N-dimensional Hilbert space with N ജ 2k. Our construction is applied to several important unitary gates such as the Hadamard gate, the CNOT gate, and the two-qubit discrete Fourier transformation gate. Controllers for these gates are explicitly constructed.