中文摘要：

从离散的角度研究带边界的1＋1维经典标量场和Dirac场的正则量子化问题．与以往不同的是，这里将时间和空间两个变量同时进行变步长的离散，应用变步长离散的变分原理，得到离散形式的运动方程、边界条件和能量守恒的表达式．然后，根据Dirac理论，将边界条件当作初级约束，将边界条件和内在约束统一处理．研究表明，采用此方法，不仅在每个离散的时空格点上能够建立起Dirac括号，从而可以完成该模型的正则量子化；而且，该方法还保持了离散情况下的能量守恒．

英文摘要：

We study the problem of canonical quantization of classical scalar and Dirac field theories in the finite volumes respectively in this paper. Unlike previous studies, we work in a completely discrete version. We discretize both the space and time variables in variable steps and use the difference discrete variational principle with variable steps to obtain the equations of motion and boundary conditions as well as the conservation of energy in discrete form. For the case of classical scalar field, the quantization procedure is simpler since it does not contain any intrinsic constraint. We take the boundary conditions as primary Dirac constraints and use the Dirac theory to construct Dirac brackets directly. However, for the case of classical Dirac field in a finite volume, things are complex since, besides boundary conditions, it contains intrinsic constraints which are introduced by the singularity of the Lagrangian. Furthermore, these two kinds of constraints are entangled at the spatial boundaries. In order to simplify the process of calculation, we calculate the final Dirac brackets in two steps. We calculate the intermediate Dirac brackets by using intrinsic constraints. And then, we obtain the final Dirac brackets by bracketing the boundary conditions. Our studies show that we can not only construct well-defined Dirac brackets at each discrete space-time lattice but also keep the conservation of energy discretely at the same time