Title: Path integral polymer propagator of relativistic and non-relativistic particles Show Chinese title

Title: 相对论和非相对论粒子的路径积分聚合子

Abstract: A recent proposal to connect the loop quantization with the spin foam model for cosmology via the path integral is hereby adapted to the case of mechanical systems within the framework of the so called polymer quantum mechanics. The mechanical models we consider are deparametrized and thus the group averaging technique is used to deal with the corresponding constraints. The transition amplitudes are written in a vertex expansion form used in the spin foam models, where here a vertex is actually a jump in position. Polymer Propagators previously obtained by spectral methods for a nonrelativistic polymer particle, both free and in a box, are regained with this method. Remarkably, the approach is also shown to yield the polymer propagator of the relativistic particle. This reduces to the standard form in the continuum limit for which the length scale parameter of the polymer quantization is taken to be small. Some possible future developments are commented upon. Show Chinese abstract

Abstract: 通过路径积分将环量子化与宇宙学的自旋泡沫模型相连接的最近提议，现已被适应于所谓聚合物量子力学框架下的机械系统情况。 我们考虑的机械模型是去参数化的，因此使用群平均技术来处理相应的约束条件。 过渡振幅以自旋泡沫模型中使用的顶点展开形式书写，其中这里的顶点实际上是一个位置跳跃。 通过这种方法重新获得了之前通过谱方法获得的非相对论聚合物粒子（自由和在盒中）的聚合物传播子。 值得注意的是，该方法也被证明可以得到相对论粒子的聚合物传播子。 在连续极限下，聚合物量子化的长度尺度参数取为很小值时，这会退化为标准形式。 对一些可能的未来发展方向进行了评论。