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

Title: 相对论性和非相对论性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 and as a new result we obtain the polymer propagator of the relativistic particle. All of them reduce to their standard form in the continuum limit for which the length scale parameter of the polymer quantization is taken to be small. Our results are robust thanks to their analytic and exact character which in turn come from the fact that presented models are solvable. They lend support to the vertex expansion scheme of the polymer path integral explored before in a formal way for cosmological models. Some possible future developments are commented upon in the discussion. Show Chinese abstract

Abstract: 近期关于通过路径积分将环量子化与宇宙学的自旋泡沫模型连接起来的提议，在聚合物量子力学框架内被应用于机械系统的情形。我们考虑的机械模型是非参数化的，因此使用了群平均技术来处理相应的约束条件。转移振幅以自旋泡沫模型中使用的顶点展开形式书写，这里一个顶点实际上是一个位置跳跃。用这种方法重新获得了之前通过谱方法为非相对论性聚合粒子（自由和受限于盒子内）得到的聚合传播子，并且作为新的结果，我们得到了相对论性粒子的聚合传播子。所有这些在聚合量子化的长度标度参数趋于小值的连续极限下都还原为其标准形式。我们的结果是稳健的，因为它们具有分析性和精确性，而这反过来又是因为所提出的模型是可以解析求解的。它们支持了之前为宇宙学模型形式上探索过的聚合路径积分的顶点展开方案。讨论部分还评论了一些可能的未来发展。