弯曲空间中微扰定义的有效经典势
Perturbatively Defined Effective Classical Potential in Curved Space
摘要：
平坦空间中量子统计系统的配分函数总是可以写成对经典玻尔兹曼因子$\exp[ -βV^{\rm eff cl({\bf x}_0)]$的积分，其中$V^{\rm eff cl({\bf x}_0)$是包含所有量子涨落效应的所谓有效经典势。积分变量是时间路径平均值${\bf x_0\equiv β^{-1}\int_0^ βdτ{\bf x}(τ)$。我们展示了如何将这个概念推广到具有度量$g_{μν(q)$的弯曲空间中的路径$q^μ(τ)$，并微扰地计算了$V^{\rm eff cl(q_0)$的高温膨胀。在坐标变换$q^μ(τ)\to q'^μ(τ)$下的独立性要求在路径平均值$q_0^μ$的定义和处理中引入了微妙之处，并且协方差仅在适当的Faddeev-Popov过程的帮助下实现。
英文摘要：
The partition function of a quantum statistical system in flat space can always be written as an integral over a classical Boltzmann factor $\exp[ -βV^{\rm eff cl({\bf x}_0)]$, where $V^{\rm eff cl({\bf x}_0)$ is the so-called effective classical potential containing the effects of all quantum fluctuations. The variable of integration is the temporal path average ${\bf x_0\equiv β^{-1}\int_0^ βdτ{\bf x}(τ)$. We show how to generalize this concept to paths $q^μ(τ)$ in curved space with metric $g_{μν(q)$, and calculate perturbatively the high-temperature expansion of $V^{\rm eff cl(q_0)$. The requirement of independence under coordinate transformations $q^μ(τ)\to q'^μ(τ)$ introduces subtleties in the definition and treatment of the path average $q_0^μ$, and covariance is achieved only with the help of a suitable Faddeev-Popov procedure.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
