A system consisting of a point particle coupled to gravity is investigated. The set of constraints is derived. It was found that a suitable superposition of those constraints is the generator of the infinitesimal transformations of the time coordinate t ≡ x0 and serves as the Hamiltonian which gives the correct equations of motion. Besides that, the system satisfies the mass shell constraint, pμp μ − m2 = 0, which is the generator of the worldline reparametrizations, where the momenta pμ, μ = 0, 1, 2, 3, generate infinitesimal changes of the particle’s position Xμ in spacetime. Consequently, the Hamiltonian contains p0, which upon quantization becomes the operator − i∂/∂T, occurring on the right-hand side of the Wheeler–DeWitt equation. Here, the role of time has the particle coordinate X0 ≡ T, which is a distinct concept than the spacetime coordinate x0 ≡ t. It is also shown how the ordering ambiguities can be avoided if a quadratic form of the momenta is cast into the form that instead of the metric contains the basis vectors.

中文翻译：

---

关于相对论粒子与引力的耦合和惠勒-德维特量子化

研究了由与重力耦合的点粒子组成的系统。导出约束集。发现这些约束的合适叠加是时间坐标的无穷小变换的生成器吨 ≡ X0并用作给出正确运动方程的哈密顿量。除此之外，系统满足质量壳约束，pμp μ - 米2 = 0，它是世界线重新参数化的生成器，其中动量pμ,μ = 0, 1, 2, 3, 产生粒子位置的无穷小变化Xμ在时空中。因此，哈密顿量包含p0, 量化后成为算子 - 一世∂/∂吨，出现在 Wheeler-DeWitt 方程的右侧。在这里，时间的作用有粒子坐标X0 ≡ 吨，这是一个不同于时空坐标的概念X0 ≡ 吨. 它还显示了如果将动量的二次形式转换为包含基向量而不是度量的形式，如何避免排序模糊。