标题： 精确可解的量子引力部分 显示英文标题

标题： Exactly Soluble Sector of Quantum Gravity

摘要： 卡坦的时空重新表述是牛顿引力理论的广义协变伽利略相对论极限形式，称为牛顿-卡坦理论，这是爱因斯坦引力理论的一种形式。 根据该理论，空间是平直的，时间是绝对的，具有瞬时因果影响，时空的退化度规结构保持固定，包含两个相互正交的非动力学度规，一个是空间的，另一个是时间的。 然而，根据该理论的时空是弯曲的，严格遵守等效原理，非度规的引力连接场是动态的，因为它由物质分布决定。 在这里，这种可能具有非零宇宙常数的广义协变但伽利略相对论的引力理论，被视为一种参数化的规范理论，其引力矢量势与复数薛定谔场（玻色子或费米子）最小耦合，首次成功地被转化为显式协变的拉格朗日形式。 然后，利用牛顿-卡坦时空本质上是全局双曲的且具有固定的因果结构这一事实，该理论被重新表述为3+1维无约束的哈密顿形式，以及4维中具有非退化辛结构的显式协变约化相空间形式。 接下来，使用标准的C*-代数技术结合显式协变相空间量子化的几何方法，对该牛顿-卡坦-薛定谔系统进行了非微扰量子化。 由此产生的牛顿引力与伽利略相对论物质耦合的幺正量子场论不仅具有广义协变性，而且是精确可解的。 显示英文摘要

摘要： Cartan's spacetime reformulation of the Newtonian theory of gravity is a generally-covariant Galilean-relativistic limit-form of Einstein's theory of gravity known as the Newton-Cartan theory. According to this theory, space is flat, time is absolute with instantaneous causal influences, and the degenerate `metric' structure of spacetime remains fixed with two mutually orthogonal non-dynamical metrics, one spatial and the other temporal. The spacetime according to this theory is, nevertheless, curved, duly respecting the principle of equivalence, and the non-metric gravitational connection-field is dynamical in the sense that it is determined by matter distributions. Here, this generally-covariant but Galilean-relativistic theory of gravity with a possible non-zero cosmological constant, viewed as a parameterized gauge theory of a gravitational vector-potential minimally coupled to a complex Schroedinger-field (bosonic or fermionic), is successfully cast -- for the first time -- into a manifestly covariant Lagrangian form. Then, exploiting the fact that Newton-Cartan spacetime is intrinsically globally-hyperbolic with a fixed causal structure, the theory is recast both into a constraint-free Hamiltonian form in 3+1-dimensions and into a manifestly covariant reduced phase-space form with non-degenerate symplectic structure in 4-dimensions. Next, this Newton-Cartan-Schroedinger system is non-perturbatively quantized using the standard C*-algebraic technique combined with the geometric procedure of manifestly covariant phase-space quantization. The ensuing unitary quantum field theory of Newtonian gravity coupled to Galilean-relativistic matter is not only generally-covariant, but also exactly soluble.