Quantum Gravity by the Complex Canonical Theory and its Application to Cosmology

两年来的研究结果总结如下：1)通过扩展量子力学的传统框架，我们可以仅从量子约束的角度来构造包括广义相对论在内的完全约束系统的正则量子理论的一致框架。2)如果形式化地量子化复杂的正则理论，则正则运动方程一般与实际情况不符。然而，在1)中解释的方法中，通过将复正则理论作为实三元形式的特殊表示来构造相容理论是可能的。3)对于具有不为零的宇宙学常数的纯引力情况，我找到了复正则理论中所有量子约束方程的精确解。这个解是描述一般时空量子行为的第一个精确解。虽然它对应于空间均匀区域中的Hartle-Hawking波函数，但它有一个重要的新特征，它延伸到经典禁区并从那里掉下来。4)DeSitter时空是这个解所描述的唯一的洛伦兹WKB轨道。在宇宙学的背景下，这意味着洛伦兹度规所描述的宇宙是从欧里德时空系综中创造出来的。

The results of the research during the period of two years are summarized as follows.1) We can construct a consistent framework for canonical quantum theories of totally constrained systems including general relativity only in terms of quantum constraints by extending the conventional framework of quantum mechanics.2) The canonical equation of motion is in general inconsistent with the reality condition if one formally quantizes the complex canonical theory. In the approach explained in 1), however, it is possible to construct a consistent theory by regarding the complex canonical theory as a special representation of the real triad formalism.3) I found an exact solution to all the quantum constraint equations in the complex canonical theory for the case of pure gravity with non-vanishing cosmological constant. This solution is the first exact solution that describes the quantum behavior of generic spacetimes. Though it corresponds to the Hartle-Hawking's wavefunction in the spatially homogeneous sector, it has one important new feature that it extends to the classically forbidden region and falls off there.4) The DeSitter spacetime is the unique Lorentzian WKB orbit described by this solution. In the cosmological context this suggests a picture that the universe described by a Lorenzian metric is created from an ensemble of Eulidean spacetimes.