A version of the quantum theory of gravity based on the concept of the wave functional of the universe is proposed. To determine the physical wave functional, the quantum principle of least action is formulated as a secular equation for the corresponding action operator. Its solution, the wave functional, is an invariant of general covariant transformations of spacetime. In the new formulation, the history of the evolution of the universe is described in terms of coordinate time together with arbitrary lapse and shift functions, which makes this description close to the formulation of the principle of general covariance in the classical theory of Einstein’s gravity. In the new formulation of quantum theory, an invariant parameter of the evolutionary time of the universe is defined, which is a generalization of the classical geodesic time measured by a standard clock along time-like geodesics.

The complex canonical transformation introduced by Ashtekar (1987) in general relativity is extended to simple supergravity.

A new form of covariant quantum theory based on a quantum version of the action principle is considered for the case of a free bosonic string. The central idea of the new approach is to delay conditions of stationarity of the classical action with respect to Lagrangian multipliers up to the quantum level where delayed conditions of stationarity are imposed on a quantum action. Physical states of the ordinary covariant quantum theory are replaced by well defined stationary states as those which obey quantum action principle. The stationary states have well defined energies.

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The statistical distribution for the case of an adiabatically isolated body was obtained in the framework of covariant quantum theory and Wick's rotation in the complex time plane. The covariant formulation of the mechanics of an isolated system lies in the rejection of absolute time and the introduction of proper time as an independent dynamic variable. The equation of motion of proper time is the law of conservation of energy. In this case, the energy of an isolated system is an external parameter for the modified distribution instead of temperature.

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To define time in the homogeneous anisotropic Bianchi-IX model of the universe, we propose a classical equation of motion of the proper time of the universe as an additional gauge condition. This equation is the law of conservation of energy. As a result, a new parameter, called a “mass” of the universe, appears. This parameter is added to the anisotropy energy and regarded as an observed quantity. The “mass” of the universe is decisive when it comes to the dynamics of its origin.

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