Abstract
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.
Highlights
The idea of universe expansion, originated at the Big Bang, is widely accepted today, being well founded on clear observational evidence
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
To formulate new rules for quantizing the theory of gravity, let us consider the transition to quantum principle of least action (QPLA) in ordinary quantum mechanics, where there is no problem of time
Summary
The idea of universe expansion, originated at the Big Bang, is widely accepted today, being well founded on clear observational evidence. Since the wave function of the universe does not depend on any external time parameter, the problem arises of its dynamic interpretation in terms of the fundamental dynamic variables gik (and matter fields) This problem was solved by Hartle and Hawking, using the representation of the wave function in the form of a Euclidean functional integral [5] (no-boundary wave function). Hartle, Hawking, and Hertog [6,7] obtained a classical picture of an expanding universe with a scalar field within the framework of the semiclassical approximation for a functional integral, in which the proper time (Equation (7)) takes on the meaning of an evolution parameter. By definition, this time is an invariant of general covariant transformations
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