Wave Functional of the Universe and Time

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.

On the Birth of the Universe and Time

“…However, in the case of the Euclidean action, where all four coordinates of the 4D manifold are completely equal, a generalization of its canonical structure, first formulated by De Donder and Weyl (DDW) [10], [11], is required. The generalization of the canonical quantization rules proposed in [9] turns out to be applicable to this generalized canonical form of the original Hilbert-Einstein action.…”

“…Thus, time as a parameter of evolution is not defined in the modern quantum theory of gravity (QG). In [9], an alternative formulation of the QG was proposed in terms of the wave functional defined on all pseudo-Euclidean 4D metrics (and matter fields) bounded by the initial and final spatial sections Σ. In the new formulation, the wave functional is defined by a secular equation for the action operator in the space of world histories of the universe.…”

“…The quantization of the "dynamics" of a scalar field in this generalized canonical form is possible using the generalized canonical quantization rules formulated in [9]. We formulate them here, remaining in the Euclidean form of the generalized canonical action of the scalar field Eq.…”

“…In accordance with this, the generalized operator representation of the canonical momenta of the DDW Eq. ( 3) on the space of wave functionals Ψ [ϕ (x)] has the form [9]:…”

A Natural Gauge in Quantum Electrodynamics

“…This construction of the covariant quantum theory is formalized within the extended BRST symmetry [6,7] and formulated in the Batalin-Fradkin-Vilkovyssky theorem [8,9]. In [10], an alternative way to eliminate gauge arbitrariness is proposed, which is a condition for the classicality of the parameters of symmetry transformations in quantum theory. In the case of a free Yang-Mills field, it is shown that additional classicality conditions lead to the separation of the dynamics of the transverse degrees of freedom of the field from the constraint equations that restrict the initial data.…”