Covariance, relativity, and the proper mass of the universe in the no-boundary wave function

A discrete class of privileged reference frames in a closed universe with identical equations of motion for physical degrees of freedom was found. A representation of the quantum state of the universe in a privileged reference frame was obtained as a Euclidean functional integral with no-boundary conditions. The boundary condition at the Pole, in addition to the smoothness conditions, is the infinite asymptotic behavior of the Hubble parameter. This makes it possible to regularize the functional integral by changing the sign of the expansion energy of the universe. The proposed construction also allows for the addition of a non-zero proper mass of the universe.

💡 Research Summary

The paper proposes a novel framework for describing a closed universe by introducing a discrete set of “privileged reference frames” derived from the spectrum of a Hermitian differential operator, the Witten operator (c_W = \Delta - D^2). Here (\Delta) is the sum of the Beltrami–Laplace operator (including metric coefficients) and the energy‑momentum tensor of matter fields, while (D) is a three‑dimensional Dirac operator defined on the spatial slice (\Sigma).

The authors first rewrite the gravitational constraints of the ADM formulation in terms of the Witten operator using the Witten identity. Solving the secular equation (c_W \chi = W \chi) yields eigenvalues (W) and eigenvectors (\chi). The eigenvalues are invariant under three‑dimensional coordinate transformations, and their Poisson brackets close with the momentum constraints (e_H^i) of the ADM algebra, forming a closed algebra without any appearance of the eigenvalues themselves. Consequently, the eigenvalues can be shifted by arbitrary constants without breaking the constraint algebra. The authors interpret these constant shifts as a “proper mass of the universe” associated with each privileged frame.

Fixing a particular eigenvector (\chi) selects a privileged reference frame. In that frame the Hamiltonian function of the universe is identified with the expectation value (\langle \chi | c_W | \chi \rangle). The quantum state of the universe (\Psi) is then expressed as a Euclidean functional integral (the no‑boundary wave function) over all canonical variables ((Q,P)) of geometry and matter: \