Quantum Gravity: A Self-Adjoint Time Operator as a Quantum Observable

Background:In quantum mechanics energy is an observable, whereas time is a parameter. Pauli's theorem and the problem of time in canonical quantum gravity reflect this asymmetry.Material and methods:We define the time operator t̂=x̂/c and form, with the relativistic energy ÊR, Einstein's constant κ and the time t̂₀ in the rest system, operators of gravitational-field and curvature type. Commutation relations, wave equations and spectra are derived in the time and in the energy representation.Results:The commutator [t̂,ÊR]=i/cv̂ₓ is bounded with norm and is not a multiple of the identity, so Pauli's theorem is not contradicted. The sum c\,t̂+̂R is unitarily equivalent to the time operator and has a continuous spectrum with explicit eigenfunctions. A postulated field equation, Ĝ₀+Λ̂=̂R, has the structure of Einstein's equation; the term Λ̂ is an operator with continuous spectrum, the Ricci scalar operator contains the energy, and for small velocities the equation becomes the Schr\"odinger equation of a particle in a homogeneous field.Conclusion:Based on the postulated field equation, a quantization of the gravitational-field and curvature terms is derived: they become operators with explicit commutation relations and spectra. The metric tensor of Einstein's field equation has not been quantized.