Unimodular theory of canonical quantum gravity

Abstract

Einstein's theory of gravity is reformulated so that the cosmological constant becomes an integration constant of the theory, rather than a "coupling" constant. However, in the Hamiltonian form of the theory, the Hamiltonian constraint is missing, while the usual momentum constraints are still present. Replacing the Hamiltonian constraint is a secondary constraint, which introduces the cosmological constant. The quantum version has a normal "Schr\"odinger" form of time development, and the wave function does not obey the usual "Wheeler-DeWitt" equation, making the interpretation of the theory much simpler. The small value of the cosmological constant in the Universe at present becomes a genuine question of initial conditions, rather than a question of why one of the coupling constants has a particular value. The key "weakness" of this formulation is that one must introduce a nondynamic background spacetime volume element.