The Quantization of a Globally Hyperbolic Spacetime

The quantization of gravity is hampered by the fact that the Einstein-Hilbert Lagrangian is singular. Switching to a Hamiltonian setting requires to impose two constraints, the Hamilton constraint and the diffeomorphism constraint. Though we were able to eliminate the diffeomorphism constraint in [23], the Hamilton constraint is a serious obstacle. Quantization of a Hamiltonian setting requires a model in which the quantized variables, which turn into operators, act, and, in case of constraints, preferably given as an equation, to quantize this equation. However, quantizing the Hamilton constraint means that only the normal Einstein equation is quantized while the tangential Einstein equations are ignored. Therefore we quantize a scalar evolution equation which incorporates both the Hamilton equations as well as the Hamilton constraint to obtain a hyperbolic equation in a fiber bundle which can be looked at a quantized version of the full Einstein equations.