Question

Wheeler de Witt for a general Cauchy surface?

Answer 1

Then we sort of take this foliation to be a time coordinate and derive the Wheeler-de-Witt equation which turns out no to depend on time.

But in the spirit of the Tomonaga-Schwinger equation where we define the input fields on an arbitrary space-like manifold, if we instead took a different space-like manifold and Σ that crossed over many foliations, how would we write such an equation?

Answer 2

The Wheeler–DeWitt theory, also known as quantum geometrodynamics or quantum General Relativity, is the foundational model.