The ADM canonical formulation of the gravitational field is extended to four-dimensional space-times embedded in a higher-dimensional bulk space. The embedding is justified as a means to remove the ambiguity of the Riemann curvature. Using Nash's perturbative embedding theorem, we derive a simple generalization of the ADM canonical structure without breaking the diffeomorphism invariance and with a non-vanishing Hamiltonian. Nash's perturbations also allows us to define the functional derivative in Schwinger's equation, leading to a Schr?dinger-like quantum equation describing the wave function of the embedded space-time.
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