Euclidean Quantum Gravity - Relation To ADM Formalism

Relation To ADM Formalism

Euclidean Quantum Gravity does relate back to ADM formalism used in canonical quantum gravity and recovers the Wheeler–DeWitt equation under various circumstances. If we have some matter field, then the path integral reads

where integration over includes an integration over the three-metric, the lapse function, and shift vector . But we demand that be independent of the lapse function and shift vector at the boundaries, so we obtain

where is the three-dimensional boundary. Observe that this expression vanishes implies the functional derivative vanishes, giving us the Wheeler-DeWitt equation. A similar statement may be made for the Diffeomorphism constraint (take functional derivative with respect to the shift functions instead).

Read more about this topic:  Euclidean Quantum Gravity

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