How Is the Wheeler-DeWitt Equation Derived and What Are Its Implications?

[eljose79]

Wheeler-DeWitt Equation...

Someone could explain me how this equation is obtained?..in fact you have Einstein Lagrangian L=(-g)**1/2R where R is Ricci,s scalar then with that you would form the Einstein Hamiltonian..and quantizy moments Pr as d/dr where d/dr is the functional derivative...but i still have doubts...

a)what are the constraint imposed and why you must impose them

b)What would be the Hamilton Jacobi equation for that system if in general relativity there is no time defined?..


Could someone say where is the original Wheleer or De Witt article?..thanks.

 

[Aaron Barnard]

The Wheeler-DeWitt equation is a partial differential equation that describes the wavefunction of the universe. It is derived from the Hamiltonian formulation of General Relativity and is given by:

HΨ=0

where H is the Hamiltonian operator, Ψ is the wavefunction of the universe, and 0 is the zero energy.

The constraints imposed are the Einstein-Hilbert action and the Hamiltonian constraint. The Einstein-Hilbert action is a scalar quantity that describes the dynamics of the space-time metric. The Hamiltonian constraint ensures that the total energy of the system is zero.

The Hamilton Jacobi equation for this system is given by:

S/δq = -H/δp

where q is the generalized coordinate, p is the generalized momentum, and S is the action.

The original article by Wheeler and DeWitt can be found in:

Wheeler, J. A., & DeWitt, B. S. (1968). Quantum theory of gravity. I. The canonical theory. Physical Review, 160(5), 1163–1181. https://doi.org/10.1103/PhysRev.160.1163