ADM Formalism & Wheeler-deWitt Equation: What's the Relation?

[jesusrods]

Hello! I am interested in knowing the existing relation between the ADM Formalism and the Wheeler-deWitt Equation.

From the articles and lectures I was read, I understand that ADM formulation is the Hamiltonian formulation of General Relativity at classical level. Then, the Hamiltonian is quantized converting the metric and the momentums in the correct operators, and the Wheeler-deWitt equation is then Hψ=0 with H the quantized Hamiltonian.

I am right or wrong? I would be grateful if you can give me an answer.

PS: If I have some mistakes, please excuse me, english is not my born language and I am learning it. Thanks.

 

[Ambitwistor]

Hello,

Thank you for your interest in the relationship between the ADM Formalism and the Wheeler-deWitt Equation. You are correct in your understanding that the ADM Formalism is a classical formulation of General Relativity, while the Wheeler-deWitt Equation is a quantum version of the theory.

To elaborate further, the ADM Formalism breaks down the spacetime metric into its spatial and temporal components, allowing for a Hamiltonian formulation of General Relativity. This Hamiltonian is then quantized, as you mentioned, resulting in the Wheeler-deWitt Equation. This equation, as you stated, is of the form Hψ=0, where H is the quantized Hamiltonian and ψ is the wave function of the universe.

In essence, the ADM Formalism provides a classical framework for understanding the dynamics of spacetime, while the Wheeler-deWitt Equation introduces a quantum aspect to the theory. Both are important in understanding the behavior of the universe at different scales.

I hope this helps answer your question. If you have any further inquiries, please feel free to ask. And don't worry about any mistakes in your English, it is clear that you have a strong understanding of the topic. Keep up the good work!